WEBVTT

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This<00:00:00.480><c> single</c><00:00:00.800><c> equation</c><00:00:01.280><c> helped</c><00:00:01.600><c> unlock</c><00:00:02.080><c> the</c>

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This single equation helped unlock the
 

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This single equation helped unlock the
modern<00:00:02.720><c> world.</c><00:00:03.200><c> It</c><00:00:03.360><c> gave</c><00:00:03.600><c> us</c><00:00:03.840><c> computer</c><00:00:04.160><c> chips,</c>

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modern world. It gave us computer chips,
 

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modern world. It gave us computer chips,
electron<00:00:05.120><c> microscopes,</c><00:00:05.839><c> atomic</c><00:00:06.240><c> clocks,</c>

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electron microscopes, atomic clocks,
 

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electron microscopes, atomic clocks,
GPS,<00:00:07.520><c> high-speed</c><00:00:08.080><c> internet,</c><00:00:08.720><c> the</c><00:00:08.960><c> list</c><00:00:09.440><c> goes</c>

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GPS, high-speed internet, the list goes
 

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GPS, high-speed internet, the list goes
on.<00:00:10.400><c> But</c><00:00:10.639><c> where</c><00:00:10.880><c> did</c><00:00:11.040><c> this</c><00:00:11.200><c> equation</c><00:00:11.679><c> come</c>

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on. But where did this equation come
 

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on. But where did this equation come
from?<00:00:12.480><c> Well,</c><00:00:12.719><c> according</c><00:00:13.040><c> to</c><00:00:13.200><c> Richard</c>

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from? Well, according to Richard
 

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from? Well, according to Richard
Feineman,<00:00:14.080><c> it</c><00:00:14.320><c> comes</c><00:00:14.559><c> from</c>

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Feineman, it comes from
 

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Feineman, it comes from
&gt;&gt; nowhere.<00:00:16.320><c> Out</c><00:00:16.560><c> of</c><00:00:16.720><c> man's</c><00:00:17.279><c> imagination,</c>

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&gt;&gt; nowhere. Out of man's imagination,
 

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&gt;&gt; nowhere. Out of man's imagination,
struggles<00:00:18.640><c> with</c><00:00:18.880><c> the</c><00:00:19.279><c> details</c><00:00:19.680><c> of</c><00:00:19.920><c> experiment</c>

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struggles with the details of experiment
 

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struggles with the details of experiment
and<00:00:20.640><c> all</c><00:00:20.800><c> kinds</c><00:00:20.960><c> of</c><00:00:21.199><c> mysteries.</c>

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and all kinds of mysteries.
 

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and all kinds of mysteries.
&gt;&gt; That<00:00:23.359><c> man</c><00:00:23.600><c> was</c><00:00:23.840><c> Irvin</c><00:00:24.320><c> Schroinger.</c><00:00:25.039><c> He</c>

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&gt;&gt; That man was Irvin Schroinger. He
 

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&gt;&gt; That man was Irvin Schroinger. He
derived<00:00:25.600><c> it</c><00:00:25.760><c> in</c><00:00:26.000><c> 1925</c><00:00:27.039><c> while</c><00:00:27.439><c> vacationing</c><00:00:27.840><c> in</c>

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derived it in 1925 while vacationing in
 

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derived it in 1925 while vacationing in
the<00:00:28.080><c> Swiss</c><00:00:28.400><c> Alps</c><00:00:29.359><c> with</c><00:00:29.599><c> his</c><00:00:29.840><c> mistress.</c>

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the Swiss Alps with his mistress.
 

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the Swiss Alps with his mistress.
So<00:00:31.760><c> I</c><00:00:32.000><c> have</c><00:00:32.160><c> two</c><00:00:32.399><c> questions.</c><00:00:33.120><c> First,</c><00:00:33.760><c> how</c><00:00:33.920><c> can</c>

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So I have two questions. First, how can
 

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So I have two questions. First, how can
we<00:00:34.640><c> intuitively</c><00:00:35.600><c> build</c><00:00:35.920><c> this</c><00:00:36.160><c> equation</c>

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we intuitively build this equation
 

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we intuitively build this equation
ourselves<00:00:37.200><c> from</c><00:00:37.600><c> scratch?</c><00:00:38.719><c> But</c><00:00:38.960><c> secondly,</c>

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ourselves from scratch? But secondly,
 

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ourselves from scratch? But secondly,
why<00:00:40.960><c> is</c><00:00:41.200><c> an</c><00:00:41.440><c> equation</c><00:00:41.840><c> with</c><00:00:42.160><c> such</c><00:00:42.719><c> real</c><00:00:43.200><c> impact</c>

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why is an equation with such real impact
 

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why is an equation with such real impact
built<00:00:44.480><c> using</c><00:00:44.719><c> an</c><00:00:44.960><c> imaginary</c><00:00:45.520><c> number?</c><00:00:46.399><c> What's</c>

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built using an imaginary number? What's
 

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built using an imaginary number? What's
I<00:00:47.520><c> doing</c><00:00:47.920><c> there?</c><00:00:49.120><c> If</c><00:00:49.360><c> you're</c><00:00:49.600><c> ready,</c><00:00:49.920><c> let's</c>

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I doing there? If you're ready, let's
 

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I doing there? If you're ready, let's
find<00:00:50.480><c> out.</c>

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find out.
 

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find out.
It<00:00:53.039><c> all</c><00:00:53.280><c> starts</c><00:00:53.520><c> in</c><00:00:53.680><c> 1853</c><00:00:54.559><c> when</c><00:00:54.800><c> the</c><00:00:54.960><c> physicist</c>

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It all starts in 1853 when the physicist
 

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It all starts in 1853 when the physicist
Anders<00:00:56.079><c> Enstrom</c><00:00:57.039><c> finds</c><00:00:57.280><c> that</c><00:00:57.680><c> hot</c><00:00:58.000><c> hydrogen</c>

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Anders Enstrom finds that hot hydrogen
 

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Anders Enstrom finds that hot hydrogen
gas<00:00:59.120><c> gives</c><00:00:59.359><c> out</c><00:00:59.680><c> very</c><00:01:00.079><c> specific</c><00:01:00.480><c> colors</c><00:01:00.800><c> of</c>

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gas gives out very specific colors of
 

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gas gives out very specific colors of
light.<00:01:01.680><c> We</c><00:01:01.920><c> soon</c><00:01:02.160><c> figured</c><00:01:02.399><c> out</c><00:01:02.719><c> it's</c><00:01:03.039><c> not</c><00:01:03.199><c> just</c>

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light. We soon figured out it's not just
 

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light. We soon figured out it's not just
hydrogen.<00:01:04.479><c> Every</c><00:01:04.799><c> hot</c><00:01:05.119><c> element</c><00:01:05.600><c> gives</c><00:01:05.840><c> out</c>

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hydrogen. Every hot element gives out
 

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hydrogen. Every hot element gives out
its<00:01:06.400><c> own</c><00:01:06.640><c> signature</c><00:01:07.200><c> spectrum.</c><00:01:08.000><c> And</c><00:01:08.240><c> suddenly</c>

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its own signature spectrum. And suddenly
 

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its own signature spectrum. And suddenly
this<00:01:08.960><c> became</c><00:01:09.360><c> a</c><00:01:09.680><c> powerful</c><00:01:10.320><c> tool</c><00:01:10.799><c> to</c><00:01:11.119><c> discover</c>

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this became a powerful tool to discover
 

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this became a powerful tool to discover
brand<00:01:11.840><c> new</c><00:01:12.080><c> elements</c><00:01:12.640><c> just</c><00:01:12.960><c> by</c><00:01:13.200><c> looking</c><00:01:13.600><c> at</c>

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brand new elements just by looking at
 

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brand new elements just by looking at
their<00:01:14.479><c> light.</c><00:01:15.360><c> This</c><00:01:15.520><c> is</c><00:01:15.600><c> how</c><00:01:15.840><c> we</c><00:01:16.159><c> discovered</c>

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their light. This is how we discovered
 

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their light. This is how we discovered
helium<00:01:17.280><c> for</c><00:01:17.520><c> the</c><00:01:17.680><c> very</c><00:01:17.840><c> first</c><00:01:18.080><c> time</c><00:01:18.479><c> in</c><00:01:18.960><c> the</c>

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helium for the very first time in the
 

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helium for the very first time in the
sun.<00:01:20.400><c> So</c><00:01:20.640><c> in</c><00:01:20.880><c> his</c><00:01:21.040><c> honor,</c><00:01:21.360><c> the</c><00:01:21.600><c> unit</c><00:01:21.920><c> for</c>

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sun. So in his honor, the unit for
 

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sun. So in his honor, the unit for
measuring<00:01:22.560><c> these</c><00:01:22.880><c> wavelengths</c><00:01:23.600><c> was</c><00:01:23.920><c> named</c>

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measuring these wavelengths was named
 

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measuring these wavelengths was named
the<00:01:24.479><c> Enstrom.</c>

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the Enstrom.
 

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the Enstrom.
But<00:01:26.400><c> nobody</c><00:01:26.880><c> knew</c><00:01:27.360><c> why</c><00:01:27.840><c> these</c><00:01:28.240><c> elements</c><00:01:28.799><c> gave</c>

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But nobody knew why these elements gave
 

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But nobody knew why these elements gave
out<00:01:29.280><c> those</c><00:01:29.680><c> specific</c><00:01:30.320><c> colors</c><00:01:30.720><c> of</c><00:01:30.960><c> light.</c><00:01:31.680><c> The</c>

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out those specific colors of light. The
 

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out those specific colors of light. The
first<00:01:32.159><c> clue</c><00:01:32.479><c> actually</c><00:01:32.799><c> came</c><00:01:33.040><c> a</c><00:01:33.280><c> few</c><00:01:33.439><c> decades</c>

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first clue actually came a few decades
 

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first clue actually came a few decades
later<00:01:34.479><c> from</c><00:01:34.799><c> a</c><00:01:35.040><c> Swiss</c><00:01:35.439><c> math</c><00:01:35.759><c> teacher</c><00:01:36.079><c> named</c>

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later from a Swiss math teacher named
 

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later from a Swiss math teacher named
John<00:01:37.439><c> Balmer.</c><00:01:38.560><c> Balmer</c><00:01:39.119><c> was</c><00:01:39.360><c> obsessed</c><00:01:39.920><c> with</c>

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John Balmer. Balmer was obsessed with
 

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John Balmer. Balmer was obsessed with
numbers<00:01:40.560><c> and</c><00:01:40.720><c> patterns.</c><00:01:41.439><c> And</c><00:01:41.600><c> he</c><00:01:41.840><c> finds</c><00:01:42.320><c> a</c>

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numbers and patterns. And he finds a
 

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numbers and patterns. And he finds a
surprisingly<00:01:43.520><c> simple</c><00:01:44.000><c> formula</c><00:01:44.880><c> for</c><00:01:45.200><c> the</c>

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surprisingly simple formula for the
 

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surprisingly simple formula for the
hydrogen<00:01:45.840><c> spectrum</c><00:01:46.399><c> just</c><00:01:46.640><c> by</c><00:01:46.799><c> trial</c><00:01:47.119><c> and</c>

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hydrogen spectrum just by trial and
 

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hydrogen spectrum just by trial and
error.<00:01:47.920><c> And</c><00:01:48.079><c> the</c><00:01:48.240><c> formula</c><00:01:48.640><c> predicts</c><00:01:49.040><c> there</c>

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error. And the formula predicts there
 

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error. And the formula predicts there
should<00:01:49.439><c> be</c><00:01:50.079><c> more</c><00:01:50.560><c> lines.</c><00:01:51.439><c> And</c><00:01:51.680><c> we</c><00:01:52.159><c> actually</c>

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should be more lines. And we actually
 

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should be more lines. And we actually
confirm<00:01:53.280><c> it.</c><00:01:54.079><c> But</c><00:01:54.399><c> nobody</c><00:01:54.880><c> had</c><00:01:55.119><c> any</c><00:01:55.439><c> clue</c><00:01:55.840><c> what</c>

01:56.069 --> 01:56.079 align:start position:0%
confirm it. But nobody had any clue what
 

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confirm it. But nobody had any clue what
this<00:01:56.320><c> formula</c><00:01:56.799><c> meant.</c><00:01:57.439><c> Why</c><00:01:57.680><c> did</c><00:01:57.840><c> it</c><00:01:58.079><c> work?</c>

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this formula meant. Why did it work?
 

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this formula meant. Why did it work?
To<00:02:00.640><c> answer</c><00:02:00.880><c> this,</c><00:02:01.200><c> we</c><00:02:01.600><c> needed</c><00:02:02.000><c> a</c><00:02:02.399><c> good</c><00:02:02.960><c> theory</c>

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To answer this, we needed a good theory
 

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To answer this, we needed a good theory
of<00:02:03.680><c> light.</c><00:02:04.399><c> By</c><00:02:04.640><c> now,</c><00:02:04.880><c> we</c><00:02:05.119><c> knew</c><00:02:05.360><c> light</c><00:02:05.680><c> is</c><00:02:05.840><c> a</c>

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of light. By now, we knew light is a
 

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of light. By now, we knew light is a
wave<00:02:06.320><c> confirmed</c><00:02:06.719><c> by</c><00:02:06.880><c> the</c><00:02:07.040><c> interference</c>

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wave confirmed by the interference
 

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wave confirmed by the interference
pattern.<00:02:08.399><c> And</c><00:02:08.720><c> Maxwell</c><00:02:09.360><c> showed</c><00:02:09.679><c> that</c><00:02:10.239><c> light</c>

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pattern. And Maxwell showed that light
 

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pattern. And Maxwell showed that light
is<00:02:11.039><c> basically</c><00:02:12.080><c> a</c><00:02:12.480><c> ripple</c><00:02:12.879><c> in</c><00:02:13.040><c> the</c>

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is basically a ripple in the
 

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is basically a ripple in the
electromagnetic<00:02:14.080><c> field</c><00:02:14.800><c> produced</c><00:02:15.280><c> by</c>

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electromagnetic field produced by
 

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electromagnetic field produced by
accelerating<00:02:16.720><c> charges.</c><00:02:17.840><c> So</c><00:02:18.160><c> wiggling</c>

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accelerating charges. So wiggling
 

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accelerating charges. So wiggling
charges<00:02:19.599><c> produce</c><00:02:20.400><c> light.</c><00:02:21.120><c> Wiggling</c><00:02:21.520><c> it</c>

02:21.750 --> 02:21.760 align:start position:0%
charges produce light. Wiggling it
 

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charges produce light. Wiggling it
slowly<00:02:22.319><c> gives</c><00:02:22.640><c> us</c><00:02:23.040><c> low</c><00:02:23.360><c> frequency</c><00:02:23.920><c> light</c><00:02:24.640><c> and</c>

02:24.949 --> 02:24.959 align:start position:0%
slowly gives us low frequency light and
 

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slowly gives us low frequency light and
wiggling<00:02:25.360><c> it</c><00:02:25.599><c> faster</c><00:02:26.239><c> gets</c><00:02:26.480><c> us</c><00:02:26.879><c> high</c>

02:27.270 --> 02:27.280 align:start position:0%
wiggling it faster gets us high
 

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wiggling it faster gets us high
frequency<00:02:27.920><c> light.</c><00:02:29.120><c> And</c><00:02:29.440><c> these</c><00:02:29.760><c> EM</c><00:02:30.160><c> waves</c><00:02:30.480><c> or</c>

02:30.710 --> 02:30.720 align:start position:0%
frequency light. And these EM waves or
 

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frequency light. And these EM waves or
light<00:02:31.120><c> could</c><00:02:31.520><c> wiggle</c><00:02:32.000><c> other</c><00:02:32.400><c> charges</c><00:02:33.440><c> which</c>

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light could wiggle other charges which
 

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light could wiggle other charges which
meant<00:02:34.319><c> wireless</c><00:02:35.280><c> communication.</c>

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meant wireless communication.
 

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meant wireless communication.
It<00:02:37.280><c> was</c><00:02:37.519><c> a</c><00:02:38.000><c> breakthrough</c><00:02:38.640><c> in</c><00:02:38.959><c> communication</c>

02:39.589 --> 02:39.599 align:start position:0%
It was a breakthrough in communication
 

02:39.599 --> 02:43.350 align:start position:0%
It was a breakthrough in communication
technology,<00:02:40.959><c> but</c><00:02:41.760><c> it</c><00:02:42.080><c> couldn't</c><00:02:42.480><c> explain</c><00:02:43.040><c> the</c>

02:43.350 --> 02:43.360 align:start position:0%
technology, but it couldn't explain the
 

02:43.360 --> 02:46.309 align:start position:0%
technology, but it couldn't explain the
hydrogen<00:02:44.000><c> spectrum.</c>

02:46.309 --> 02:46.319 align:start position:0%
hydrogen spectrum.
 

02:46.319 --> 02:48.229 align:start position:0%
hydrogen spectrum.
See,<00:02:46.560><c> according</c><00:02:46.879><c> to</c><00:02:47.040><c> Maxwell,</c><00:02:47.519><c> a</c><00:02:47.680><c> hot</c><00:02:47.920><c> glowing</c>

02:48.229 --> 02:48.239 align:start position:0%
See, according to Maxwell, a hot glowing
 

02:48.239 --> 02:50.309 align:start position:0%
See, according to Maxwell, a hot glowing
gas<00:02:48.480><c> has</c><00:02:48.800><c> billions</c><00:02:49.200><c> of</c><00:02:49.360><c> randomly</c><00:02:49.840><c> jiggling</c>

02:50.309 --> 02:50.319 align:start position:0%
gas has billions of randomly jiggling
 

02:50.319 --> 02:52.550 align:start position:0%
gas has billions of randomly jiggling
charges<00:02:50.879><c> from</c><00:02:51.200><c> very</c><00:02:51.519><c> low</c><00:02:51.760><c> frequencies</c><00:02:52.239><c> to</c>

02:52.550 --> 02:52.560 align:start position:0%
charges from very low frequencies to
 

02:52.560 --> 02:55.509 align:start position:0%
charges from very low frequencies to
very<00:02:52.879><c> high</c><00:02:53.120><c> frequencies,</c><00:02:54.160><c> which</c><00:02:54.480><c> meant</c><00:02:55.120><c> they</c>

02:55.509 --> 02:55.519 align:start position:0%
very high frequencies, which meant they
 

02:55.519 --> 02:57.750 align:start position:0%
very high frequencies, which meant they
should<00:02:55.760><c> be</c><00:02:55.920><c> giving</c><00:02:56.239><c> out</c><00:02:56.640><c> every</c><00:02:57.120><c> color</c><00:02:57.440><c> of</c>

02:57.750 --> 02:57.760 align:start position:0%
should be giving out every color of
 

02:57.760 --> 02:59.430 align:start position:0%
should be giving out every color of
light,

02:59.430 --> 02:59.440 align:start position:0%
light,
 

02:59.440 --> 03:02.309 align:start position:0%
light,
but<00:02:59.920><c> they</c><00:03:00.239><c> didn't.</c><00:03:01.200><c> So</c><00:03:01.680><c> something</c><00:03:02.000><c> was</c>

03:02.309 --> 03:02.319 align:start position:0%
but they didn't. So something was
 

03:02.319 --> 03:05.990 align:start position:0%
but they didn't. So something was
horribly<00:03:03.120><c> wrong</c><00:03:03.920><c> with</c><00:03:04.239><c> Maxwell's</c><00:03:04.800><c> theory.</c>

03:05.990 --> 03:06.000 align:start position:0%
horribly wrong with Maxwell's theory.
 

03:06.000 --> 03:08.949 align:start position:0%
horribly wrong with Maxwell's theory.
But<00:03:06.400><c> it</c><00:03:06.800><c> got</c><00:03:07.120><c> worse.</c><00:03:08.159><c> Pretty</c><00:03:08.480><c> soon</c><00:03:08.720><c> we</c>

03:08.949 --> 03:08.959 align:start position:0%
But it got worse. Pretty soon we
 

03:08.959 --> 03:11.030 align:start position:0%
But it got worse. Pretty soon we
discovered<00:03:09.280><c> that</c><00:03:09.519><c> atoms</c><00:03:10.000><c> had</c><00:03:10.159><c> a</c><00:03:10.480><c> positive</c>

03:11.030 --> 03:11.040 align:start position:0%
discovered that atoms had a positive
 

03:11.040 --> 03:13.350 align:start position:0%
discovered that atoms had a positive
nuclear<00:03:11.599><c> core.</c><00:03:12.239><c> So</c><00:03:12.400><c> we</c><00:03:12.640><c> thought</c><00:03:12.800><c> the</c><00:03:12.959><c> negative</c>

03:13.350 --> 03:13.360 align:start position:0%
nuclear core. So we thought the negative
 

03:13.360 --> 03:15.509 align:start position:0%
nuclear core. So we thought the negative
electrons<00:03:13.840><c> must</c><00:03:14.080><c> be</c><00:03:14.239><c> orbiting</c><00:03:14.720><c> this</c><00:03:14.959><c> nucleus,</c>

03:15.509 --> 03:15.519 align:start position:0%
electrons must be orbiting this nucleus,
 

03:15.519 --> 03:17.910 align:start position:0%
electrons must be orbiting this nucleus,
making<00:03:15.840><c> the</c><00:03:16.000><c> atoms</c><00:03:16.560><c> stable.</c><00:03:17.280><c> But</c><00:03:17.440><c> if</c><00:03:17.599><c> they</c><00:03:17.760><c> did</c>

03:17.910 --> 03:17.920 align:start position:0%
making the atoms stable. But if they did
 

03:17.920 --> 03:19.509 align:start position:0%
making the atoms stable. But if they did
that,<00:03:18.400><c> they</c><00:03:18.640><c> would</c><00:03:18.800><c> be</c><00:03:18.959><c> constantly</c>

03:19.509 --> 03:19.519 align:start position:0%
that, they would be constantly
 

03:19.519 --> 03:21.350 align:start position:0%
that, they would be constantly
accelerating<00:03:20.080><c> and</c><00:03:20.319><c> accelerating</c><00:03:20.879><c> charges</c>

03:21.350 --> 03:21.360 align:start position:0%
accelerating and accelerating charges
 

03:21.360 --> 03:23.750 align:start position:0%
accelerating and accelerating charges
radiate<00:03:21.840><c> EM</c><00:03:22.239><c> waves.</c><00:03:22.879><c> So</c><00:03:23.040><c> they</c><00:03:23.200><c> would</c><00:03:23.440><c> lose</c>

03:23.750 --> 03:23.760 align:start position:0%
radiate EM waves. So they would lose
 

03:23.760 --> 03:26.790 align:start position:0%
radiate EM waves. So they would lose
energy<00:03:24.159><c> and</c><00:03:24.480><c> collapse.</c><00:03:25.680><c> So</c><00:03:25.920><c> now</c><00:03:26.319><c> we</c><00:03:26.560><c> couldn't</c>

03:26.790 --> 03:26.800 align:start position:0%
energy and collapse. So now we couldn't
 

03:26.800 --> 03:30.390 align:start position:0%
energy and collapse. So now we couldn't
even<00:03:27.040><c> explain</c><00:03:27.519><c> why</c><00:03:27.920><c> atoms</c><00:03:28.480><c> were</c><00:03:29.040><c> stable.</c>

03:30.390 --> 03:30.400 align:start position:0%
even explain why atoms were stable.
 

03:30.400 --> 03:33.589 align:start position:0%
even explain why atoms were stable.
Physics<00:03:30.959><c> seemed</c><00:03:31.280><c> to</c><00:03:31.360><c> be</c><00:03:31.440><c> in</c><00:03:31.680><c> crisis.</c><00:03:32.799><c> But</c><00:03:33.360><c> a</c>

03:33.589 --> 03:33.599 align:start position:0%
Physics seemed to be in crisis. But a
 

03:33.599 --> 03:37.190 align:start position:0%
Physics seemed to be in crisis. But a
few<00:03:33.760><c> years</c><00:03:34.000><c> later,</c><00:03:34.480><c> everything</c><00:03:34.879><c> changed.</c>

03:37.190 --> 03:37.200 align:start position:0%
few years later, everything changed.
 

03:37.200 --> 03:39.670 align:start position:0%
few years later, everything changed.
If<00:03:37.440><c> you</c><00:03:37.599><c> shine</c><00:03:38.239><c> UV</c><00:03:38.640><c> light</c><00:03:38.879><c> on</c><00:03:39.200><c> zinc,</c><00:03:39.680><c> for</c>

03:39.670 --> 03:39.680 align:start position:0%
If you shine UV light on zinc, for
 

03:39.680 --> 03:42.070 align:start position:0%
If you shine UV light on zinc, for
example,<00:03:40.480><c> electrons</c><00:03:40.959><c> come</c><00:03:41.120><c> out.</c><00:03:41.680><c> That</c><00:03:41.920><c> makes</c>

03:42.070 --> 03:42.080 align:start position:0%
example, electrons come out. That makes
 

03:42.080 --> 03:43.990 align:start position:0%
example, electrons come out. That makes
sense.<00:03:42.640><c> Electrons</c><00:03:43.120><c> receive</c><00:03:43.440><c> energy</c><00:03:43.760><c> from</c><00:03:43.920><c> the</c>

03:43.990 --> 03:44.000 align:start position:0%
sense. Electrons receive energy from the
 

03:44.000 --> 03:46.309 align:start position:0%
sense. Electrons receive energy from the
EM<00:03:44.400><c> waves.</c><00:03:45.120><c> But</c><00:03:45.280><c> if</c><00:03:45.440><c> you</c><00:03:45.519><c> shine</c><00:03:45.760><c> a</c><00:03:46.000><c> brighter</c>

03:46.309 --> 03:46.319 align:start position:0%
EM waves. But if you shine a brighter
 

03:46.319 --> 03:48.710 align:start position:0%
EM waves. But if you shine a brighter
visible<00:03:46.799><c> light,</c><00:03:47.280><c> no</c><00:03:47.519><c> electrons</c><00:03:48.080><c> came</c><00:03:48.239><c> out.</c>

03:48.710 --> 03:48.720 align:start position:0%
visible light, no electrons came out.
 

03:48.720 --> 03:50.070 align:start position:0%
visible light, no electrons came out.
That<00:03:48.879><c> didn't</c><00:03:49.120><c> make</c><00:03:49.200><c> any</c><00:03:49.440><c> sense.</c><00:03:49.680><c> I</c><00:03:49.920><c> mean,</c>

03:50.070 --> 03:50.080 align:start position:0%
That didn't make any sense. I mean,
 

03:50.080 --> 03:51.750 align:start position:0%
That didn't make any sense. I mean,
you're<00:03:50.239><c> pumping</c><00:03:50.480><c> in</c><00:03:50.720><c> more</c><00:03:50.959><c> energy.</c><00:03:51.440><c> So,</c><00:03:51.599><c> we</c>

03:51.750 --> 03:51.760 align:start position:0%
you're pumping in more energy. So, we
 

03:51.760 --> 03:53.270 align:start position:0%
you're pumping in more energy. So, we
would<00:03:51.920><c> expect</c><00:03:52.159><c> electrons</c><00:03:52.640><c> to</c><00:03:52.799><c> come</c><00:03:52.959><c> out</c><00:03:53.120><c> with</c>

03:53.270 --> 03:53.280 align:start position:0%
would expect electrons to come out with
 

03:53.280 --> 03:56.869 align:start position:0%
would expect electrons to come out with
more<00:03:53.519><c> energy.</c><00:03:54.959><c> So,</c><00:03:55.280><c> why</c><00:03:55.599><c> didn't</c><00:03:55.920><c> they</c><00:03:56.400><c> why</c><00:03:56.720><c> did</c>

03:56.869 --> 03:56.879 align:start position:0%
more energy. So, why didn't they why did
 

03:56.879 --> 03:59.429 align:start position:0%
more energy. So, why didn't they why did
the<00:03:57.040><c> color</c><00:03:57.360><c> matter</c><00:03:57.680><c> and</c><00:03:57.920><c> not</c><00:03:58.159><c> the</c><00:03:58.400><c> brightness?</c>

03:59.429 --> 03:59.439 align:start position:0%
the color matter and not the brightness?
 

03:59.439 --> 04:01.589 align:start position:0%
the color matter and not the brightness?
To<00:03:59.680><c> explain</c><00:04:00.000><c> this,</c><00:04:00.480><c> Albert</c><00:04:00.959><c> Einstein</c>

04:01.589 --> 04:01.599 align:start position:0%
To explain this, Albert Einstein
 

04:01.599 --> 04:04.789 align:start position:0%
To explain this, Albert Einstein
proposed<00:04:02.159><c> something</c><00:04:02.799><c> radical.</c><00:04:03.840><c> What</c><00:04:04.080><c> if</c>

04:04.789 --> 04:04.799 align:start position:0%
proposed something radical. What if
 

04:04.799 --> 04:06.630 align:start position:0%
proposed something radical. What if
light<00:04:05.200><c> doesn't</c><00:04:05.599><c> deliver</c><00:04:06.080><c> energy</c>

04:06.630 --> 04:06.640 align:start position:0%
light doesn't deliver energy
 

04:06.640 --> 04:10.789 align:start position:0%
light doesn't deliver energy
continuously<00:04:08.080><c> but</c><00:04:08.400><c> in</c><00:04:08.799><c> discrete</c><00:04:09.840><c> chunks</c>

04:10.789 --> 04:10.799 align:start position:0%
continuously but in discrete chunks
 

04:10.799 --> 04:13.990 align:start position:0%
continuously but in discrete chunks
called<00:04:11.519><c> photons?</c><00:04:12.959><c> And</c><00:04:13.200><c> what</c><00:04:13.360><c> if</c><00:04:13.519><c> the</c><00:04:13.760><c> energy</c>

04:13.990 --> 04:14.000 align:start position:0%
called photons? And what if the energy
 

04:14.000 --> 04:16.789 align:start position:0%
called photons? And what if the energy
of<00:04:14.239><c> each</c><00:04:14.400><c> photon</c><00:04:14.959><c> depends</c><00:04:15.519><c> only</c><00:04:16.000><c> on</c><00:04:16.320><c> its</c>

04:16.789 --> 04:16.799 align:start position:0%
of each photon depends only on its
 

04:16.799 --> 04:20.150 align:start position:0%
of each photon depends only on its
frequency?<00:04:18.079><c> Then</c><00:04:18.560><c> a</c><00:04:18.880><c> bright</c><00:04:19.440><c> visible</c><00:04:19.840><c> light</c>

04:20.150 --> 04:20.160 align:start position:0%
frequency? Then a bright visible light
 

04:20.160 --> 04:22.150 align:start position:0%
frequency? Then a bright visible light
will<00:04:20.560><c> deliver</c><00:04:20.880><c> a</c><00:04:21.199><c> lot</c><00:04:21.359><c> of</c><00:04:21.440><c> photons</c><00:04:21.919><c> per</c>

04:22.150 --> 04:22.160 align:start position:0%
will deliver a lot of photons per
 

04:22.160 --> 04:24.390 align:start position:0%
will deliver a lot of photons per
second,<00:04:22.720><c> but</c><00:04:22.960><c> each</c><00:04:23.199><c> one's</c><00:04:23.440><c> too</c><00:04:23.680><c> weak</c><00:04:23.919><c> to</c><00:04:24.160><c> budge</c>

04:24.390 --> 04:24.400 align:start position:0%
second, but each one's too weak to budge
 

04:24.400 --> 04:26.550 align:start position:0%
second, but each one's too weak to budge
an<00:04:24.560><c> electron.</c><00:04:25.280><c> It's</c><00:04:25.520><c> like</c><00:04:25.759><c> throwing</c><00:04:26.240><c> ping</c>

04:26.550 --> 04:26.560 align:start position:0%
an electron. It's like throwing ping
 

04:26.560 --> 04:29.270 align:start position:0%
an electron. It's like throwing ping
pong<00:04:26.720><c> balls</c><00:04:27.040><c> at</c><00:04:27.199><c> a</c><00:04:27.360><c> bowling</c><00:04:27.759><c> ball.</c><00:04:28.800><c> But</c><00:04:29.040><c> on</c><00:04:29.199><c> the</c>

04:29.270 --> 04:29.280 align:start position:0%
pong balls at a bowling ball. But on the
 

04:29.280 --> 04:31.990 align:start position:0%
pong balls at a bowling ball. But on the
other<00:04:29.360><c> hand,</c><00:04:29.600><c> a</c><00:04:29.919><c> dim</c><00:04:30.320><c> UV</c><00:04:30.720><c> light</c><00:04:31.199><c> will</c><00:04:31.520><c> deliver</c>

04:31.990 --> 04:32.000 align:start position:0%
other hand, a dim UV light will deliver
 

04:32.000 --> 04:35.110 align:start position:0%
other hand, a dim UV light will deliver
fewer<00:04:32.400><c> photons</c><00:04:32.960><c> per</c><00:04:33.280><c> second,</c><00:04:33.840><c> but</c><00:04:34.240><c> each</c><00:04:34.639><c> one</c>

04:35.110 --> 04:35.120 align:start position:0%
fewer photons per second, but each one
 

04:35.120 --> 04:38.150 align:start position:0%
fewer photons per second, but each one
is<00:04:35.520><c> strong</c><00:04:35.840><c> enough</c><00:04:36.240><c> to</c><00:04:36.560><c> knock</c><00:04:36.880><c> it</c><00:04:37.040><c> off</c><00:04:37.520><c> like</c><00:04:37.840><c> a</c>

04:38.150 --> 04:38.160 align:start position:0%
is strong enough to knock it off like a
 

04:38.160 --> 04:40.230 align:start position:0%
is strong enough to knock it off like a
single<00:04:38.639><c> cannonball.</c>

04:40.230 --> 04:40.240 align:start position:0%
single cannonball.
 

04:40.240 --> 04:42.230 align:start position:0%
single cannonball.
So<00:04:40.639><c> this</c><00:04:40.960><c> explained</c><00:04:41.360><c> the</c><00:04:41.520><c> photoelectric</c>

04:42.230 --> 04:42.240 align:start position:0%
So this explained the photoelectric
 

04:42.240 --> 04:45.590 align:start position:0%
So this explained the photoelectric
mystery<00:04:43.120><c> beautifully</c><00:04:44.000><c> and</c><00:04:44.320><c> Einstein</c><00:04:45.199><c> won</c><00:04:45.440><c> the</c>

04:45.590 --> 04:45.600 align:start position:0%
mystery beautifully and Einstein won the
 

04:45.600 --> 04:48.469 align:start position:0%
mystery beautifully and Einstein won the
Nobel<00:04:46.000><c> Prize</c><00:04:46.240><c> for</c><00:04:46.479><c> it.</c><00:04:46.960><c> And</c><00:04:47.199><c> this</c><00:04:47.360><c> constant</c><00:04:47.919><c> is</c>

04:48.469 --> 04:48.479 align:start position:0%
Nobel Prize for it. And this constant is
 

04:48.479 --> 04:50.950 align:start position:0%
Nobel Prize for it. And this constant is
you<00:04:48.720><c> probably</c><00:04:48.960><c> know</c><00:04:49.360><c> the</c><00:04:49.520><c> planks</c><00:04:50.000><c> constant.</c>

04:50.950 --> 04:50.960 align:start position:0%
you probably know the planks constant.
 

04:50.960 --> 04:52.790 align:start position:0%
you probably know the planks constant.
But<00:04:51.120><c> a</c><00:04:51.360><c> few</c><00:04:51.520><c> years</c><00:04:51.759><c> later,</c><00:04:52.160><c> a</c><00:04:52.400><c> Danish</c>

04:52.790 --> 04:52.800 align:start position:0%
But a few years later, a Danish
 

04:52.800 --> 04:55.510 align:start position:0%
But a few years later, a Danish
physicist<00:04:53.280><c> named</c><00:04:53.680><c> Neils</c><00:04:54.160><c> Boore</c><00:04:54.880><c> pushed</c><00:04:55.280><c> this</c>

04:55.510 --> 04:55.520 align:start position:0%
physicist named Neils Boore pushed this
 

04:55.520 --> 04:59.189 align:start position:0%
physicist named Neils Boore pushed this
idea<00:04:56.000><c> even</c><00:04:56.400><c> further.</c>

04:59.189 --> 04:59.199 align:start position:0%
idea even further.
 

04:59.199 --> 05:02.070 align:start position:0%
idea even further.
Bore<00:04:59.600><c> wondered</c><00:05:00.240><c> if</c><00:05:00.639><c> light</c><00:05:01.040><c> is</c><00:05:01.280><c> absorbed</c><00:05:01.759><c> in</c>

05:02.070 --> 05:02.080 align:start position:0%
Bore wondered if light is absorbed in
 

05:02.080 --> 05:04.310 align:start position:0%
Bore wondered if light is absorbed in
chunks,<00:05:02.960><c> it</c><00:05:03.199><c> must</c><00:05:03.440><c> also</c><00:05:03.680><c> be</c><00:05:03.840><c> emitted</c><00:05:04.160><c> in</c>

05:04.310 --> 05:04.320 align:start position:0%
chunks, it must also be emitted in
 

05:04.320 --> 05:07.270 align:start position:0%
chunks, it must also be emitted in
chunks,<00:05:04.800><c> right?</c><00:05:05.600><c> Which</c><00:05:05.840><c> meant</c><00:05:06.320><c> electrons</c><00:05:06.960><c> had</c>

05:07.270 --> 05:07.280 align:start position:0%
chunks, right? Which meant electrons had
 

05:07.280 --> 05:10.710 align:start position:0%
chunks, right? Which meant electrons had
to<00:05:07.600><c> transition</c><00:05:08.639><c> from</c><00:05:08.960><c> a</c><00:05:09.280><c> higher</c><00:05:09.600><c> energy</c><00:05:10.000><c> level</c>

05:10.710 --> 05:10.720 align:start position:0%
to transition from a higher energy level
 

05:10.720 --> 05:13.749 align:start position:0%
to transition from a higher energy level
straight<00:05:11.360><c> to</c><00:05:11.840><c> a</c><00:05:12.080><c> lower</c><00:05:12.320><c> energy</c><00:05:12.720><c> one</c><00:05:13.360><c> to</c>

05:13.749 --> 05:13.759 align:start position:0%
straight to a lower energy one to
 

05:13.759 --> 05:15.830 align:start position:0%
straight to a lower energy one to
release<00:05:14.160><c> these</c><00:05:14.400><c> energy</c><00:05:14.800><c> chunks.</c><00:05:15.440><c> So</c><00:05:15.600><c> he</c>

05:15.830 --> 05:15.840 align:start position:0%
release these energy chunks. So he
 

05:15.840 --> 05:17.670 align:start position:0%
release these energy chunks. So he
postulated<00:05:16.400><c> that</c><00:05:16.639><c> electrons</c><00:05:17.199><c> orbit</c><00:05:17.520><c> the</c>

05:17.670 --> 05:17.680 align:start position:0%
postulated that electrons orbit the
 

05:17.680 --> 05:20.390 align:start position:0%
postulated that electrons orbit the
nucleus<00:05:18.240><c> only</c><00:05:18.639><c> at</c><00:05:19.039><c> some</c><00:05:19.440><c> special</c><00:05:19.919><c> energy</c>

05:20.390 --> 05:20.400 align:start position:0%
nucleus only at some special energy
 

05:20.400 --> 05:23.670 align:start position:0%
nucleus only at some special energy
levels,<00:05:21.199><c> nowhere</c><00:05:21.680><c> in</c><00:05:21.919><c> between.</c><00:05:22.880><c> And</c><00:05:23.120><c> in</c><00:05:23.360><c> these</c>

05:23.670 --> 05:23.680 align:start position:0%
levels, nowhere in between. And in these
 

05:23.680 --> 05:26.070 align:start position:0%
levels, nowhere in between. And in these
special<00:05:24.000><c> orbits,</c><00:05:24.560><c> he</c><00:05:24.800><c> said</c><00:05:25.280><c> they</c><00:05:25.759><c> don't</c>

05:26.070 --> 05:26.080 align:start position:0%
special orbits, he said they don't
 

05:26.080 --> 05:28.629 align:start position:0%
special orbits, he said they don't
radiate<00:05:26.639><c> energy.</c><00:05:27.600><c> It</c><00:05:27.840><c> seemed</c><00:05:28.080><c> like</c><00:05:28.240><c> he</c><00:05:28.479><c> was</c>

05:28.629 --> 05:28.639 align:start position:0%
radiate energy. It seemed like he was
 

05:28.639 --> 05:31.189 align:start position:0%
radiate energy. It seemed like he was
just<00:05:28.960><c> making</c><00:05:29.360><c> stuff</c><00:05:29.600><c> up.</c><00:05:30.560><c> But</c><00:05:30.800><c> look</c><00:05:30.960><c> what</c>

05:31.189 --> 05:31.199 align:start position:0%
just making stuff up. But look what
 

05:31.199 --> 05:33.749 align:start position:0%
just making stuff up. But look what
happens<00:05:32.000><c> now.</c><00:05:32.400><c> When</c><00:05:32.560><c> you</c><00:05:32.720><c> heat</c><00:05:32.960><c> up</c><00:05:33.039><c> a</c><00:05:33.280><c> gas,</c>

05:33.749 --> 05:33.759 align:start position:0%
happens now. When you heat up a gas,
 

05:33.759 --> 05:35.990 align:start position:0%
happens now. When you heat up a gas,
electrons<00:05:34.240><c> jump</c><00:05:34.479><c> to</c><00:05:34.639><c> a</c><00:05:34.800><c> higher</c><00:05:35.360><c> allowed</c>

05:35.990 --> 05:36.000 align:start position:0%
electrons jump to a higher allowed
 

05:36.000 --> 05:38.629 align:start position:0%
electrons jump to a higher allowed
level.<00:05:36.880><c> But</c><00:05:37.039><c> when</c><00:05:37.280><c> they</c><00:05:37.520><c> fall</c><00:05:37.759><c> back</c><00:05:38.080><c> down,</c>

05:38.629 --> 05:38.639 align:start position:0%
level. But when they fall back down,
 

05:38.639 --> 05:40.950 align:start position:0%
level. But when they fall back down,
they<00:05:39.039><c> release</c><00:05:39.440><c> the</c><00:05:39.680><c> energy</c><00:05:40.080><c> difference</c><00:05:40.400><c> as</c><00:05:40.720><c> a</c>

05:40.950 --> 05:40.960 align:start position:0%
they release the energy difference as a
 

05:40.960 --> 05:44.629 align:start position:0%
they release the energy difference as a
photon<00:05:41.360><c> of</c><00:05:41.600><c> a</c><00:05:41.840><c> specific</c><00:05:42.400><c> frequency.</c><00:05:43.759><c> This</c>

05:44.629 --> 05:44.639 align:start position:0%
photon of a specific frequency. This
 

05:44.639 --> 05:47.749 align:start position:0%
photon of a specific frequency. This
explained<00:05:45.360><c> the</c><00:05:45.759><c> hydrogen</c><00:05:46.479><c> spectrum.</c><00:05:47.360><c> And</c><00:05:47.520><c> the</c>

05:47.749 --> 05:47.759 align:start position:0%
explained the hydrogen spectrum. And the
 

05:47.759 --> 05:50.310 align:start position:0%
explained the hydrogen spectrum. And the
photon's<00:05:48.320><c> energy</c><00:05:48.800><c> is</c><00:05:49.120><c> basically</c><00:05:49.520><c> the</c><00:05:49.759><c> energy</c>

05:50.310 --> 05:50.320 align:start position:0%
photon's energy is basically the energy
 

05:50.320 --> 05:52.790 align:start position:0%
photon's energy is basically the energy
lost<00:05:50.800><c> by</c><00:05:51.039><c> the</c><00:05:51.280><c> electron.</c><00:05:52.080><c> From</c><00:05:52.320><c> this</c><00:05:52.639><c> he</c>

05:52.790 --> 05:52.800 align:start position:0%
lost by the electron. From this he
 

05:52.800 --> 05:54.790 align:start position:0%
lost by the electron. From this he
derived<00:05:53.199><c> an</c><00:05:53.360><c> expression</c><00:05:53.680><c> for</c><00:05:53.919><c> the</c><00:05:54.080><c> frequency</c>

05:54.790 --> 05:54.800 align:start position:0%
derived an expression for the frequency
 

05:54.800 --> 05:57.909 align:start position:0%
derived an expression for the frequency
and<00:05:55.280><c> got</c><00:05:55.520><c> the</c><00:05:55.840><c> Balmer's</c><00:05:56.560><c> formula.</c><00:05:57.520><c> So</c><00:05:57.680><c> his</c>

05:57.909 --> 05:57.919 align:start position:0%
and got the Balmer's formula. So his
 

05:57.919 --> 05:59.270 align:start position:0%
and got the Balmer's formula. So his
postulate<00:05:58.479><c> explained</c><00:05:58.800><c> the</c><00:05:58.880><c> hydrogen</c>

05:59.270 --> 05:59.280 align:start position:0%
postulate explained the hydrogen
 

05:59.280 --> 06:01.189 align:start position:0%
postulate explained the hydrogen
spectrum,<00:05:59.759><c> the</c><00:05:59.919><c> Balmer's</c><00:06:00.400><c> formula</c><00:06:00.800><c> and</c><00:06:01.039><c> the</c>

06:01.189 --> 06:01.199 align:start position:0%
spectrum, the Balmer's formula and the
 

06:01.199 --> 06:04.230 align:start position:0%
spectrum, the Balmer's formula and the
atomic<00:06:01.680><c> stability</c><00:06:02.400><c> all</c><00:06:02.720><c> at</c><00:06:02.960><c> once.</c><00:06:03.759><c> This</c><00:06:04.000><c> was</c>

06:04.230 --> 06:04.240 align:start position:0%
atomic stability all at once. This was
 

06:04.240 --> 06:05.749 align:start position:0%
atomic stability all at once. This was
huge.

06:05.749 --> 06:05.759 align:start position:0%
huge.
 

06:05.759 --> 06:08.309 align:start position:0%
huge.
Later<00:06:06.080><c> in</c><00:06:06.240><c> an</c><00:06:06.479><c> interview,</c><00:06:07.039><c> he</c><00:06:07.360><c> says,</c><00:06:07.600><c> "As</c><00:06:08.000><c> soon</c>

06:08.309 --> 06:08.319 align:start position:0%
Later in an interview, he says, "As soon
 

06:08.319 --> 06:10.629 align:start position:0%
Later in an interview, he says, "As soon
as<00:06:08.560><c> I</c><00:06:08.720><c> saw</c><00:06:08.960><c> Balmer's</c><00:06:09.520><c> formula,</c><00:06:10.160><c> the</c><00:06:10.479><c> whole</c>

06:10.629 --> 06:10.639 align:start position:0%
as I saw Balmer's formula, the whole
 

06:10.639 --> 06:13.270 align:start position:0%
as I saw Balmer's formula, the whole
thing<00:06:10.880><c> was</c><00:06:11.120><c> immediately</c><00:06:11.759><c> clear</c><00:06:12.000><c> to</c><00:06:12.240><c> me."</c><00:06:13.039><c> He</c>

06:13.270 --> 06:13.280 align:start position:0%
thing was immediately clear to me." He
 

06:13.280 --> 06:15.510 align:start position:0%
thing was immediately clear to me." He
was<00:06:13.440><c> probably</c><00:06:13.759><c> showing</c><00:06:14.080><c> off,</c><00:06:14.639><c> but</c><00:06:14.960><c> there</c><00:06:15.199><c> were</c>

06:15.510 --> 06:15.520 align:start position:0%
was probably showing off, but there were
 

06:15.520 --> 06:18.230 align:start position:0%
was probably showing off, but there were
many<00:06:15.840><c> unanswered</c><00:06:16.639><c> questions</c><00:06:17.039><c> here.</c><00:06:17.759><c> I</c><00:06:18.000><c> mean,</c>

06:18.230 --> 06:18.240 align:start position:0%
many unanswered questions here. I mean,
 

06:18.240 --> 06:20.230 align:start position:0%
many unanswered questions here. I mean,
why<00:06:18.479><c> were</c><00:06:18.720><c> electrons</c><00:06:19.199><c> restricted</c><00:06:19.680><c> to</c><00:06:19.840><c> those</c>

06:20.230 --> 06:20.240 align:start position:0%
why were electrons restricted to those
 

06:20.240 --> 06:22.469 align:start position:0%
why were electrons restricted to those
specific<00:06:20.720><c> orbits?</c><00:06:21.520><c> and</c><00:06:21.840><c> why</c><00:06:22.080><c> don't</c><00:06:22.240><c> they</c>

06:22.469 --> 06:22.479 align:start position:0%
specific orbits? and why don't they
 

06:22.479 --> 06:24.950 align:start position:0%
specific orbits? and why don't they
radiate<00:06:23.120><c> while</c><00:06:23.520><c> they</c><00:06:23.759><c> are</c><00:06:23.919><c> there?</c><00:06:24.639><c> Boore</c>

06:24.950 --> 06:24.960 align:start position:0%
radiate while they are there? Boore
 

06:24.960 --> 06:27.510 align:start position:0%
radiate while they are there? Boore
basically<00:06:25.280><c> said,</c><00:06:25.520><c> "Trust</c><00:06:25.840><c> me."</c><00:06:26.639><c> But</c><00:06:26.800><c> a</c><00:06:27.120><c> French</c>

06:27.510 --> 06:27.520 align:start position:0%
basically said, "Trust me." But a French
 

06:27.520 --> 06:30.390 align:start position:0%
basically said, "Trust me." But a French
PhD<00:06:28.080><c> student</c><00:06:28.319><c> named</c><00:06:28.880><c> Louis</c><00:06:29.199><c> De</c><00:06:29.440><c> Bruy</c><00:06:30.080><c> came</c><00:06:30.240><c> up</c>

06:30.390 --> 06:30.400 align:start position:0%
PhD student named Louis De Bruy came up
 

06:30.400 --> 06:33.749 align:start position:0%
PhD student named Louis De Bruy came up
with<00:06:30.560><c> an</c><00:06:30.880><c> answer.</c><00:06:31.360><c> He</c><00:06:31.600><c> pushed</c><00:06:31.919><c> Bor's</c><00:06:32.400><c> idea</c><00:06:33.280><c> to</c>

06:33.749 --> 06:33.759 align:start position:0%
with an answer. He pushed Bor's idea to
 

06:33.759 --> 06:37.189 align:start position:0%
with an answer. He pushed Bor's idea to
the<00:06:34.080><c> limit.</c><00:06:35.120><c> Light</c><00:06:35.680><c> moves</c><00:06:35.919><c> as</c><00:06:36.160><c> a</c><00:06:36.319><c> wave</c><00:06:36.800><c> but</c>

06:37.189 --> 06:37.199 align:start position:0%
the limit. Light moves as a wave but
 

06:37.199 --> 06:40.230 align:start position:0%
the limit. Light moves as a wave but
interacts<00:06:37.759><c> with</c><00:06:38.080><c> stuff</c><00:06:38.880><c> like</c><00:06:39.120><c> a</c><00:06:39.440><c> particle,</c>

06:40.230 --> 06:40.240 align:start position:0%
interacts with stuff like a particle,
 

06:40.240 --> 06:43.749 align:start position:0%
interacts with stuff like a particle,
right?<00:06:41.120><c> Dual</c><00:06:41.520><c> nature.</c><00:06:42.479><c> So</c><00:06:42.720><c> he</c><00:06:42.960><c> wondered,</c><00:06:43.600><c> what</c>

06:43.749 --> 06:43.759 align:start position:0%
right? Dual nature. So he wondered, what
 

06:43.759 --> 06:46.469 align:start position:0%
right? Dual nature. So he wondered, what
if<00:06:44.160><c> matter</c><00:06:44.639><c> behaves</c><00:06:44.960><c> the</c><00:06:45.199><c> same</c><00:06:45.520><c> way?</c><00:06:46.080><c> What</c><00:06:46.319><c> if</c>

06:46.469 --> 06:46.479 align:start position:0%
if matter behaves the same way? What if
 

06:46.479 --> 06:50.550 align:start position:0%
if matter behaves the same way? What if
matter<00:06:47.039><c> 2</c><00:06:47.919><c> interacts</c><00:06:48.479><c> like</c><00:06:48.720><c> a</c><00:06:48.880><c> particle</c><00:06:49.919><c> but</c>

06:50.550 --> 06:50.560 align:start position:0%
matter 2 interacts like a particle but
 

06:50.560 --> 06:55.189 align:start position:0%
matter 2 interacts like a particle but
moves<00:06:51.440><c> as</c><00:06:51.759><c> a</c><00:06:52.080><c> wave?</c><00:06:53.440><c> Then</c><00:06:54.319><c> electrons</c><00:06:54.880><c> inside</c>

06:55.189 --> 06:55.199 align:start position:0%
moves as a wave? Then electrons inside
 

06:55.199 --> 06:58.230 align:start position:0%
moves as a wave? Then electrons inside
an<00:06:55.520><c> atom</c><00:06:56.000><c> could</c><00:06:56.240><c> form</c><00:06:56.880><c> standing</c><00:06:57.360><c> waves.</c><00:06:58.080><c> And</c>

06:58.230 --> 06:58.240 align:start position:0%
an atom could form standing waves. And
 

06:58.240 --> 07:00.070 align:start position:0%
an atom could form standing waves. And
just<00:06:58.479><c> like</c><00:06:58.639><c> a</c><00:06:58.800><c> guitar</c><00:06:59.199><c> string</c><00:06:59.520><c> can</c><00:06:59.759><c> only</c>

07:00.070 --> 07:00.080 align:start position:0%
just like a guitar string can only
 

07:00.080 --> 07:02.550 align:start position:0%
just like a guitar string can only
vibrate<00:07:00.479><c> with</c><00:07:00.800><c> say</c><00:07:01.280><c> three</c><00:07:01.520><c> loops</c><00:07:01.919><c> or</c><00:07:02.319><c> four</c>

07:02.550 --> 07:02.560 align:start position:0%
vibrate with say three loops or four
 

07:02.560 --> 07:06.150 align:start position:0%
vibrate with say three loops or four
loops<00:07:03.120><c> or</c><00:07:03.440><c> five</c><00:07:03.840><c> but</c><00:07:04.240><c> nothing</c><00:07:04.639><c> in</c><00:07:04.960><c> between.</c>

07:06.150 --> 07:06.160 align:start position:0%
loops or five but nothing in between.
 

07:06.160 --> 07:08.790 align:start position:0%
loops or five but nothing in between.
Electron<00:07:06.639><c> waves</c><00:07:07.039><c> can</c><00:07:07.440><c> only</c><00:07:07.759><c> vibrate</c><00:07:08.319><c> with</c>

07:08.790 --> 07:08.800 align:start position:0%
Electron waves can only vibrate with
 

07:08.800 --> 07:13.110 align:start position:0%
Electron waves can only vibrate with
three<00:07:09.280><c> or</c><00:07:09.840><c> four</c><00:07:10.639><c> or</c><00:07:11.120><c> five</c><00:07:11.520><c> loops</c><00:07:11.919><c> but</c><00:07:12.560><c> nothing</c>

07:13.110 --> 07:13.120 align:start position:0%
three or four or five loops but nothing
 

07:13.120 --> 07:16.469 align:start position:0%
three or four or five loops but nothing
in<00:07:13.520><c> between.</c>

07:16.469 --> 07:16.479 align:start position:0%
in between.
 

07:16.479 --> 07:20.629 align:start position:0%
in between.
This<00:07:17.120><c> explained</c><00:07:17.919><c> why</c><00:07:19.039><c> electrons</c><00:07:19.759><c> can</c><00:07:20.080><c> only</c>

07:20.629 --> 07:20.639 align:start position:0%
This explained why electrons can only
 

07:20.639 --> 07:23.430 align:start position:0%
This explained why electrons can only
exist<00:07:21.280><c> at</c><00:07:21.599><c> those</c><00:07:22.000><c> specific</c><00:07:22.720><c> distances</c><00:07:23.199><c> from</c>

07:23.430 --> 07:23.440 align:start position:0%
exist at those specific distances from
 

07:23.440 --> 07:26.390 align:start position:0%
exist at those specific distances from
the<00:07:23.599><c> nucleus</c><00:07:24.160><c> as</c><00:07:24.479><c> Boore</c><00:07:24.880><c> said.</c><00:07:25.840><c> And</c><00:07:26.000><c> it</c><00:07:26.160><c> also</c>

07:26.390 --> 07:26.400 align:start position:0%
the nucleus as Boore said. And it also
 

07:26.400 --> 07:28.309 align:start position:0%
the nucleus as Boore said. And it also
explains<00:07:26.800><c> why</c><00:07:27.039><c> they</c><00:07:27.280><c> don't</c><00:07:27.520><c> radiate</c><00:07:28.000><c> while</c>

07:28.309 --> 07:28.319 align:start position:0%
explains why they don't radiate while
 

07:28.319 --> 07:30.230 align:start position:0%
explains why they don't radiate while
sitting<00:07:28.560><c> at</c><00:07:28.800><c> those</c><00:07:28.960><c> energy</c><00:07:29.280><c> levels</c><00:07:29.680><c> because</c>

07:30.230 --> 07:30.240 align:start position:0%
sitting at those energy levels because
 

07:30.240 --> 07:31.909 align:start position:0%
sitting at those energy levels because
they're<00:07:30.479><c> not</c><00:07:30.720><c> orbiting</c><00:07:31.199><c> particles.</c><00:07:31.759><c> They're</c>

07:31.909 --> 07:31.919 align:start position:0%
they're not orbiting particles. They're
 

07:31.919 --> 07:34.230 align:start position:0%
they're not orbiting particles. They're
not<00:07:32.080><c> accelerating.</c><00:07:32.880><c> They're</c><00:07:33.280><c> stationary</c>

07:34.230 --> 07:34.240 align:start position:0%
not accelerating. They're stationary
 

07:34.240 --> 07:37.670 align:start position:0%
not accelerating. They're stationary
waves<00:07:34.800><c> now.</c><00:07:35.680><c> They</c><00:07:36.080><c> only</c><00:07:36.479><c> radiate</c><00:07:36.800><c> a</c><00:07:36.960><c> photon</c>

07:37.670 --> 07:37.680 align:start position:0%
waves now. They only radiate a photon
 

07:37.680 --> 07:40.550 align:start position:0%
waves now. They only radiate a photon
when<00:07:38.000><c> they</c><00:07:38.240><c> transition</c><00:07:38.720><c> from</c><00:07:38.960><c> a</c><00:07:39.280><c> higher</c><00:07:40.000><c> to</c><00:07:40.240><c> a</c>

07:40.550 --> 07:40.560 align:start position:0%
when they transition from a higher to a
 

07:40.560 --> 07:43.029 align:start position:0%
when they transition from a higher to a
lower<00:07:41.280><c> level.</c>

07:43.029 --> 07:43.039 align:start position:0%
lower level.
 

07:43.039 --> 07:47.110 align:start position:0%
lower level.
This<00:07:43.680><c> was</c><00:07:44.000><c> so</c><00:07:44.560><c> radical.</c><00:07:45.199><c> Matter</c><00:07:46.080><c> behaving</c><00:07:46.720><c> as</c>

07:47.110 --> 07:47.120 align:start position:0%
This was so radical. Matter behaving as
 

07:47.120 --> 07:49.990 align:start position:0%
This was so radical. Matter behaving as
waves.<00:07:48.000><c> And</c><00:07:48.240><c> that's</c><00:07:48.479><c> not</c><00:07:48.639><c> all.</c><00:07:49.120><c> For</c><00:07:49.360><c> his</c><00:07:49.599><c> PhD</c>

07:49.990 --> 07:50.000 align:start position:0%
waves. And that's not all. For his PhD
 

07:50.000 --> 07:53.670 align:start position:0%
waves. And that's not all. For his PhD
thesis,<00:07:50.800><c> he</c><00:07:51.120><c> derived</c><00:07:51.680><c> an</c><00:07:52.000><c> expression</c><00:07:52.720><c> for</c><00:07:53.199><c> its</c>

07:53.670 --> 07:53.680 align:start position:0%
thesis, he derived an expression for its
 

07:53.680 --> 07:56.550 align:start position:0%
thesis, he derived an expression for its
wavelength<00:07:54.560><c> using</c><00:07:54.960><c> special</c><00:07:55.360><c> relativity.</c><00:07:56.319><c> He</c>

07:56.550 --> 07:56.560 align:start position:0%
wavelength using special relativity. He
 

07:56.560 --> 07:59.749 align:start position:0%
wavelength using special relativity. He
had<00:07:56.800><c> found</c><00:07:57.120><c> wavelength</c><00:07:57.680><c> of</c><00:07:58.080><c> matter.</c><00:07:59.120><c> It</c><00:07:59.440><c> seems</c>

07:59.749 --> 07:59.759 align:start position:0%
had found wavelength of matter. It seems
 

07:59.759 --> 08:02.790 align:start position:0%
had found wavelength of matter. It seems
so<00:08:00.000><c> bizarre.</c><00:08:00.800><c> His</c><00:08:01.120><c> examiners</c><00:08:01.919><c> only</c><00:08:02.319><c> approved</c>

08:02.790 --> 08:02.800 align:start position:0%
so bizarre. His examiners only approved
 

08:02.800 --> 08:05.670 align:start position:0%
so bizarre. His examiners only approved
it<00:08:03.199><c> after</c><00:08:03.759><c> confirming</c><00:08:04.319><c> it</c><00:08:04.720><c> with</c><00:08:05.039><c> Einstein</c>

08:05.670 --> 08:05.680 align:start position:0%
it after confirming it with Einstein
 

08:05.680 --> 08:09.029 align:start position:0%
it after confirming it with Einstein
himself.<00:08:06.639><c> Finally</c><00:08:07.280><c> in</c><00:08:07.440><c> the</c><00:08:07.680><c> following</c><00:08:08.000><c> year</c><00:08:08.800><c> a</c>

08:09.029 --> 08:09.039 align:start position:0%
himself. Finally in the following year a
 

08:09.039 --> 08:10.790 align:start position:0%
himself. Finally in the following year a
professor<00:08:09.360><c> at</c><00:08:09.599><c> the</c><00:08:09.759><c> University</c><00:08:10.080><c> of</c><00:08:10.240><c> Zurich</c>

08:10.790 --> 08:10.800 align:start position:0%
professor at the University of Zurich
 

08:10.800 --> 08:13.589 align:start position:0%
professor at the University of Zurich
gave<00:08:10.960><c> a</c><00:08:11.120><c> seminar</c><00:08:11.520><c> on</c><00:08:11.759><c> this</c><00:08:12.000><c> very</c><00:08:12.400><c> topic</c><00:08:13.199><c> and</c>

08:13.589 --> 08:13.599 align:start position:0%
gave a seminar on this very topic and
 

08:13.599 --> 08:15.430 align:start position:0%
gave a seminar on this very topic and
after<00:08:13.919><c> the</c><00:08:14.160><c> talk</c><00:08:14.560><c> a</c><00:08:14.800><c> colleague</c><00:08:15.039><c> in</c><00:08:15.280><c> the</c>

08:15.430 --> 08:15.440 align:start position:0%
after the talk a colleague in the
 

08:15.440 --> 08:18.629 align:start position:0%
after the talk a colleague in the
audience<00:08:15.840><c> asked</c><00:08:16.400><c> if</c><00:08:16.720><c> matter</c><00:08:17.039><c> is</c><00:08:17.199><c> a</c><00:08:17.440><c> wave</c><00:08:18.319><c> where</c>

08:18.629 --> 08:18.639 align:start position:0%
audience asked if matter is a wave where
 

08:18.639 --> 08:21.589 align:start position:0%
audience asked if matter is a wave where
is<00:08:18.800><c> the</c><00:08:18.960><c> wave</c><00:08:19.360><c> equation.</c><00:08:20.400><c> The</c><00:08:20.639><c> professor</c><00:08:21.280><c> was</c>

08:21.589 --> 08:21.599 align:start position:0%
is the wave equation. The professor was
 

08:21.599 --> 08:23.830 align:start position:0%
is the wave equation. The professor was
Irvin<00:08:22.160><c> Schroinger</c><00:08:22.960><c> and</c><00:08:23.199><c> he</c><00:08:23.440><c> took</c><00:08:23.599><c> that</c>

08:23.830 --> 08:23.840 align:start position:0%
Irvin Schroinger and he took that
 

08:23.840 --> 08:26.710 align:start position:0%
Irvin Schroinger and he took that
question<00:08:24.400><c> seriously.</c><00:08:25.599><c> So</c><00:08:25.919><c> he</c><00:08:26.240><c> went</c><00:08:26.400><c> on</c><00:08:26.560><c> a</c>

08:26.710 --> 08:26.720 align:start position:0%
question seriously. So he went on a
 

08:26.720 --> 08:28.790 align:start position:0%
question seriously. So he went on a
vacation<00:08:27.120><c> to</c><00:08:27.360><c> the</c><00:08:27.520><c> Swiss</c><00:08:27.840><c> Alps</c><00:08:28.400><c> over</c>

08:28.790 --> 08:28.800 align:start position:0%
vacation to the Swiss Alps over
 

08:28.800 --> 08:31.670 align:start position:0%
vacation to the Swiss Alps over
Christmas<00:08:29.759><c> and</c><00:08:30.080><c> came</c><00:08:30.319><c> back</c><00:08:30.960><c> with</c><00:08:31.120><c> the</c><00:08:31.360><c> wave</c>

08:31.670 --> 08:31.680 align:start position:0%
Christmas and came back with the wave
 

08:31.680 --> 08:36.790 align:start position:0%
Christmas and came back with the wave
equation.<00:08:32.719><c> But</c><00:08:33.039><c> how</c><00:08:33.599><c> did</c><00:08:33.839><c> he</c><00:08:34.080><c> do</c><00:08:34.240><c> it?</c>

08:36.790 --> 08:36.800 align:start position:0%
equation. But how did he do it?
 

08:36.800 --> 08:38.310 align:start position:0%
equation. But how did he do it?
I<00:08:37.039><c> went</c><00:08:37.200><c> through</c><00:08:37.360><c> Shinger's</c><00:08:38.000><c> original</c>

08:38.310 --> 08:38.320 align:start position:0%
I went through Shinger's original
 

08:38.320 --> 08:40.870 align:start position:0%
I went through Shinger's original
derivation<00:08:39.120><c> and</c><00:08:39.760><c> I</c><00:08:40.000><c> couldn't</c><00:08:40.320><c> understand</c><00:08:40.640><c> a</c>

08:40.870 --> 08:40.880 align:start position:0%
derivation and I couldn't understand a
 

08:40.880 --> 08:43.350 align:start position:0%
derivation and I couldn't understand a
thing.<00:08:41.440><c> In</c><00:08:41.599><c> fact,</c><00:08:41.839><c> he</c><00:08:42.080><c> himself</c><00:08:42.719><c> called</c><00:08:43.039><c> it</c>

08:43.350 --> 08:43.360 align:start position:0%
thing. In fact, he himself called it
 

08:43.360 --> 08:46.550 align:start position:0%
thing. In fact, he himself called it
unintelligible<00:08:44.560><c> later</c><00:08:44.880><c> on.</c><00:08:45.600><c> So,</c><00:08:45.760><c> I</c><00:08:45.920><c> looked</c><00:08:46.160><c> up</c>

08:46.550 --> 08:46.560 align:start position:0%
unintelligible later on. So, I looked up
 

08:46.560 --> 08:49.190 align:start position:0%
unintelligible later on. So, I looked up
Fineman's<00:08:47.279><c> derivation</c><00:08:48.080><c> and</c><00:08:48.320><c> that</c><00:08:48.720><c> was</c><00:08:48.880><c> also</c>

08:49.190 --> 08:49.200 align:start position:0%
Fineman's derivation and that was also
 

08:49.200 --> 08:51.350 align:start position:0%
Fineman's derivation and that was also
quite<00:08:49.600><c> mathy.</c><00:08:50.480><c> But</c><00:08:50.640><c> after</c><00:08:50.959><c> a</c><00:08:51.120><c> lot</c><00:08:51.200><c> of</c>

08:51.350 --> 08:51.360 align:start position:0%
quite mathy. But after a lot of
 

08:51.360 --> 08:53.990 align:start position:0%
quite mathy. But after a lot of
searching,<00:08:52.240><c> I</c><00:08:52.560><c> found</c><00:08:52.800><c> a</c><00:08:53.040><c> modern</c><00:08:53.440><c> version</c>

08:53.990 --> 08:54.000 align:start position:0%
searching, I found a modern version
 

08:54.000 --> 08:56.550 align:start position:0%
searching, I found a modern version
which<00:08:54.480><c> seemed</c><00:08:54.880><c> pretty</c><00:08:55.200><c> intuitive.</c><00:08:56.080><c> It</c><00:08:56.240><c> starts</c>

08:56.550 --> 08:56.560 align:start position:0%
which seemed pretty intuitive. It starts
 

08:56.560 --> 08:59.190 align:start position:0%
which seemed pretty intuitive. It starts
by<00:08:56.720><c> asking</c><00:08:57.360><c> what</c><00:08:57.680><c> rule</c><00:08:58.160><c> should</c><00:08:58.480><c> matter</c><00:08:58.880><c> waves</c>

08:59.190 --> 08:59.200 align:start position:0%
by asking what rule should matter waves
 

08:59.200 --> 09:02.150 align:start position:0%
by asking what rule should matter waves
obey<00:08:59.839><c> in</c><00:09:00.399><c> general.</c><00:09:01.279><c> The</c><00:09:01.519><c> most</c><00:09:01.680><c> fundamental</c>

09:02.150 --> 09:02.160 align:start position:0%
obey in general. The most fundamental
 

09:02.160 --> 09:04.230 align:start position:0%
obey in general. The most fundamental
one<00:09:02.320><c> we</c><00:09:02.560><c> know</c><00:09:02.800><c> is</c><00:09:03.200><c> energy</c><00:09:03.600><c> conservation,</c>

09:04.230 --> 09:04.240 align:start position:0%
one we know is energy conservation,
 

09:04.240 --> 09:06.710 align:start position:0%
one we know is energy conservation,
right?<00:09:04.640><c> Total</c><00:09:05.040><c> energy</c><00:09:05.680><c> equals</c><00:09:06.240><c> kinetic</c>

09:06.710 --> 09:06.720 align:start position:0%
right? Total energy equals kinetic
 

09:06.720 --> 09:09.509 align:start position:0%
right? Total energy equals kinetic
energy<00:09:07.279><c> plus</c><00:09:07.680><c> potential</c><00:09:08.160><c> energy.</c><00:09:08.959><c> Okay,</c><00:09:09.279><c> that</c>

09:09.509 --> 09:09.519 align:start position:0%
energy plus potential energy. Okay, that
 

09:09.519 --> 09:12.470 align:start position:0%
energy plus potential energy. Okay, that
makes<00:09:09.760><c> sense.</c><00:09:10.640><c> Um,</c><00:09:10.959><c> kinetic</c><00:09:11.440><c> energy</c><00:09:11.760><c> is</c><00:09:12.160><c> half</c>

09:12.470 --> 09:12.480 align:start position:0%
makes sense. Um, kinetic energy is half
 

09:12.480 --> 09:14.870 align:start position:0%
makes sense. Um, kinetic energy is half
mv<00:09:12.880><c> squared.</c><00:09:13.519><c> So</c><00:09:13.680><c> if</c><00:09:13.920><c> you</c><00:09:14.080><c> multiply</c><00:09:14.480><c> the</c><00:09:14.720><c> top</c>

09:14.870 --> 09:14.880 align:start position:0%
mv squared. So if you multiply the top
 

09:14.880 --> 09:17.750 align:start position:0%
mv squared. So if you multiply the top
and<00:09:15.120><c> the</c><00:09:15.279><c> bottom</c><00:09:15.760><c> by</c><00:09:16.080><c> m,</c><00:09:16.480><c> we</c><00:09:16.720><c> can</c><00:09:16.880><c> write</c><00:09:17.120><c> it</c><00:09:17.279><c> as</c>

09:17.750 --> 09:17.760 align:start position:0%
and the bottom by m, we can write it as
 

09:17.760 --> 09:19.829 align:start position:0%
and the bottom by m, we can write it as
momentum<00:09:18.320><c> squared</c><00:09:18.640><c> over</c><00:09:18.880><c> 2m.</c><00:09:19.360><c> And</c><00:09:19.440><c> I'm</c><00:09:19.600><c> like,</c>

09:19.829 --> 09:19.839 align:start position:0%
momentum squared over 2m. And I'm like,
 

09:19.839 --> 09:21.829 align:start position:0%
momentum squared over 2m. And I'm like,
perfect.<00:09:20.399><c> This</c><00:09:20.640><c> is</c><00:09:20.959><c> all</c><00:09:21.200><c> starting</c><00:09:21.519><c> to</c><00:09:21.680><c> make</c>

09:21.829 --> 09:21.839 align:start position:0%
perfect. This is all starting to make
 

09:21.839 --> 09:25.829 align:start position:0%
perfect. This is all starting to make
sense.<00:09:22.959><c> And</c><00:09:23.120><c> in</c><00:09:23.360><c> the</c><00:09:23.600><c> final</c><00:09:24.000><c> step,</c><00:09:24.560><c> it</c><00:09:24.800><c> says</c><00:09:25.360><c> to</c>

09:25.829 --> 09:25.839 align:start position:0%
sense. And in the final step, it says to
 

09:25.839 --> 09:29.509 align:start position:0%
sense. And in the final step, it says to
make<00:09:26.000><c> it</c><00:09:26.320><c> quantum,</c><00:09:27.360><c> replace</c><00:09:27.920><c> energy</c><00:09:28.800><c> with</c>

09:29.509 --> 09:29.519 align:start position:0%
make it quantum, replace energy with
 

09:29.519 --> 09:33.030 align:start position:0%
make it quantum, replace energy with
energy<00:09:30.000><c> operator.</c><00:09:30.880><c> Wait,</c><00:09:31.600><c> what?</c><00:09:32.640><c> kinetic</c>

09:33.030 --> 09:33.040 align:start position:0%
energy operator. Wait, what? kinetic
 

09:33.040 --> 09:35.430 align:start position:0%
energy operator. Wait, what? kinetic
energy<00:09:33.360><c> with</c><00:09:33.680><c> kinetic</c><00:09:34.160><c> energy</c><00:09:34.480><c> operator.</c>

09:35.430 --> 09:35.440 align:start position:0%
energy with kinetic energy operator.
 

09:35.440 --> 09:38.389 align:start position:0%
energy with kinetic energy operator.
What?<00:09:35.760><c> What's</c><00:09:36.160><c> going</c><00:09:36.320><c> on?</c><00:09:37.360><c> And</c><00:09:37.680><c> they</c><00:09:38.000><c> act</c><00:09:38.240><c> on</c>

09:38.389 --> 09:38.399 align:start position:0%
What? What's going on? And they act on
 

09:38.399 --> 09:40.389 align:start position:0%
What? What's going on? And they act on
the<00:09:38.560><c> wave</c><00:09:38.800><c> function</c><00:09:39.120><c> sigh,</c><00:09:39.680><c> giving</c><00:09:40.000><c> us</c><00:09:40.160><c> the</c>

09:40.389 --> 09:40.399 align:start position:0%
the wave function sigh, giving us the
 

09:40.399 --> 09:43.190 align:start position:0%
the wave function sigh, giving us the
Schroinger's<00:09:41.040><c> equation.</c><00:09:42.240><c> What</c><00:09:42.800><c> just</c>

09:43.190 --> 09:43.200 align:start position:0%
Schroinger's equation. What just
 

09:43.200 --> 09:45.590 align:start position:0%
Schroinger's equation. What just
happened?

09:45.590 --> 09:45.600 align:start position:0%
happened?
 

09:45.600 --> 09:47.509 align:start position:0%
happened?
After<00:09:45.920><c> calming</c><00:09:46.160><c> down</c><00:09:46.399><c> a</c><00:09:46.560><c> bit,</c><00:09:46.800><c> I</c><00:09:47.040><c> realized</c><00:09:47.360><c> I</c>

09:47.509 --> 09:47.519 align:start position:0%
After calming down a bit, I realized I
 

09:47.519 --> 09:49.590 align:start position:0%
After calming down a bit, I realized I
just<00:09:47.680><c> had</c><00:09:47.839><c> to</c><00:09:47.920><c> answer</c><00:09:48.240><c> two</c><00:09:48.480><c> questions.</c><00:09:49.360><c> First</c>

09:49.590 --> 09:49.600 align:start position:0%
just had to answer two questions. First
 

09:49.600 --> 09:53.190 align:start position:0%
just had to answer two questions. First
of<00:09:49.760><c> all,</c><00:09:50.240><c> why</c><00:09:50.560><c> do</c><00:09:50.720><c> we</c><00:09:50.959><c> need</c><00:09:51.519><c> these</c><00:09:52.080><c> operators?</c>

09:53.190 --> 09:53.200 align:start position:0%
of all, why do we need these operators?
 

09:53.200 --> 09:55.509 align:start position:0%
of all, why do we need these operators?
And<00:09:53.519><c> second</c><00:09:53.760><c> of</c><00:09:54.000><c> all,</c><00:09:54.320><c> how</c><00:09:54.560><c> do</c><00:09:54.640><c> I</c><00:09:54.959><c> build</c><00:09:55.279><c> them</c>

09:55.509 --> 09:55.519 align:start position:0%
And second of all, how do I build them
 

09:55.519 --> 09:57.990 align:start position:0%
And second of all, how do I build them
myself<00:09:56.080><c> intuitively?</c><00:09:57.279><c> So,</c><00:09:57.519><c> let's</c><00:09:57.680><c> start</c><00:09:57.839><c> with</c>

09:57.990 --> 09:58.000 align:start position:0%
myself intuitively? So, let's start with
 

09:58.000 --> 10:01.269 align:start position:0%
myself intuitively? So, let's start with
the<00:09:58.240><c> first</c><00:09:58.399><c> question.</c><00:09:59.120><c> We</c><00:09:59.360><c> know</c><00:09:59.680><c> E</c><00:10:00.080><c> equals</c><00:10:00.640><c> HF.</c>

10:01.269 --> 10:01.279 align:start position:0%
the first question. We know E equals HF.
 

10:01.279 --> 10:05.430 align:start position:0%
the first question. We know E equals HF.
And<00:10:01.519><c> we</c><00:10:01.839><c> also</c><00:10:02.080><c> know</c><00:10:02.399><c> P=</c><00:10:03.200><c> H</c><00:10:03.519><c> /</c><00:10:03.680><c> lambda</c><00:10:04.160><c> deoy.</c><00:10:05.200><c> Why</c>

10:05.430 --> 10:05.440 align:start position:0%
And we also know P= H / lambda deoy. Why
 

10:05.440 --> 10:08.150 align:start position:0%
And we also know P= H / lambda deoy. Why
not<00:10:05.600><c> just</c><00:10:06.080><c> substitute</c><00:10:06.720><c> them</c><00:10:07.200><c> directly</c><00:10:07.760><c> into</c>

10:08.150 --> 10:08.160 align:start position:0%
not just substitute them directly into
 

10:08.160 --> 10:10.470 align:start position:0%
not just substitute them directly into
this<00:10:08.399><c> equation,</c><00:10:09.040><c> right?</c><00:10:09.680><c> That</c><00:10:09.920><c> was</c><00:10:10.080><c> what</c><00:10:10.240><c> I</c>

10:10.470 --> 10:10.480 align:start position:0%
this equation, right? That was what I
 

10:10.480 --> 10:12.550 align:start position:0%
this equation, right? That was what I
was<00:10:10.560><c> thinking.</c><00:10:11.360><c> Well,</c><00:10:11.680><c> the</c><00:10:12.000><c> answer</c><00:10:12.320><c> is</c>

10:12.550 --> 10:12.560 align:start position:0%
was thinking. Well, the answer is
 

10:12.560 --> 10:14.870 align:start position:0%
was thinking. Well, the answer is
actually<00:10:12.880><c> right</c><00:10:13.120><c> in</c><00:10:13.279><c> front</c><00:10:13.440><c> of</c><00:10:13.519><c> us.</c><00:10:14.240><c> You</c><00:10:14.399><c> see,</c>

10:14.870 --> 10:14.880 align:start position:0%
actually right in front of us. You see,
 

10:14.880 --> 10:17.509 align:start position:0%
actually right in front of us. You see,
that<00:10:15.120><c> would</c><00:10:15.440><c> only</c><00:10:15.760><c> work</c><00:10:16.160><c> for</c><00:10:16.480><c> an</c><00:10:16.800><c> infinitely</c>

10:17.509 --> 10:17.519 align:start position:0%
that would only work for an infinitely
 

10:17.519 --> 10:21.190 align:start position:0%
that would only work for an infinitely
long<00:10:18.079><c> pure</c><00:10:19.120><c> sine</c><00:10:19.440><c> wave</c><00:10:20.079><c> because</c><00:10:20.480><c> look,</c><00:10:20.880><c> only</c>

10:21.190 --> 10:21.200 align:start position:0%
long pure sine wave because look, only
 

10:21.200 --> 10:24.790 align:start position:0%
long pure sine wave because look, only
then<00:10:22.000><c> it</c><00:10:22.240><c> would</c><00:10:22.480><c> have</c><00:10:23.360><c> one</c><00:10:23.760><c> single</c><00:10:24.240><c> definite</c>

10:24.790 --> 10:24.800 align:start position:0%
then it would have one single definite
 

10:24.800 --> 10:27.590 align:start position:0%
then it would have one single definite
veil<00:10:25.200><c> length</c><00:10:25.519><c> and</c><00:10:26.079><c> frequency.</c>

10:27.590 --> 10:27.600 align:start position:0%
veil length and frequency.
 

10:27.600 --> 10:31.750 align:start position:0%
veil length and frequency.
But<00:10:27.920><c> a</c><00:10:28.320><c> general</c><00:10:29.040><c> wave</c><00:10:29.680><c> doesn't</c><00:10:30.240><c> have</c><00:10:30.720><c> a</c><00:10:31.120><c> single</c>

10:31.750 --> 10:31.760 align:start position:0%
But a general wave doesn't have a single
 

10:31.760 --> 10:34.230 align:start position:0%
But a general wave doesn't have a single
wavelength<00:10:32.399><c> or</c><00:10:32.560><c> frequency.</c><00:10:33.519><c> It</c><00:10:33.680><c> can</c><00:10:33.839><c> have</c><00:10:34.000><c> a</c>

10:34.230 --> 10:34.240 align:start position:0%
wavelength or frequency. It can have a
 

10:34.240 --> 10:38.310 align:start position:0%
wavelength or frequency. It can have a
whole<00:10:34.480><c> range</c><00:10:35.360><c> all</c><00:10:35.680><c> mixed</c><00:10:36.160><c> together.</c>

10:38.310 --> 10:38.320 align:start position:0%
whole range all mixed together.
 

10:38.320 --> 10:40.389 align:start position:0%
whole range all mixed together.
Now<00:10:38.560><c> for</c><00:10:38.800><c> ordinary</c><00:10:39.279><c> waves,</c><00:10:39.680><c> that's</c><00:10:39.920><c> not</c><00:10:40.079><c> a</c><00:10:40.240><c> big</c>

10:40.389 --> 10:40.399 align:start position:0%
Now for ordinary waves, that's not a big
 

10:40.399 --> 10:42.550 align:start position:0%
Now for ordinary waves, that's not a big
deal.<00:10:40.880><c> But</c><00:10:41.040><c> for</c><00:10:41.360><c> matter</c><00:10:41.760><c> waves,</c><00:10:42.160><c> think</c><00:10:42.320><c> about</c>

10:42.550 --> 10:42.560 align:start position:0%
deal. But for matter waves, think about
 

10:42.560 --> 10:44.550 align:start position:0%
deal. But for matter waves, think about
what<00:10:42.720><c> it</c><00:10:42.880><c> means.</c><00:10:43.440><c> This</c><00:10:43.680><c> means</c><00:10:43.920><c> the</c><00:10:44.079><c> particle</c>

10:44.550 --> 10:44.560 align:start position:0%
what it means. This means the particle
 

10:44.560 --> 10:46.949 align:start position:0%
what it means. This means the particle
doesn't<00:10:44.800><c> even</c><00:10:45.120><c> have</c><00:10:45.279><c> a</c><00:10:45.600><c> single</c><00:10:46.240><c> value</c><00:10:46.640><c> of</c>

10:46.949 --> 10:46.959 align:start position:0%
doesn't even have a single value of
 

10:46.959 --> 10:49.990 align:start position:0%
doesn't even have a single value of
momentum<00:10:47.600><c> or</c><00:10:48.160><c> energy.</c><00:10:49.040><c> At</c><00:10:49.279><c> this</c><00:10:49.519><c> point,</c><00:10:49.760><c> we</c>

10:49.990 --> 10:50.000 align:start position:0%
momentum or energy. At this point, we
 

10:50.000 --> 10:51.750 align:start position:0%
momentum or energy. At this point, we
don't<00:10:50.160><c> even</c><00:10:50.399><c> know</c><00:10:50.560><c> how</c><00:10:50.720><c> to</c><00:10:50.880><c> interpret</c><00:10:51.200><c> that.</c><00:10:51.519><c> I</c>

10:51.750 --> 10:51.760 align:start position:0%
don't even know how to interpret that. I
 

10:51.760 --> 10:53.670 align:start position:0%
don't even know how to interpret that. I
mean,<00:10:51.920><c> what</c><00:10:52.160><c> does</c><00:10:52.320><c> it</c><00:10:52.480><c> even</c><00:10:52.720><c> mean</c><00:10:53.040><c> for</c><00:10:53.440><c> an</c>

10:53.670 --> 10:53.680 align:start position:0%
mean, what does it even mean for an
 

10:53.680 --> 10:57.509 align:start position:0%
mean, what does it even mean for an
electron<00:10:54.480><c> to</c><00:10:54.800><c> not</c><00:10:55.040><c> have</c><00:10:55.600><c> a</c><00:10:55.920><c> definite</c><00:10:56.560><c> energy?</c>

10:57.509 --> 10:57.519 align:start position:0%
electron to not have a definite energy?
 

10:57.519 --> 10:59.829 align:start position:0%
electron to not have a definite energy?
Well,<00:10:57.839><c> let's</c><00:10:58.240><c> keep</c><00:10:58.560><c> that</c><00:10:58.959><c> question</c><00:10:59.360><c> aside.</c>

10:59.829 --> 10:59.839 align:start position:0%
Well, let's keep that question aside.
 

10:59.839 --> 11:01.350 align:start position:0%
Well, let's keep that question aside.
We'll<00:11:00.000><c> come</c><00:11:00.079><c> back</c><00:11:00.240><c> to</c><00:11:00.320><c> it</c><00:11:00.560><c> later.</c><00:11:01.040><c> But</c><00:11:01.200><c> for</c>

11:01.350 --> 11:01.360 align:start position:0%
We'll come back to it later. But for
 

11:01.360 --> 11:04.389 align:start position:0%
We'll come back to it later. But for
now,<00:11:01.760><c> what's</c><00:11:02.000><c> important</c><00:11:02.320><c> is</c><00:11:03.040><c> because</c><00:11:04.000><c> mow</c>

11:04.389 --> 11:04.399 align:start position:0%
now, what's important is because mow
 

11:04.399 --> 11:06.470 align:start position:0%
now, what's important is because mow
waves<00:11:04.800><c> don't</c><00:11:05.120><c> have</c><00:11:05.279><c> a</c><00:11:05.600><c> single</c><00:11:05.920><c> value</c><00:11:06.240><c> of</c>

11:06.470 --> 11:06.480 align:start position:0%
waves don't have a single value of
 

11:06.480 --> 11:09.350 align:start position:0%
waves don't have a single value of
energy<00:11:06.800><c> or</c><00:11:07.120><c> momentum,</c><00:11:08.000><c> we</c><00:11:08.240><c> can't</c><00:11:08.560><c> substitute</c>

11:09.350 --> 11:09.360 align:start position:0%
energy or momentum, we can't substitute
 

11:09.360 --> 11:12.230 align:start position:0%
energy or momentum, we can't substitute
directly.<00:11:10.640><c> Instead,</c><00:11:11.200><c> we</c><00:11:11.519><c> need</c><00:11:11.760><c> something</c>

11:12.230 --> 11:12.240 align:start position:0%
directly. Instead, we need something
 

11:12.240 --> 11:17.269 align:start position:0%
directly. Instead, we need something
that<00:11:12.880><c> extracts</c><00:11:13.680><c> the</c><00:11:14.160><c> entire</c><00:11:15.360><c> mixture.</c>

11:17.269 --> 11:17.279 align:start position:0%
that extracts the entire mixture.
 

11:17.279 --> 11:21.430 align:start position:0%
that extracts the entire mixture.
That's<00:11:17.839><c> what</c><00:11:18.480><c> these</c><00:11:19.120><c> operators</c><00:11:19.920><c> do.</c><00:11:21.120><c> The</c>

11:21.430 --> 11:21.440 align:start position:0%
That's what these operators do. The
 

11:21.440 --> 11:24.150 align:start position:0%
That's what these operators do. The
energy<00:11:21.839><c> operator</c><00:11:22.399><c> over</c><00:11:22.720><c> here</c><00:11:23.440><c> pulls</c><00:11:23.839><c> the</c>

11:24.150 --> 11:24.160 align:start position:0%
energy operator over here pulls the
 

11:24.160 --> 11:25.829 align:start position:0%
energy operator over here pulls the
entire<00:11:24.480><c> range</c><00:11:24.720><c> of</c><00:11:24.880><c> energy</c><00:11:25.279><c> hiding</c><00:11:25.519><c> in</c><00:11:25.680><c> the</c>

11:25.829 --> 11:25.839 align:start position:0%
entire range of energy hiding in the
 

11:25.839 --> 11:27.430 align:start position:0%
entire range of energy hiding in the
wave.<00:11:26.320><c> Similarly,</c><00:11:26.640><c> the</c><00:11:26.800><c> kinetic</c><00:11:27.120><c> energy</c>

11:27.430 --> 11:27.440 align:start position:0%
wave. Similarly, the kinetic energy
 

11:27.440 --> 11:29.509 align:start position:0%
wave. Similarly, the kinetic energy
operator<00:11:28.000><c> pulls</c><00:11:28.640><c> the</c><00:11:28.880><c> entire</c><00:11:29.200><c> range</c><00:11:29.360><c> of</c>

11:29.509 --> 11:29.519 align:start position:0%
operator pulls the entire range of
 

11:29.519 --> 11:31.910 align:start position:0%
operator pulls the entire range of
kinetic<00:11:29.920><c> energy.</c><00:11:30.800><c> That's</c><00:11:31.120><c> why</c><00:11:31.279><c> in</c><00:11:31.519><c> quantum</c>

11:31.910 --> 11:31.920 align:start position:0%
kinetic energy. That's why in quantum
 

11:31.920 --> 11:34.150 align:start position:0%
kinetic energy. That's why in quantum
mechanics<00:11:32.720><c> we</c><00:11:33.040><c> always</c><00:11:33.440><c> talk</c><00:11:33.760><c> about</c>

11:34.150 --> 11:34.160 align:start position:0%
mechanics we always talk about
 

11:34.160 --> 11:37.509 align:start position:0%
mechanics we always talk about
operators,<00:11:35.200><c> right?</c><00:11:35.600><c> Because</c><00:11:36.880><c> matter</c><00:11:37.200><c> waves</c>

11:37.509 --> 11:37.519 align:start position:0%
operators, right? Because matter waves
 

11:37.519 --> 11:40.230 align:start position:0%
operators, right? Because matter waves
don't<00:11:37.839><c> have</c><00:11:38.320><c> single</c><00:11:38.880><c> values</c><00:11:39.279><c> for</c><00:11:39.600><c> energy</c><00:11:40.000><c> or</c>

11:40.230 --> 11:40.240 align:start position:0%
don't have single values for energy or
 

11:40.240 --> 11:42.630 align:start position:0%
don't have single values for energy or
momentum<00:11:40.640><c> or</c><00:11:40.880><c> whatever.</c><00:11:41.279><c> They</c><00:11:41.600><c> have</c><00:11:42.000><c> a</c><00:11:42.320><c> whole</c>

11:42.630 --> 11:42.640 align:start position:0%
momentum or whatever. They have a whole
 

11:42.640 --> 11:46.389 align:start position:0%
momentum or whatever. They have a whole
range.<00:11:43.440><c> Okay,</c><00:11:43.680><c> so</c><00:11:44.000><c> first</c><00:11:44.320><c> question</c><00:11:44.640><c> answered.</c>

11:46.389 --> 11:46.399 align:start position:0%
range. Okay, so first question answered.
 

11:46.399 --> 11:48.470 align:start position:0%
range. Okay, so first question answered.
On<00:11:46.720><c> to</c><00:11:47.120><c> the</c><00:11:47.440><c> second</c><00:11:47.680><c> one</c><00:11:47.839><c> and</c><00:11:48.079><c> the</c><00:11:48.240><c> most</c>

11:48.470 --> 11:48.480 align:start position:0%
On to the second one and the most
 

11:48.480 --> 11:50.630 align:start position:0%
On to the second one and the most
important<00:11:48.880><c> one.</c><00:11:49.519><c> How</c><00:11:49.760><c> do</c><00:11:49.839><c> I</c><00:11:50.079><c> build</c><00:11:50.320><c> these</c>

11:50.630 --> 11:50.640 align:start position:0%
important one. How do I build these
 

11:50.640 --> 11:53.430 align:start position:0%
important one. How do I build these
operators<00:11:51.360><c> myself?</c><00:11:52.560><c> Here's</c><00:11:52.880><c> a</c><00:11:53.120><c> powerful</c>

11:53.430 --> 11:53.440 align:start position:0%
operators myself? Here's a powerful
 

11:53.440 --> 11:55.110 align:start position:0%
operators myself? Here's a powerful
problem<00:11:53.760><c> solving</c><00:11:54.079><c> principle.</c><00:11:54.720><c> When</c><00:11:54.959><c> you're</c>

11:55.110 --> 11:55.120 align:start position:0%
problem solving principle. When you're
 

11:55.120 --> 11:57.190 align:start position:0%
problem solving principle. When you're
trying<00:11:55.279><c> to</c><00:11:55.440><c> solve</c><00:11:55.680><c> a</c><00:11:56.079><c> hard</c><00:11:56.399><c> problem,</c><00:11:56.880><c> first</c>

11:57.190 --> 11:57.200 align:start position:0%
trying to solve a hard problem, first
 

11:57.200 --> 11:59.430 align:start position:0%
trying to solve a hard problem, first
see<00:11:57.440><c> if</c><00:11:57.600><c> you</c><00:11:57.839><c> can</c><00:11:58.399><c> create</c><00:11:58.720><c> a</c><00:11:58.880><c> simpler</c><00:11:59.279><c> version</c>

11:59.430 --> 11:59.440 align:start position:0%
see if you can create a simpler version
 

11:59.440 --> 12:01.590 align:start position:0%
see if you can create a simpler version
of<00:11:59.600><c> that</c><00:11:59.839><c> problem</c><00:12:00.079><c> and</c><00:12:00.399><c> try</c><00:12:00.560><c> to</c><00:12:00.720><c> solve</c><00:12:01.040><c> that.</c>

12:01.590 --> 12:01.600 align:start position:0%
of that problem and try to solve that.
 

12:01.600 --> 12:03.829 align:start position:0%
of that problem and try to solve that.
So<00:12:01.760><c> in</c><00:12:01.920><c> our</c><00:12:02.160><c> case,</c><00:12:02.640><c> the</c><00:12:02.880><c> hard</c><00:12:03.200><c> problem</c><00:12:03.440><c> is</c><00:12:03.680><c> to</c>

12:03.829 --> 12:03.839 align:start position:0%
So in our case, the hard problem is to
 

12:03.839 --> 12:07.350 align:start position:0%
So in our case, the hard problem is to
build<00:12:04.160><c> operators</c><00:12:04.959><c> that</c><00:12:05.360><c> work</c><00:12:05.920><c> in</c><00:12:06.320><c> general.</c>

12:07.350 --> 12:07.360 align:start position:0%
build operators that work in general.
 

12:07.360 --> 12:09.430 align:start position:0%
build operators that work in general.
The<00:12:07.600><c> simpler</c><00:12:08.079><c> version</c><00:12:08.560><c> would</c><00:12:08.800><c> be</c><00:12:08.880><c> to</c><00:12:09.120><c> try</c><00:12:09.279><c> and</c>

12:09.430 --> 12:09.440 align:start position:0%
The simpler version would be to try and
 

12:09.440 --> 12:11.190 align:start position:0%
The simpler version would be to try and
build<00:12:09.680><c> these</c><00:12:09.920><c> operators</c><00:12:10.320><c> for</c><00:12:10.560><c> the</c><00:12:10.800><c> simplest</c>

12:11.190 --> 12:11.200 align:start position:0%
build these operators for the simplest
 

12:11.200 --> 12:13.269 align:start position:0%
build these operators for the simplest
wave<00:12:11.600><c> possible,</c>

12:13.269 --> 12:13.279 align:start position:0%
wave possible,
 

12:13.279 --> 12:17.110 align:start position:0%
wave possible,
a<00:12:13.600><c> sine</c><00:12:14.000><c> wave.</c>

12:17.110 --> 12:17.120 align:start position:0%
 
 

12:17.120 --> 12:19.269 align:start position:0%
 
So<00:12:17.440><c> maybe</c><00:12:17.760><c> if</c><00:12:18.000><c> I</c><00:12:18.160><c> can</c><00:12:18.399><c> build</c><00:12:18.560><c> an</c><00:12:18.800><c> operator</c><00:12:19.120><c> for</c>

12:19.269 --> 12:19.279 align:start position:0%
So maybe if I can build an operator for
 

12:19.279 --> 12:21.590 align:start position:0%
So maybe if I can build an operator for
a<00:12:19.440><c> sine</c><00:12:19.680><c> wave,</c><00:12:20.560><c> I</c><00:12:20.800><c> can</c><00:12:20.880><c> then</c><00:12:21.120><c> use</c><00:12:21.360><c> that</c>

12:21.590 --> 12:21.600 align:start position:0%
a sine wave, I can then use that
 

12:21.600 --> 12:23.750 align:start position:0%
a sine wave, I can then use that
intuition<00:12:22.079><c> to</c><00:12:22.399><c> generalize</c><00:12:22.959><c> it.</c><00:12:23.440><c> First</c><00:12:23.600><c> of</c>

12:23.750 --> 12:23.760 align:start position:0%
intuition to generalize it. First of
 

12:23.760 --> 12:27.509 align:start position:0%
intuition to generalize it. First of
all,<00:12:24.079><c> this</c><00:12:24.399><c> could</c><00:12:24.639><c> be</c><00:12:24.880><c> a</c><00:12:25.440><c> traveling</c><00:12:25.920><c> wave</c><00:12:26.959><c> or</c>

12:27.509 --> 12:27.519 align:start position:0%
all, this could be a traveling wave or
 

12:27.519 --> 12:29.910 align:start position:0%
all, this could be a traveling wave or
you<00:12:27.760><c> know</c><00:12:28.000><c> it</c><00:12:28.240><c> could</c><00:12:28.399><c> be</c><00:12:28.560><c> a</c><00:12:29.040><c> standing</c><00:12:29.440><c> wave</c>

12:29.910 --> 12:29.920 align:start position:0%
you know it could be a standing wave
 

12:29.920 --> 12:32.790 align:start position:0%
you know it could be a standing wave
like<00:12:30.320><c> you</c><00:12:30.480><c> know</c><00:12:30.639><c> de</c><00:12:30.880><c> Bruy</c><00:12:31.360><c> described.</c><00:12:32.240><c> I</c><00:12:32.560><c> like</c>

12:32.790 --> 12:32.800 align:start position:0%
like you know de Bruy described. I like
 

12:32.800 --> 12:34.710 align:start position:0%
like you know de Bruy described. I like
standing<00:12:33.200><c> waves.</c><00:12:33.760><c> The</c><00:12:33.920><c> math</c><00:12:34.160><c> feels</c><00:12:34.480><c> slightly</c>

12:34.710 --> 12:34.720 align:start position:0%
standing waves. The math feels slightly
 

12:34.720 --> 12:37.269 align:start position:0%
standing waves. The math feels slightly
more<00:12:34.880><c> intuitive.</c><00:12:35.760><c> So</c><00:12:36.160><c> let's</c><00:12:36.480><c> go</c><00:12:36.639><c> with</c><00:12:36.800><c> that.</c>

12:37.269 --> 12:37.279 align:start position:0%
more intuitive. So let's go with that.
 

12:37.279 --> 12:39.110 align:start position:0%
more intuitive. So let's go with that.
Let's<00:12:37.519><c> draw</c><00:12:37.760><c> a</c><00:12:37.920><c> couple</c><00:12:38.079><c> of</c><00:12:38.240><c> axes.</c><00:12:38.800><c> You</c><00:12:38.959><c> have</c>

12:39.110 --> 12:39.120 align:start position:0%
Let's draw a couple of axes. You have
 

12:39.120 --> 12:41.590 align:start position:0%
Let's draw a couple of axes. You have
x-axis<00:12:39.680><c> and</c><00:12:39.839><c> you</c><00:12:40.000><c> have</c><00:12:40.160><c> s</c><00:12:40.959><c> which</c><00:12:41.120><c> is</c><00:12:41.360><c> you</c><00:12:41.440><c> know</c>

12:41.590 --> 12:41.600 align:start position:0%
x-axis and you have s which is you know
 

12:41.600 --> 12:43.910 align:start position:0%
x-axis and you have s which is you know
shinger's<00:12:42.240><c> own</c><00:12:42.480><c> notation.</c><00:12:43.279><c> So</c><00:12:43.519><c> the</c><00:12:43.680><c> first</c>

12:43.910 --> 12:43.920 align:start position:0%
shinger's own notation. So the first
 

12:43.920 --> 12:48.069 align:start position:0%
shinger's own notation. So the first
question<00:12:44.079><c> is</c><00:12:44.480><c> what</c><00:12:44.720><c> is</c><00:12:44.800><c> the</c><00:12:45.040><c> equation</c><00:12:45.440><c> for</c><00:12:45.920><c> s?</c>

12:48.069 --> 12:48.079 align:start position:0%
question is what is the equation for s?
 

12:48.079 --> 12:50.790 align:start position:0%
question is what is the equation for s?
Well,<00:12:48.320><c> let's</c><00:12:48.720><c> pause</c><00:12:49.040><c> the</c><00:12:49.200><c> animation.</c><00:12:50.240><c> We</c><00:12:50.480><c> can</c>

12:50.790 --> 12:50.800 align:start position:0%
Well, let's pause the animation. We can
 

12:50.800 --> 12:54.389 align:start position:0%
Well, let's pause the animation. We can
write<00:12:51.279><c> s=</c><00:12:52.399><c> sinx.</c><00:12:53.360><c> I</c><00:12:53.519><c> mean</c><00:12:53.680><c> we</c><00:12:53.920><c> could</c><00:12:54.079><c> also</c>

12:54.389 --> 12:54.399 align:start position:0%
write s= sinx. I mean we could also
 

12:54.399 --> 12:56.870 align:start position:0%
write s= sinx. I mean we could also
write<00:12:54.880><c> s=</c><00:12:55.519><c> cos</c><00:12:55.839><c> x.</c><00:12:56.079><c> It's</c><00:12:56.240><c> just</c><00:12:56.399><c> a</c><00:12:56.560><c> matter</c><00:12:56.720><c> of</c>

12:56.870 --> 12:56.880 align:start position:0%
write s= cos x. It's just a matter of
 

12:56.880 --> 12:59.350 align:start position:0%
write s= cos x. It's just a matter of
where<00:12:57.040><c> we</c><00:12:57.200><c> put</c><00:12:57.279><c> the</c><00:12:57.519><c> origin.</c><00:12:58.399><c> But</c><00:12:58.639><c> let's</c><00:12:59.040><c> stick</c>

12:59.350 --> 12:59.360 align:start position:0%
where we put the origin. But let's stick
 

12:59.360 --> 13:01.750 align:start position:0%
where we put the origin. But let's stick
to<00:12:59.680><c> sinx.</c><00:13:00.480><c> That's</c><00:13:00.720><c> the</c><00:13:00.880><c> simplest</c><00:13:01.279><c> equation,</c>

13:01.750 --> 13:01.760 align:start position:0%
to sinx. That's the simplest equation,
 

13:01.760 --> 13:04.629 align:start position:0%
to sinx. That's the simplest equation,
right?<00:13:02.240><c> But</c><00:13:02.480><c> guess</c><00:13:02.639><c> what?</c><00:13:03.120><c> S's</c><00:13:03.600><c> height</c><00:13:04.320><c> swings</c>

13:04.629 --> 13:04.639 align:start position:0%
right? But guess what? S's height swings
 

13:04.639 --> 13:07.829 align:start position:0%
right? But guess what? S's height swings
between<00:13:04.959><c> +1</c><00:13:05.440><c> and</c><00:13:05.680><c> minus1.</c><00:13:06.720><c> I</c><00:13:07.040><c> want</c><00:13:07.279><c> our</c><00:13:07.600><c> height</c>

13:07.829 --> 13:07.839 align:start position:0%
between +1 and minus1. I want our height
 

13:07.839 --> 13:09.750 align:start position:0%
between +1 and minus1. I want our height
to<00:13:08.000><c> be</c><00:13:08.160><c> slightly</c><00:13:08.560><c> more</c><00:13:08.720><c> general.</c><00:13:09.360><c> Let's</c><00:13:09.600><c> call</c>

13:09.750 --> 13:09.760 align:start position:0%
to be slightly more general. Let's call
 

13:09.760 --> 13:12.710 align:start position:0%
to be slightly more general. Let's call
it<00:13:10.240><c> a.</c><00:13:10.800><c> So</c><00:13:11.040><c> how</c><00:13:11.200><c> do</c><00:13:11.279><c> we</c><00:13:11.440><c> do</c><00:13:11.600><c> that?</c><00:13:12.160><c> Well,</c><00:13:12.480><c> we</c>

13:12.710 --> 13:12.720 align:start position:0%
it a. So how do we do that? Well, we
 

13:12.720 --> 13:16.150 align:start position:0%
it a. So how do we do that? Well, we
scale<00:13:13.120><c> this</c><00:13:13.600><c> by</c><00:13:14.000><c> a.</c><00:13:14.560><c> That's</c><00:13:14.880><c> the</c><00:13:15.200><c> amplitude.</c>

13:16.150 --> 13:16.160 align:start position:0%
scale this by a. That's the amplitude.
 

13:16.160 --> 13:18.790 align:start position:0%
scale this by a. That's the amplitude.
Now<00:13:16.480><c> I</c><00:13:16.720><c> can</c><00:13:16.880><c> control</c><00:13:17.200><c> the</c><00:13:17.440><c> amplitude.</c>

13:18.790 --> 13:18.800 align:start position:0%
Now I can control the amplitude.
 

13:18.800 --> 13:22.389 align:start position:0%
Now I can control the amplitude.
Perfect.<00:13:19.600><c> But</c><00:13:20.079><c> sign</c><00:13:20.639><c> always</c><00:13:21.120><c> resets</c><00:13:21.680><c> after</c><00:13:22.079><c> 2</c>

13:22.389 --> 13:22.399 align:start position:0%
Perfect. But sign always resets after 2
 

13:22.399 --> 13:25.190 align:start position:0%
Perfect. But sign always resets after 2
pi.<00:13:23.440><c> Which</c><00:13:23.600><c> means</c><00:13:24.000><c> right</c><00:13:24.240><c> now</c><00:13:24.399><c> our</c><00:13:24.720><c> wavelength</c>

13:25.190 --> 13:25.200 align:start position:0%
pi. Which means right now our wavelength
 

13:25.200 --> 13:28.389 align:start position:0%
pi. Which means right now our wavelength
is<00:13:25.440><c> locked</c><00:13:26.079><c> exactly</c><00:13:26.720><c> at</c><00:13:27.040><c> 2</c><00:13:27.279><c> pi.</c><00:13:27.920><c> I</c><00:13:28.160><c> don't</c><00:13:28.240><c> want</c>

13:28.389 --> 13:28.399 align:start position:0%
is locked exactly at 2 pi. I don't want
 

13:28.399 --> 13:31.269 align:start position:0%
is locked exactly at 2 pi. I don't want
that.<00:13:29.200><c> I</c><00:13:29.519><c> want</c><00:13:30.000><c> to</c><00:13:30.240><c> be</c><00:13:30.399><c> able</c><00:13:30.560><c> to</c><00:13:30.800><c> have</c><00:13:30.959><c> any</c>

13:31.269 --> 13:31.279 align:start position:0%
that. I want to be able to have any
 

13:31.279 --> 13:33.829 align:start position:0%
that. I want to be able to have any
wavelength.<00:13:31.760><c> I</c><00:13:32.000><c> want</c><00:13:32.399><c> lambda.</c><00:13:33.200><c> So</c><00:13:33.440><c> what</c><00:13:33.600><c> do</c><00:13:33.680><c> we</c>

13:33.829 --> 13:33.839 align:start position:0%
wavelength. I want lambda. So what do we
 

13:33.839 --> 13:36.710 align:start position:0%
wavelength. I want lambda. So what do we
do?<00:13:34.399><c> Well,</c><00:13:34.639><c> we</c><00:13:34.959><c> multiply</c><00:13:35.440><c> x</c><00:13:35.680><c> by</c><00:13:36.000><c> 2</c><00:13:36.240><c> pi</c><00:13:36.480><c> over</c>

13:36.710 --> 13:36.720 align:start position:0%
do? Well, we multiply x by 2 pi over
 

13:36.720 --> 13:38.949 align:start position:0%
do? Well, we multiply x by 2 pi over
lambda.<00:13:37.200><c> I</c><00:13:37.360><c> mean,</c><00:13:37.600><c> think</c><00:13:37.760><c> about</c><00:13:37.839><c> it.</c><00:13:38.399><c> Now</c><00:13:38.720><c> when</c>

13:38.949 --> 13:38.959 align:start position:0%
lambda. I mean, think about it. Now when
 

13:38.959 --> 13:42.069 align:start position:0%
lambda. I mean, think about it. Now when
x<00:13:39.279><c> equals</c><00:13:39.600><c> lambda</c><00:13:40.560><c> the</c><00:13:41.120><c> lambda</c><00:13:41.519><c> cancels</c><00:13:41.920><c> out</c>

13:42.069 --> 13:42.079 align:start position:0%
x equals lambda the lambda cancels out
 

13:42.079 --> 13:44.790 align:start position:0%
x equals lambda the lambda cancels out
and<00:13:42.240><c> the</c><00:13:42.399><c> argument</c><00:13:42.880><c> becomes</c><00:13:43.200><c> 2</c><00:13:43.440><c> pi</c><00:13:44.160><c> and</c><00:13:44.560><c> the</c>

13:44.790 --> 13:44.800 align:start position:0%
and the argument becomes 2 pi and the
 

13:44.800 --> 13:48.949 align:start position:0%
and the argument becomes 2 pi and the
wave<00:13:45.519><c> resets.</c><00:13:46.720><c> So</c><00:13:47.040><c> now</c><00:13:47.920><c> lambda</c><00:13:48.480><c> has</c><00:13:48.639><c> become</c>

13:48.949 --> 13:48.959 align:start position:0%
wave resets. So now lambda has become
 

13:48.959 --> 13:51.990 align:start position:0%
wave resets. So now lambda has become
our<00:13:49.120><c> wavelength.</c><00:13:50.560><c> But</c><00:13:50.959><c> this</c><00:13:51.279><c> isn't</c><00:13:51.519><c> a</c><00:13:51.760><c> wave</c>

13:51.990 --> 13:52.000 align:start position:0%
our wavelength. But this isn't a wave
 

13:52.000 --> 13:56.230 align:start position:0%
our wavelength. But this isn't a wave
yet.<00:13:52.399><c> It's</c><00:13:52.720><c> a</c><00:13:53.040><c> frozen</c><00:13:53.839><c> picture.</c>

13:56.230 --> 13:56.240 align:start position:0%
yet. It's a frozen picture.
 

13:56.240 --> 13:58.550 align:start position:0%
yet. It's a frozen picture.
A<00:13:56.480><c> standing</c><00:13:56.880><c> wave</c><00:13:57.360><c> means</c><00:13:57.680><c> the</c><00:13:58.000><c> amplitude</c>

13:58.550 --> 13:58.560 align:start position:0%
A standing wave means the amplitude
 

13:58.560 --> 14:02.629 align:start position:0%
A standing wave means the amplitude
itself<00:13:59.519><c> changes</c><00:14:00.160><c> over</c><00:14:00.480><c> time</c><00:14:01.519><c> periodically.</c>

14:02.629 --> 14:02.639 align:start position:0%
itself changes over time periodically.
 

14:02.639 --> 14:04.790 align:start position:0%
itself changes over time periodically.
So<00:14:02.959><c> a</c><00:14:03.199><c> itself</c><00:14:03.600><c> needs</c><00:14:03.839><c> to</c><00:14:04.000><c> be</c><00:14:04.079><c> some</c><00:14:04.320><c> periodic</c>

14:04.790 --> 14:04.800 align:start position:0%
So a itself needs to be some periodic
 

14:04.800 --> 14:06.710 align:start position:0%
So a itself needs to be some periodic
function<00:14:05.040><c> of</c><00:14:05.199><c> time.</c><00:14:05.680><c> Again</c><00:14:06.160><c> we'll</c><00:14:06.399><c> choose</c><00:14:06.560><c> the</c>

14:06.710 --> 14:06.720 align:start position:0%
function of time. Again we'll choose the
 

14:06.720 --> 14:10.230 align:start position:0%
function of time. Again we'll choose the
simplest<00:14:07.120><c> function</c><00:14:07.839><c> s</c><00:14:08.959><c> but</c><00:14:09.199><c> just</c><00:14:09.440><c> like</c><00:14:09.600><c> before</c>

14:10.230 --> 14:10.240 align:start position:0%
simplest function s but just like before
 

14:10.240 --> 14:13.670 align:start position:0%
simplest function s but just like before
s<00:14:10.880><c> has</c><00:14:11.199><c> a</c><00:14:11.440><c> period</c><00:14:11.680><c> of</c><00:14:11.839><c> 2</c><00:14:12.079><c> pi.</c><00:14:12.399><c> So</c><00:14:12.560><c> right</c><00:14:12.800><c> now</c><00:14:12.959><c> our</c>

14:13.670 --> 14:13.680 align:start position:0%
s has a period of 2 pi. So right now our
 

14:13.680 --> 14:16.069 align:start position:0%
s has a period of 2 pi. So right now our
you<00:14:13.839><c> know</c><00:14:14.000><c> time</c><00:14:14.320><c> period</c><00:14:14.560><c> is</c><00:14:14.800><c> locked</c><00:14:15.040><c> at</c><00:14:15.199><c> 2</c><00:14:15.440><c> pi.</c>

14:16.069 --> 14:16.079 align:start position:0%
you know time period is locked at 2 pi.
 

14:16.079 --> 14:17.750 align:start position:0%
you know time period is locked at 2 pi.
So<00:14:16.240><c> we</c><00:14:16.399><c> want</c><00:14:16.560><c> the</c><00:14:16.720><c> period</c><00:14:16.959><c> to</c><00:14:17.120><c> be</c><00:14:17.360><c> let's</c><00:14:17.600><c> say</c>

14:17.750 --> 14:17.760 align:start position:0%
So we want the period to be let's say
 

14:17.760 --> 14:19.590 align:start position:0%
So we want the period to be let's say
capital<00:14:18.160><c> t.</c><00:14:18.399><c> So</c><00:14:18.560><c> we</c><00:14:18.720><c> use</c><00:14:18.880><c> the</c><00:14:19.040><c> same</c><00:14:19.199><c> trick</c><00:14:19.360><c> as</c>

14:19.590 --> 14:19.600 align:start position:0%
capital t. So we use the same trick as
 

14:19.600 --> 14:21.509 align:start position:0%
capital t. So we use the same trick as
before.<00:14:19.920><c> We</c><00:14:20.079><c> multiply</c><00:14:20.480><c> this</c><00:14:20.639><c> by</c><00:14:20.800><c> 2</c><00:14:21.040><c> pi</c><00:14:21.279><c> over</c>

14:21.509 --> 14:21.519 align:start position:0%
before. We multiply this by 2 pi over
 

14:21.519 --> 14:24.710 align:start position:0%
before. We multiply this by 2 pi over
capital<00:14:21.920><c> t.</c><00:14:22.560><c> And</c><00:14:22.880><c> we</c><00:14:23.199><c> are</c><00:14:23.360><c> done.</c><00:14:24.160><c> We</c><00:14:24.399><c> just</c><00:14:24.560><c> need</c>

14:24.710 --> 14:24.720 align:start position:0%
capital t. And we are done. We just need
 

14:24.720 --> 14:28.310 align:start position:0%
capital t. And we are done. We just need
a<00:14:25.120><c> bit</c><00:14:25.279><c> of</c><00:14:25.440><c> cleaning</c><00:14:25.760><c> up</c><00:14:25.920><c> over</c><00:14:26.160><c> here.</c><00:14:26.880><c> 1</c><00:14:27.120><c> /</c><00:14:27.440><c> t</c><00:14:27.760><c> is</c>

14:28.310 --> 14:28.320 align:start position:0%
a bit of cleaning up over here. 1 / t is
 

14:28.320 --> 14:32.069 align:start position:0%
a bit of cleaning up over here. 1 / t is
frequency.<00:14:29.519><c> And</c><00:14:29.760><c> physicists</c><00:14:30.720><c> hate</c><00:14:31.120><c> writing</c><00:14:31.839><c> 2</c>

14:32.069 --> 14:32.079 align:start position:0%
frequency. And physicists hate writing 2
 

14:32.079 --> 14:34.710 align:start position:0%
frequency. And physicists hate writing 2
pi<00:14:32.560><c> over</c><00:14:32.800><c> and</c><00:14:33.040><c> over</c><00:14:33.360><c> again.</c><00:14:34.000><c> So</c><00:14:34.320><c> we</c><00:14:34.560><c> will</c>

14:34.710 --> 14:34.720 align:start position:0%
pi over and over again. So we will
 

14:34.720 --> 14:38.710 align:start position:0%
pi over and over again. So we will
define<00:14:35.519><c> 2</c><00:14:35.760><c> pi</c><00:14:36.160><c> f</c><00:14:36.560><c> as</c><00:14:37.040><c> a</c><00:14:37.199><c> new</c><00:14:37.440><c> variable</c><00:14:38.160><c> omega</c>

14:38.710 --> 14:38.720 align:start position:0%
define 2 pi f as a new variable omega
 

14:38.720 --> 14:41.110 align:start position:0%
define 2 pi f as a new variable omega
and<00:14:38.880><c> similarly</c><00:14:39.279><c> 2</c><00:14:39.440><c> pi</c><00:14:39.680><c> over</c><00:14:39.839><c> lambda</c><00:14:40.480><c> as</c><00:14:40.720><c> a</c><00:14:40.959><c> new</c>

14:41.110 --> 14:41.120 align:start position:0%
and similarly 2 pi over lambda as a new
 

14:41.120 --> 14:43.509 align:start position:0%
and similarly 2 pi over lambda as a new
variable<00:14:41.760><c> kappa.</c><00:14:42.320><c> We're</c><00:14:42.560><c> only</c><00:14:42.800><c> doing</c><00:14:43.040><c> this</c><00:14:43.360><c> so</c>

14:43.509 --> 14:43.519 align:start position:0%
variable kappa. We're only doing this so
 

14:43.519 --> 14:44.870 align:start position:0%
variable kappa. We're only doing this so
that<00:14:43.760><c> we</c><00:14:43.920><c> don't</c><00:14:44.000><c> have</c><00:14:44.160><c> to</c><00:14:44.240><c> write</c><00:14:44.399><c> two</c><00:14:44.560><c> pies</c>

14:44.870 --> 14:44.880 align:start position:0%
that we don't have to write two pies
 

14:44.880 --> 14:47.030 align:start position:0%
that we don't have to write two pies
over<00:14:44.959><c> and</c><00:14:45.120><c> over</c><00:14:45.360><c> again.</c><00:14:46.160><c> Okay.</c><00:14:46.480><c> If</c><00:14:46.720><c> we</c>

14:47.030 --> 14:47.040 align:start position:0%
over and over again. Okay. If we
 

14:47.040 --> 14:51.110 align:start position:0%
over and over again. Okay. If we
substitute<00:14:47.519><c> it,</c><00:14:48.399><c> boom,</c><00:14:49.680><c> we</c><00:14:49.920><c> have</c><00:14:50.160><c> built</c><00:14:50.800><c> the</c>

14:51.110 --> 14:51.120 align:start position:0%
substitute it, boom, we have built the
 

14:51.120 --> 14:55.110 align:start position:0%
substitute it, boom, we have built the
equation<00:14:51.920><c> for</c><00:14:52.320><c> our</c><00:14:52.720><c> standing</c><00:14:53.600><c> wave.</c><00:14:54.480><c> But</c><00:14:54.720><c> this</c>

14:55.110 --> 14:55.120 align:start position:0%
equation for our standing wave. But this
 

14:55.120 --> 14:57.430 align:start position:0%
equation for our standing wave. But this
is<00:14:55.360><c> still</c><00:14:55.600><c> a</c><00:14:55.920><c> generic</c><00:14:56.320><c> wave.</c><00:14:56.720><c> There's</c><00:14:56.959><c> nothing</c>

14:57.430 --> 14:57.440 align:start position:0%
is still a generic wave. There's nothing
 

14:57.440 --> 15:01.110 align:start position:0%
is still a generic wave. There's nothing
quantum<00:14:58.240><c> about</c><00:14:58.560><c> it.</c>

15:01.110 --> 15:01.120 align:start position:0%
quantum about it.
 

15:01.120 --> 15:03.990 align:start position:0%
quantum about it.
To<00:15:01.440><c> make</c><00:15:01.600><c> it</c><00:15:01.760><c> quantum,</c><00:15:02.639><c> we</c><00:15:02.959><c> bring</c><00:15:03.120><c> in</c><00:15:03.440><c> Einstein</c>

15:03.990 --> 15:04.000 align:start position:0%
To make it quantum, we bring in Einstein
 

15:04.000 --> 15:06.870 align:start position:0%
To make it quantum, we bring in Einstein
and<00:15:04.240><c> De</c><00:15:04.399><c> Bruy</c><00:15:05.120><c> equation.</c><00:15:06.079><c> Now,</c><00:15:06.320><c> since</c><00:15:06.560><c> this</c><00:15:06.800><c> is</c>

15:06.870 --> 15:06.880 align:start position:0%
and De Bruy equation. Now, since this is
 

15:06.880 --> 15:09.269 align:start position:0%
and De Bruy equation. Now, since this is
a<00:15:07.199><c> pure</c><00:15:07.600><c> sine</c><00:15:07.920><c> wave</c><00:15:08.240><c> of</c><00:15:08.560><c> one</c><00:15:08.880><c> specific</c>

15:09.269 --> 15:09.279 align:start position:0%
a pure sine wave of one specific
 

15:09.279 --> 15:11.509 align:start position:0%
a pure sine wave of one specific
frequency<00:15:09.680><c> and</c><00:15:09.920><c> wavelength,</c><00:15:10.639><c> we</c><00:15:11.040><c> can</c>

15:11.509 --> 15:11.519 align:start position:0%
frequency and wavelength, we can
 

15:11.519 --> 15:14.870 align:start position:0%
frequency and wavelength, we can
directly<00:15:12.639><c> substitute</c><00:15:13.440><c> over</c><00:15:13.839><c> here.</c><00:15:14.639><c> But</c>

15:14.870 --> 15:14.880 align:start position:0%
directly substitute over here. But
 

15:14.880 --> 15:17.189 align:start position:0%
directly substitute over here. But
before<00:15:15.120><c> we</c><00:15:15.360><c> do</c><00:15:15.440><c> that,</c><00:15:15.760><c> we</c><00:15:16.079><c> have</c><00:15:16.240><c> to</c><00:15:16.399><c> write</c><00:15:16.880><c> this</c>

15:17.189 --> 15:17.199 align:start position:0%
before we do that, we have to write this
 

15:17.199 --> 15:19.590 align:start position:0%
before we do that, we have to write this
in<00:15:17.440><c> terms</c><00:15:17.680><c> of</c><00:15:17.920><c> omega</c><00:15:18.240><c> and</c><00:15:18.320><c> kappa</c><00:15:18.720><c> as</c><00:15:18.880><c> well.</c><00:15:19.360><c> So,</c>

15:19.590 --> 15:19.600 align:start position:0%
in terms of omega and kappa as well. So,
 

15:19.600 --> 15:21.509 align:start position:0%
in terms of omega and kappa as well. So,
let's<00:15:19.839><c> quickly</c><00:15:20.160><c> do</c><00:15:20.320><c> that.</c><00:15:20.720><c> To</c><00:15:20.959><c> do</c><00:15:21.040><c> that,</c><00:15:21.279><c> we'll</c>

15:21.509 --> 15:21.519 align:start position:0%
let's quickly do that. To do that, we'll
 

15:21.519 --> 15:23.269 align:start position:0%
let's quickly do that. To do that, we'll
just<00:15:21.680><c> multiply</c><00:15:22.000><c> and</c><00:15:22.240><c> divide</c><00:15:22.480><c> by</c><00:15:22.639><c> 2</c><00:15:22.880><c> pi</c>

15:23.269 --> 15:23.279 align:start position:0%
just multiply and divide by 2 pi
 

15:23.279 --> 15:26.150 align:start position:0%
just multiply and divide by 2 pi
everywhere.<00:15:24.320><c> And</c><00:15:24.560><c> now</c><00:15:24.800><c> look</c><00:15:24.959><c> at</c><00:15:25.120><c> what</c><00:15:25.279><c> we</c><00:15:25.440><c> get.</c>

15:26.150 --> 15:26.160 align:start position:0%
everywhere. And now look at what we get.
 

15:26.160 --> 15:29.670 align:start position:0%
everywhere. And now look at what we get.
H<00:15:26.560><c> over</c><00:15:26.880><c> 2</c><00:15:27.199><c> pi,</c><00:15:27.600><c> we'll</c><00:15:27.920><c> call</c><00:15:28.160><c> that</c><00:15:28.639><c> as</c><00:15:28.880><c> h</c><00:15:29.279><c> bar.</c>

15:29.670 --> 15:29.680 align:start position:0%
H over 2 pi, we'll call that as h bar.
 

15:29.680 --> 15:32.389 align:start position:0%
H over 2 pi, we'll call that as h bar.
We<00:15:29.920><c> call</c><00:15:30.079><c> this</c><00:15:30.240><c> the</c><00:15:30.480><c> reduced</c><00:15:31.199><c> plank</c><00:15:31.600><c> constant.</c>

15:32.389 --> 15:32.399 align:start position:0%
We call this the reduced plank constant.
 

15:32.399 --> 15:35.590 align:start position:0%
We call this the reduced plank constant.
And<00:15:32.639><c> 2</c><00:15:32.800><c> pi</c><00:15:33.120><c> f</c><00:15:33.360><c> is</c><00:15:33.760><c> omega.</c><00:15:34.560><c> And</c><00:15:34.880><c> 2</c><00:15:35.120><c> pi</c><00:15:35.360><c> over</c>

15:35.590 --> 15:35.600 align:start position:0%
And 2 pi f is omega. And 2 pi over
 

15:35.600 --> 15:38.949 align:start position:0%
And 2 pi f is omega. And 2 pi over
lambda.<00:15:36.160><c> Well,</c><00:15:36.399><c> we</c><00:15:36.639><c> have</c><00:15:37.120><c> kappa.</c><00:15:38.480><c> And</c><00:15:38.639><c> this</c>

15:38.949 --> 15:38.959 align:start position:0%
lambda. Well, we have kappa. And this
 

15:38.959 --> 15:40.870 align:start position:0%
lambda. Well, we have kappa. And this
cleaning<00:15:39.279><c> up</c><00:15:39.600><c> actually</c><00:15:40.000><c> makes</c><00:15:40.320><c> things</c><00:15:40.560><c> much</c>

15:40.870 --> 15:40.880 align:start position:0%
cleaning up actually makes things much
 

15:40.880 --> 15:42.949 align:start position:0%
cleaning up actually makes things much
more<00:15:41.279><c> beautiful.</c><00:15:42.000><c> I</c><00:15:42.160><c> mean,</c><00:15:42.399><c> think</c><00:15:42.480><c> about</c><00:15:42.639><c> it.</c>

15:42.949 --> 15:42.959 align:start position:0%
more beautiful. I mean, think about it.
 

15:42.959 --> 15:45.750 align:start position:0%
more beautiful. I mean, think about it.
What<00:15:43.199><c> exactly</c><00:15:43.600><c> is</c><00:15:44.079><c> omega</c><00:15:44.480><c> over</c><00:15:44.720><c> here?</c><00:15:45.199><c> Omega</c>

15:45.750 --> 15:45.760 align:start position:0%
What exactly is omega over here? Omega
 

15:45.760 --> 15:48.150 align:start position:0%
What exactly is omega over here? Omega
basically<00:15:46.079><c> tells</c><00:15:46.320><c> us</c><00:15:46.560><c> number</c><00:15:46.880><c> of</c><00:15:47.199><c> waves</c><00:15:47.839><c> per</c>

15:48.150 --> 15:48.160 align:start position:0%
basically tells us number of waves per
 

15:48.160 --> 15:50.710 align:start position:0%
basically tells us number of waves per
second,<00:15:48.880><c> right?</c><00:15:49.519><c> So</c><00:15:49.759><c> we</c><00:15:50.000><c> can</c><00:15:50.160><c> call</c><00:15:50.320><c> it</c><00:15:50.480><c> the</c>

15:50.710 --> 15:50.720 align:start position:0%
second, right? So we can call it the
 

15:50.720 --> 15:53.990 align:start position:0%
second, right? So we can call it the
temporal<00:15:51.360><c> frequency.</c><00:15:52.560><c> What</c><00:15:52.800><c> about</c><00:15:53.040><c> kappa?</c>

15:53.990 --> 15:54.000 align:start position:0%
temporal frequency. What about kappa?
 

15:54.000 --> 15:56.949 align:start position:0%
temporal frequency. What about kappa?
Well,<00:15:54.240><c> kappa</c><00:15:54.720><c> tells</c><00:15:54.959><c> us</c><00:15:55.279><c> the</c><00:15:55.600><c> number</c><00:15:55.839><c> of</c><00:15:56.000><c> waves</c>

15:56.949 --> 15:56.959 align:start position:0%
Well, kappa tells us the number of waves
 

15:56.959 --> 16:00.470 align:start position:0%
Well, kappa tells us the number of waves
per<00:15:57.600><c> meter.</c><00:15:58.079><c> Look</c><00:15:58.240><c> at</c><00:15:58.320><c> this.</c><00:15:58.880><c> per</c><00:15:59.199><c> meter.</c><00:16:00.240><c> So</c>

16:00.470 --> 16:00.480 align:start position:0%
per meter. Look at this. per meter. So
 

16:00.480 --> 16:05.110 align:start position:0%
per meter. Look at this. per meter. So
that<00:16:01.120><c> is</c><00:16:01.360><c> the</c><00:16:01.759><c> spatial</c><00:16:02.320><c> frequency</c>

16:05.110 --> 16:05.120 align:start position:0%
that is the spatial frequency
 

16:05.120 --> 16:07.269 align:start position:0%
that is the spatial frequency
which<00:16:05.440><c> means</c><00:16:05.839><c> for</c><00:16:06.079><c> matter</c><00:16:06.399><c> waves</c><00:16:07.040><c> the</c>

16:07.269 --> 16:07.279 align:start position:0%
which means for matter waves the
 

16:07.279 --> 16:10.629 align:start position:0%
which means for matter waves the
temporal<00:16:07.920><c> frequency</c><00:16:09.279><c> encodes</c><00:16:10.000><c> the</c><00:16:10.240><c> total</c>

16:10.629 --> 16:10.639 align:start position:0%
temporal frequency encodes the total
 

16:10.639 --> 16:13.910 align:start position:0%
temporal frequency encodes the total
energy.<00:16:11.440><c> It</c><00:16:11.600><c> lives</c><00:16:11.839><c> in</c><00:16:12.079><c> the</c><00:16:12.240><c> time</c><00:16:12.560><c> domain</c><00:16:13.600><c> and</c>

16:13.910 --> 16:13.920 align:start position:0%
energy. It lives in the time domain and
 

16:13.920 --> 16:16.550 align:start position:0%
energy. It lives in the time domain and
the<00:16:14.320><c> spatial</c><00:16:14.880><c> frequency</c><00:16:15.600><c> encodes</c><00:16:16.320><c> the</c>

16:16.550 --> 16:16.560 align:start position:0%
the spatial frequency encodes the
 

16:16.560 --> 16:19.749 align:start position:0%
the spatial frequency encodes the
momentum.<00:16:17.279><c> It</c><00:16:17.519><c> lives</c><00:16:17.759><c> in</c><00:16:17.920><c> the</c><00:16:18.240><c> space</c><00:16:18.800><c> domain.</c>

16:19.749 --> 16:19.759 align:start position:0%
momentum. It lives in the space domain.
 

16:19.759 --> 16:22.150 align:start position:0%
momentum. It lives in the space domain.
When<00:16:20.000><c> I</c><00:16:20.160><c> saw</c><00:16:20.320><c> this,</c><00:16:20.639><c> a</c><00:16:20.959><c> light</c><00:16:21.120><c> bulb</c><00:16:21.519><c> went</c><00:16:21.759><c> off.</c>

16:22.150 --> 16:22.160 align:start position:0%
When I saw this, a light bulb went off.
 

16:22.160 --> 16:23.430 align:start position:0%
When I saw this, a light bulb went off.
I<00:16:22.320><c> mean,</c><00:16:22.480><c> think</c><00:16:22.639><c> about</c><00:16:22.800><c> it.</c><00:16:23.040><c> You</c><00:16:23.279><c> probably</c>

16:23.430 --> 16:23.440 align:start position:0%
I mean, think about it. You probably
 

16:23.440 --> 16:25.350 align:start position:0%
I mean, think about it. You probably
know<00:16:23.600><c> that</c><00:16:23.759><c> in</c><00:16:23.920><c> special</c><00:16:24.240><c> relativity,</c><00:16:24.959><c> space</c>

16:25.350 --> 16:25.360 align:start position:0%
know that in special relativity, space
 

16:25.360 --> 16:27.430 align:start position:0%
know that in special relativity, space
and<00:16:25.680><c> time</c><00:16:26.079><c> are</c><00:16:26.320><c> not</c><00:16:26.560><c> two</c><00:16:26.720><c> separate</c><00:16:27.040><c> things.</c>

16:27.430 --> 16:27.440 align:start position:0%
and time are not two separate things.
 

16:27.440 --> 16:30.150 align:start position:0%
and time are not two separate things.
They're<00:16:27.680><c> just</c><00:16:28.079><c> shadows</c><00:16:28.560><c> of</c><00:16:28.800><c> the</c><00:16:29.120><c> underlying</c>

16:30.150 --> 16:30.160 align:start position:0%
They're just shadows of the underlying
 

16:30.160 --> 16:33.430 align:start position:0%
They're just shadows of the underlying
four-dimensional<00:16:31.279><c> spaceime,</c><00:16:32.240><c> right?</c><00:16:32.959><c> So,</c><00:16:33.199><c> we</c>

16:33.430 --> 16:33.440 align:start position:0%
four-dimensional spaceime, right? So, we
 

16:33.440 --> 16:35.749 align:start position:0%
four-dimensional spaceime, right? So, we
could<00:16:33.600><c> guess</c><00:16:34.079><c> that</c><00:16:34.399><c> energy</c><00:16:34.800><c> and</c><00:16:35.120><c> momentum</c>

16:35.749 --> 16:35.759 align:start position:0%
could guess that energy and momentum
 

16:35.759 --> 16:38.550 align:start position:0%
could guess that energy and momentum
aren't<00:16:36.320><c> separate</c><00:16:36.800><c> things.</c><00:16:37.759><c> Maybe</c><00:16:38.079><c> they</c><00:16:38.320><c> are</c>

16:38.550 --> 16:38.560 align:start position:0%
aren't separate things. Maybe they are
 

16:38.560 --> 16:41.110 align:start position:0%
aren't separate things. Maybe they are
just<00:16:39.199><c> two</c><00:16:39.600><c> components,</c>

16:41.110 --> 16:41.120 align:start position:0%
just two components,
 

16:41.120 --> 16:44.069 align:start position:0%
just two components,
shadows<00:16:41.759><c> of</c><00:16:42.160><c> something</c><00:16:42.800><c> much</c><00:16:43.279><c> deeper,</c><00:16:43.839><c> a</c>

16:44.069 --> 16:44.079 align:start position:0%
shadows of something much deeper, a
 

16:44.079 --> 16:46.389 align:start position:0%
shadows of something much deeper, a
four-dimensional<00:16:44.959><c> object.</c><00:16:45.839><c> And</c><00:16:46.000><c> that's</c>

16:46.389 --> 16:46.399 align:start position:0%
four-dimensional object. And that's
 

16:46.399 --> 16:48.230 align:start position:0%
four-dimensional object. And that's
exactly<00:16:47.040><c> what</c><00:16:47.360><c> we</c><00:16:47.600><c> have</c><00:16:47.759><c> in</c><00:16:47.920><c> special</c>

16:48.230 --> 16:48.240 align:start position:0%
exactly what we have in special
 

16:48.240 --> 16:51.189 align:start position:0%
exactly what we have in special
relativity.<00:16:49.040><c> It's</c><00:16:49.279><c> called</c><00:16:49.600><c> four</c><00:16:50.079><c> momentum.</c>

16:51.189 --> 16:51.199 align:start position:0%
relativity. It's called four momentum.
 

16:51.199 --> 16:52.710 align:start position:0%
relativity. It's called four momentum.
So,<00:16:51.360><c> in</c><00:16:51.600><c> relativity,</c><00:16:52.160><c> we</c><00:16:52.320><c> don't</c><00:16:52.480><c> think</c><00:16:52.560><c> of</c>

16:52.710 --> 16:52.720 align:start position:0%
So, in relativity, we don't think of
 

16:52.720 --> 16:54.870 align:start position:0%
So, in relativity, we don't think of
energy<00:16:53.040><c> and</c><00:16:53.199><c> momentum</c><00:16:53.600><c> separately.</c><00:16:54.480><c> We</c><00:16:54.720><c> just</c>

16:54.870 --> 16:54.880 align:start position:0%
energy and momentum separately. We just
 

16:54.880 --> 16:57.189 align:start position:0%
energy and momentum separately. We just
think<00:16:55.040><c> of</c><00:16:55.199><c> them</c><00:16:55.360><c> as</c><00:16:55.680><c> two</c><00:16:56.000><c> components</c><00:16:56.560><c> of</c><00:16:56.959><c> the</c>

16:57.189 --> 16:57.199 align:start position:0%
think of them as two components of the
 

16:57.199 --> 16:58.790 align:start position:0%
think of them as two components of the
underlying

16:58.790 --> 16:58.800 align:start position:0%
underlying
 

16:58.800 --> 17:00.710 align:start position:0%
underlying
object<00:16:59.199><c> called</c><00:16:59.360><c> for</c><00:16:59.680><c> momentum.</c><00:17:00.320><c> I</c><00:17:00.480><c> know</c><00:17:00.560><c> this</c>

17:00.710 --> 17:00.720 align:start position:0%
object called for momentum. I know this
 

17:00.720 --> 17:03.590 align:start position:0%
object called for momentum. I know this
is<00:17:00.800><c> a</c><00:17:00.959><c> deg</c><00:17:01.519><c> uh</c><00:17:01.759><c> this</c><00:17:02.000><c> is</c><00:17:02.079><c> a</c><00:17:02.240><c> tangent</c><00:17:02.720><c> but</c><00:17:03.199><c> oh</c><00:17:03.440><c> my</c>

17:03.590 --> 17:03.600 align:start position:0%
is a deg uh this is a tangent but oh my
 

17:03.600 --> 17:07.909 align:start position:0%
is a deg uh this is a tangent but oh my
god<00:17:03.839><c> like</c><00:17:04.160><c> that</c><00:17:04.400><c> connection</c><00:17:04.799><c> is</c><00:17:05.199><c> beautiful.</c>

17:07.909 --> 17:07.919 align:start position:0%
god like that connection is beautiful.
 

17:07.919 --> 17:10.230 align:start position:0%
god like that connection is beautiful.
Anyways<00:17:08.400><c> we</c><00:17:08.720><c> substitute</c><00:17:09.199><c> now</c><00:17:09.439><c> for</c><00:17:09.760><c> omega</c><00:17:10.079><c> and</c>

17:10.230 --> 17:10.240 align:start position:0%
Anyways we substitute now for omega and
 

17:10.240 --> 17:13.110 align:start position:0%
Anyways we substitute now for omega and
kappa<00:17:10.799><c> and</c><00:17:11.120><c> boom</c><00:17:11.679><c> we</c><00:17:11.919><c> have</c><00:17:12.079><c> built</c><00:17:12.319><c> our</c><00:17:12.640><c> very</c>

17:13.110 --> 17:13.120 align:start position:0%
kappa and boom we have built our very
 

17:13.120 --> 17:16.230 align:start position:0%
kappa and boom we have built our very
first<00:17:13.679><c> mowave</c><00:17:14.799><c> equation.</c><00:17:15.600><c> But</c><00:17:15.760><c> remember</c><00:17:16.079><c> what</c>

17:16.230 --> 17:16.240 align:start position:0%
first mowave equation. But remember what
 

17:16.240 --> 17:19.270 align:start position:0%
first mowave equation. But remember what
our<00:17:16.480><c> actual</c><00:17:16.799><c> goal</c><00:17:17.039><c> was</c><00:17:17.679><c> here</c><00:17:17.919><c> was</c><00:17:18.160><c> to</c><00:17:18.559><c> build</c>

17:19.270 --> 17:19.280 align:start position:0%
our actual goal was here was to build
 

17:19.280 --> 17:22.870 align:start position:0%
our actual goal was here was to build
energy<00:17:19.760><c> and</c><00:17:20.079><c> kinetic</c><00:17:20.559><c> energy</c><00:17:21.120><c> operators.</c><00:17:22.400><c> So</c>

17:22.870 --> 17:22.880 align:start position:0%
energy and kinetic energy operators. So
 

17:22.880 --> 17:25.510 align:start position:0%
energy and kinetic energy operators. So
let's<00:17:23.199><c> start</c><00:17:23.439><c> with</c><00:17:24.000><c> kinetic</c><00:17:24.480><c> energy.</c><00:17:25.280><c> Since</c>

17:25.510 --> 17:25.520 align:start position:0%
let's start with kinetic energy. Since
 

17:25.520 --> 17:27.829 align:start position:0%
let's start with kinetic energy. Since
it<00:17:25.760><c> has</c><00:17:26.079><c> momentum</c><00:17:26.559><c> squared</c><00:17:26.959><c> in</c><00:17:27.120><c> it,</c><00:17:27.520><c> our</c>

17:27.829 --> 17:27.839 align:start position:0%
it has momentum squared in it, our
 

17:27.839 --> 17:30.230 align:start position:0%
it has momentum squared in it, our
question<00:17:28.079><c> would</c><00:17:28.319><c> be</c><00:17:28.640><c> how</c><00:17:28.880><c> do</c><00:17:29.039><c> we</c><00:17:29.280><c> pull</c>

17:30.230 --> 17:30.240 align:start position:0%
question would be how do we pull
 

17:30.240 --> 17:34.470 align:start position:0%
question would be how do we pull
momentum<00:17:31.520><c> out</c><00:17:32.000><c> of</c><00:17:32.559><c> this</c><00:17:32.960><c> equation?</c><00:17:33.919><c> We</c><00:17:34.240><c> can</c>

17:34.470 --> 17:34.480 align:start position:0%
momentum out of this equation? We can
 

17:34.480 --> 17:37.270 align:start position:0%
momentum out of this equation? We can
differentiate<00:17:35.120><c> it</c><00:17:35.520><c> with</c><00:17:35.840><c> respect</c><00:17:36.240><c> to</c><00:17:36.480><c> x,</c>

17:37.270 --> 17:37.280 align:start position:0%
differentiate it with respect to x,
 

17:37.280 --> 17:39.430 align:start position:0%
differentiate it with respect to x,
right?<00:17:38.000><c> Well,</c><00:17:38.240><c> actually</c><00:17:38.559><c> we</c><00:17:38.720><c> need</c><00:17:38.880><c> to</c><00:17:39.039><c> do</c>

17:39.430 --> 17:39.440 align:start position:0%
right? Well, actually we need to do
 

17:39.440 --> 17:41.750 align:start position:0%
right? Well, actually we need to do
partial<00:17:39.919><c> derivative.</c><00:17:40.880><c> Then</c><00:17:41.200><c> this</c><00:17:41.440><c> part</c>

17:41.750 --> 17:41.760 align:start position:0%
partial derivative. Then this part
 

17:41.760 --> 17:45.190 align:start position:0%
partial derivative. Then this part
becomes<00:17:42.160><c> a</c><00:17:42.320><c> constant.</c><00:17:43.039><c> And</c><00:17:43.200><c> now</c><00:17:43.440><c> look,</c><00:17:44.000><c> p</c><00:17:44.320><c> or</c><00:17:44.559><c> h</c>

17:45.190 --> 17:45.200 align:start position:0%
becomes a constant. And now look, p or h
 

17:45.200 --> 17:49.750 align:start position:0%
becomes a constant. And now look, p or h
pops<00:17:45.600><c> out</c><00:17:45.760><c> in</c><00:17:46.000><c> front</c><00:17:46.799><c> and</c><00:17:47.200><c> s</c><00:17:47.679><c> turns</c><00:17:48.160><c> to</c><00:17:48.720><c> cos.</c>

17:49.750 --> 17:49.760 align:start position:0%
pops out in front and s turns to cos.
 

17:49.760 --> 17:51.270 align:start position:0%
pops out in front and s turns to cos.
But<00:17:49.919><c> we</c><00:17:50.080><c> don't</c><00:17:50.160><c> want</c><00:17:50.320><c> just</c><00:17:50.559><c> momentum.</c><00:17:50.960><c> We</c><00:17:51.120><c> want</c>

17:51.270 --> 17:51.280 align:start position:0%
But we don't want just momentum. We want
 

17:51.280 --> 17:53.190 align:start position:0%
But we don't want just momentum. We want
momentum<00:17:51.760><c> squared</c><00:17:52.160><c> because</c><00:17:52.320><c> our</c><00:17:52.640><c> goal</c><00:17:52.880><c> is</c><00:17:53.039><c> to</c>

17:53.190 --> 17:53.200 align:start position:0%
momentum squared because our goal is to
 

17:53.200 --> 17:55.830 align:start position:0%
momentum squared because our goal is to
get<00:17:53.760><c> kinetic</c><00:17:54.160><c> energy.</c><00:17:54.640><c> So</c><00:17:54.799><c> what</c><00:17:54.960><c> do</c><00:17:55.120><c> we</c><00:17:55.280><c> do?</c>

17:55.830 --> 17:55.840 align:start position:0%
get kinetic energy. So what do we do?
 

17:55.840 --> 17:57.750 align:start position:0%
get kinetic energy. So what do we do?
Well,<00:17:56.080><c> we</c><00:17:56.320><c> differentiate</c><00:17:56.880><c> again.</c><00:17:57.440><c> On</c><00:17:57.600><c> the</c>

17:57.750 --> 17:57.760 align:start position:0%
Well, we differentiate again. On the
 

17:57.760 --> 17:59.110 align:start position:0%
Well, we differentiate again. On the
left<00:17:57.919><c> hand</c><00:17:58.080><c> side,</c><00:17:58.320><c> we</c><00:17:58.480><c> get</c><00:17:58.640><c> a</c><00:17:58.880><c> second</c>

17:59.110 --> 17:59.120 align:start position:0%
left hand side, we get a second
 

17:59.120 --> 18:02.630 align:start position:0%
left hand side, we get a second
derivative.<00:18:00.160><c> And</c><00:18:00.400><c> over</c><00:18:00.640><c> here,</c><00:18:01.679><c> P</c><00:18:01.919><c> /</c><00:18:02.080><c> H</c><00:18:02.320><c> pops</c>

18:02.630 --> 18:02.640 align:start position:0%
derivative. And over here, P / H pops
 

18:02.640 --> 18:05.510 align:start position:0%
derivative. And over here, P / H pops
out<00:18:02.880><c> one</c><00:18:03.039><c> more</c><00:18:03.280><c> time.</c><00:18:04.080><c> And</c><00:18:04.400><c> cost</c><00:18:04.720><c> becomes</c>

18:05.510 --> 18:05.520 align:start position:0%
out one more time. And cost becomes
 

18:05.520 --> 18:08.310 align:start position:0%
out one more time. And cost becomes
negative<00:18:06.240><c> sign.</c><00:18:07.600><c> And</c><00:18:07.840><c> if</c><00:18:08.000><c> you</c><00:18:08.160><c> look</c>

18:08.310 --> 18:08.320 align:start position:0%
negative sign. And if you look
 

18:08.320 --> 18:11.990 align:start position:0%
negative sign. And if you look
carefully,<00:18:09.440><c> look,</c><00:18:10.000><c> this</c><00:18:10.640><c> is</c><00:18:10.880><c> our</c><00:18:11.360><c> original</c>

18:11.990 --> 18:12.000 align:start position:0%
carefully, look, this is our original
 

18:12.000 --> 18:17.270 align:start position:0%
carefully, look, this is our original
function<00:18:13.120><c> s.</c><00:18:14.320><c> So</c><00:18:14.640><c> we</c><00:18:15.120><c> have</c><00:18:15.360><c> got</c><00:18:15.840><c> sigh</c><00:18:16.320><c> back.</c>

18:17.270 --> 18:17.280 align:start position:0%
function s. So we have got sigh back.
 

18:17.280 --> 18:19.990 align:start position:0%
function s. So we have got sigh back.
And<00:18:17.520><c> if</c><00:18:17.760><c> we</c><00:18:18.240><c> rearrange,</c>

18:19.990 --> 18:20.000 align:start position:0%
And if we rearrange,
 

18:20.000 --> 18:24.070 align:start position:0%
And if we rearrange,
we<00:18:20.640><c> have</c><00:18:20.960><c> built</c><00:18:21.440><c> our</c><00:18:21.919><c> very</c><00:18:22.480><c> first</c><00:18:22.960><c> operator,</c>

18:24.070 --> 18:24.080 align:start position:0%
we have built our very first operator,
 

18:24.080 --> 18:28.789 align:start position:0%
we have built our very first operator,
the<00:18:24.720><c> momentum</c><00:18:25.760><c> squared</c><00:18:26.400><c> operator.</c>

18:28.789 --> 18:28.799 align:start position:0%
the momentum squared operator.
 

18:28.799 --> 18:31.430 align:start position:0%
the momentum squared operator.
Look,<00:18:29.120><c> when</c><00:18:29.280><c> you</c><00:18:29.440><c> do</c><00:18:29.520><c> this</c><00:18:29.760><c> operation</c><00:18:30.160><c> on</c><00:18:30.480><c> S,</c>

18:31.430 --> 18:31.440 align:start position:0%
Look, when you do this operation on S,
 

18:31.440 --> 18:33.830 align:start position:0%
Look, when you do this operation on S,
you<00:18:31.679><c> get</c><00:18:31.840><c> momentum</c><00:18:32.559><c> squared</c><00:18:33.200><c> time</c><00:18:33.360><c> S.</c><00:18:33.679><c> So</c>

18:33.830 --> 18:33.840 align:start position:0%
you get momentum squared time S. So
 

18:33.840 --> 18:36.150 align:start position:0%
you get momentum squared time S. So
momentum<00:18:34.240><c> squared</c><00:18:34.559><c> has</c><00:18:34.720><c> been</c><00:18:35.120><c> extracted.</c><00:18:36.000><c> So</c>

18:36.150 --> 18:36.160 align:start position:0%
momentum squared has been extracted. So
 

18:36.160 --> 18:39.029 align:start position:0%
momentum squared has been extracted. So
this<00:18:36.400><c> is</c><00:18:36.480><c> the</c><00:18:36.720><c> momentum</c><00:18:37.280><c> squared</c><00:18:37.840><c> operator.</c>

18:39.029 --> 18:39.039 align:start position:0%
this is the momentum squared operator.
 

18:39.039 --> 18:40.870 align:start position:0%
this is the momentum squared operator.
For<00:18:39.280><c> the</c><00:18:39.440><c> last</c><00:18:39.600><c> step,</c><00:18:39.919><c> I</c><00:18:40.160><c> need</c><00:18:40.400><c> kinetic</c>

18:40.870 --> 18:40.880 align:start position:0%
For the last step, I need kinetic
 

18:40.880 --> 18:42.630 align:start position:0%
For the last step, I need kinetic
energy,<00:18:41.360><c> right?</c><00:18:41.600><c> So</c><00:18:41.760><c> kinetic</c><00:18:42.080><c> energy</c><00:18:42.320><c> is</c><00:18:42.480><c> just</c>

18:42.630 --> 18:42.640 align:start position:0%
energy, right? So kinetic energy is just
 

18:42.640 --> 18:45.190 align:start position:0%
energy, right? So kinetic energy is just
momentum<00:18:42.960><c> square</c><00:18:43.440><c> by</c><00:18:43.520><c> 2</c><00:18:43.760><c> m.</c><00:18:44.000><c> So</c><00:18:44.240><c> let</c><00:18:44.400><c> me</c><00:18:44.559><c> just</c>

18:45.190 --> 18:45.200 align:start position:0%
momentum square by 2 m. So let me just
 

18:45.200 --> 18:47.190 align:start position:0%
momentum square by 2 m. So let me just
divide<00:18:45.679><c> by</c><00:18:45.919><c> 2</c><00:18:46.320><c> m.</c>

18:47.190 --> 18:47.200 align:start position:0%
divide by 2 m.
 

18:47.200 --> 18:50.390 align:start position:0%
divide by 2 m.
And<00:18:47.520><c> this</c><00:18:48.080><c> is</c><00:18:48.320><c> the</c><00:18:48.559><c> kinetic</c><00:18:49.039><c> energy.</c><00:18:49.919><c> So</c><00:18:50.160><c> I</c>

18:50.390 --> 18:50.400 align:start position:0%
And this is the kinetic energy. So I
 

18:50.400 --> 18:52.549 align:start position:0%
And this is the kinetic energy. So I
have<00:18:50.799><c> found</c><00:18:51.280><c> how</c><00:18:51.520><c> to</c><00:18:51.679><c> extract</c><00:18:52.160><c> kinetic</c>

18:52.549 --> 18:52.559 align:start position:0%
have found how to extract kinetic
 

18:52.559 --> 18:56.549 align:start position:0%
have found how to extract kinetic
energy.<00:18:53.360><c> So</c><00:18:53.679><c> this</c><00:18:54.559><c> must</c><00:18:54.960><c> be</c><00:18:55.600><c> the</c><00:18:55.919><c> kinetic</c>

18:56.549 --> 18:56.559 align:start position:0%
energy. So this must be the kinetic
 

18:56.559 --> 18:58.950 align:start position:0%
energy. So this must be the kinetic
energy<00:18:57.440><c> operator.</c>

18:58.950 --> 18:58.960 align:start position:0%
energy operator.
 

18:58.960 --> 19:02.870 align:start position:0%
energy operator.
Wow.<00:18:59.360><c> We've</c><00:18:59.679><c> built</c><00:19:00.000><c> it</c><00:19:00.320><c> all</c><00:19:00.640><c> by</c><00:19:01.120><c> ourselves.</c>

19:02.870 --> 19:02.880 align:start position:0%
Wow. We've built it all by ourselves.
 

19:02.880 --> 19:05.350 align:start position:0%
Wow. We've built it all by ourselves.
And<00:19:03.120><c> if</c><00:19:03.280><c> we</c><00:19:03.520><c> compare</c><00:19:03.760><c> it</c><00:19:04.000><c> to</c><00:19:04.400><c> the</c><00:19:04.799><c> actual</c>

19:05.350 --> 19:05.360 align:start position:0%
And if we compare it to the actual
 

19:05.360 --> 19:08.230 align:start position:0%
And if we compare it to the actual
Schroinger's<00:19:06.160><c> equation,</c><00:19:07.039><c> oh</c><00:19:07.280><c> my</c><00:19:07.520><c> god,</c><00:19:07.919><c> it's</c>

19:08.230 --> 19:08.240 align:start position:0%
Schroinger's equation, oh my god, it's
 

19:08.240 --> 19:11.029 align:start position:0%
Schroinger's equation, oh my god, it's
exactly<00:19:08.799><c> the</c><00:19:09.120><c> same</c><00:19:09.520><c> thing.</c>

19:11.029 --> 19:11.039 align:start position:0%
exactly the same thing.
 

19:11.039 --> 19:12.870 align:start position:0%
exactly the same thing.
Whoa.

19:12.870 --> 19:12.880 align:start position:0%
Whoa.
 

19:12.880 --> 19:15.990 align:start position:0%
Whoa.
All right,<00:19:13.200><c> let's</c><00:19:13.679><c> calm</c><00:19:13.919><c> down.</c><00:19:15.039><c> But</c><00:19:15.360><c> what</c>

19:15.990 --> 19:16.000 align:start position:0%
All right, let's calm down. But what
 

19:16.000 --> 19:19.669 align:start position:0%
All right, let's calm down. But what
does<00:19:16.240><c> it</c><00:19:16.720><c> actually</c><00:19:17.360><c> say?</c>

19:19.669 --> 19:19.679 align:start position:0%
does it actually say?
 

19:19.679 --> 19:22.230 align:start position:0%
does it actually say?
Well,<00:19:20.240><c> second</c><00:19:20.640><c> derivative</c><00:19:21.440><c> is</c><00:19:21.760><c> basically</c>

19:22.230 --> 19:22.240 align:start position:0%
Well, second derivative is basically
 

19:22.240 --> 19:25.750 align:start position:0%
Well, second derivative is basically
curvature,<00:19:23.200><c> right?</c><00:19:24.000><c> So</c><00:19:24.320><c> this</c><00:19:24.559><c> says</c><00:19:25.200><c> if</c><00:19:25.520><c> your</c>

19:25.750 --> 19:25.760 align:start position:0%
curvature, right? So this says if your
 

19:25.760 --> 19:29.510 align:start position:0%
curvature, right? So this says if your
wave<00:19:26.160><c> has</c><00:19:26.640><c> more</c><00:19:27.200><c> curvature,</c><00:19:28.559><c> then</c><00:19:29.039><c> it</c><00:19:29.280><c> will</c>

19:29.510 --> 19:29.520 align:start position:0%
wave has more curvature, then it will
 

19:29.520 --> 19:32.870 align:start position:0%
wave has more curvature, then it will
have<00:19:30.080><c> more</c><00:19:30.640><c> kinetic</c><00:19:31.120><c> energy.</c><00:19:32.080><c> And</c><00:19:32.320><c> that</c><00:19:32.559><c> makes</c>

19:32.870 --> 19:32.880 align:start position:0%
have more kinetic energy. And that makes
 

19:32.880 --> 19:35.750 align:start position:0%
have more kinetic energy. And that makes
sense<00:19:33.280><c> because</c><00:19:33.600><c> we</c><00:19:34.000><c> already</c><00:19:34.320><c> saw</c><00:19:35.120><c> that</c><00:19:35.600><c> you</c>

19:35.750 --> 19:35.760 align:start position:0%
sense because we already saw that you
 

19:35.760 --> 19:37.669 align:start position:0%
sense because we already saw that you
know<00:19:35.840><c> from</c><00:19:36.000><c> Droy's</c><00:19:36.559><c> equation</c><00:19:36.880><c> that</c><00:19:37.280><c> shorter</c>

19:37.669 --> 19:37.679 align:start position:0%
know from Droy's equation that shorter
 

19:37.679 --> 19:40.230 align:start position:0%
know from Droy's equation that shorter
wavelength<00:19:38.559><c> means</c><00:19:38.880><c> more</c><00:19:39.200><c> momentum</c><00:19:39.919><c> means</c>

19:40.230 --> 19:40.240 align:start position:0%
wavelength means more momentum means
 

19:40.240 --> 19:41.990 align:start position:0%
wavelength means more momentum means
more<00:19:40.480><c> kinetic</c><00:19:40.880><c> energy.</c><00:19:41.280><c> Shorter</c><00:19:41.600><c> wavelength</c>

19:41.990 --> 19:42.000 align:start position:0%
more kinetic energy. Shorter wavelength
 

19:42.000 --> 19:43.669 align:start position:0%
more kinetic energy. Shorter wavelength
means<00:19:42.240><c> more</c><00:19:42.480><c> curvature.</c><00:19:43.120><c> But</c><00:19:43.360><c> our</c>

19:43.669 --> 19:43.679 align:start position:0%
means more curvature. But our
 

19:43.679 --> 19:45.990 align:start position:0%
means more curvature. But our
understanding<00:19:44.160><c> has</c><00:19:44.480><c> upgraded</c><00:19:45.360><c> because</c><00:19:45.760><c> the</c>

19:45.990 --> 19:46.000 align:start position:0%
understanding has upgraded because the
 

19:46.000 --> 19:49.430 align:start position:0%
understanding has upgraded because the
idea<00:19:46.240><c> of</c><00:19:46.400><c> wavelength</c><00:19:47.120><c> only</c><00:19:47.600><c> works</c><00:19:48.000><c> for</c><00:19:48.640><c> pure</c>

19:49.430 --> 19:49.440 align:start position:0%
idea of wavelength only works for pure
 

19:49.440 --> 19:52.870 align:start position:0%
idea of wavelength only works for pure
sine<00:19:49.760><c> waves</c><00:19:50.400><c> right</c><00:19:51.280><c> but</c><00:19:52.080><c> the</c><00:19:52.320><c> idea</c><00:19:52.559><c> of</c>

19:52.870 --> 19:52.880 align:start position:0%
sine waves right but the idea of
 

19:52.880 --> 19:56.310 align:start position:0%
sine waves right but the idea of
curvature<00:19:54.080><c> can</c><00:19:54.320><c> be</c><00:19:54.559><c> defined</c>

19:56.310 --> 19:56.320 align:start position:0%
curvature can be defined
 

19:56.320 --> 20:01.510 align:start position:0%
curvature can be defined
at<00:19:56.799><c> every</c><00:19:57.600><c> single</c><00:19:58.400><c> point.</c><00:19:59.440><c> So</c><00:20:00.160><c> it</c><00:20:00.480><c> is</c><00:20:01.120><c> a</c>

20:01.510 --> 20:01.520 align:start position:0%
at every single point. So it is a
 

20:01.520 --> 20:03.990 align:start position:0%
at every single point. So it is a
general<00:20:02.000><c> way</c><00:20:02.240><c> of</c><00:20:02.400><c> thinking</c><00:20:02.720><c> about</c><00:20:02.960><c> it.</c>

20:03.990 --> 20:04.000 align:start position:0%
general way of thinking about it.
 

20:04.000 --> 20:07.350 align:start position:0%
general way of thinking about it.
Kinetic<00:20:04.480><c> energy</c><00:20:04.960><c> is</c><00:20:05.360><c> encoded</c><00:20:06.160><c> in</c><00:20:06.880><c> the</c>

20:07.350 --> 20:07.360 align:start position:0%
Kinetic energy is encoded in the
 

20:07.360 --> 20:09.029 align:start position:0%
Kinetic energy is encoded in the
curvature.

20:09.029 --> 20:09.039 align:start position:0%
curvature.
 

20:09.039 --> 20:13.750 align:start position:0%
curvature.
Oh<00:20:09.440><c> man</c><00:20:10.160><c> that</c><00:20:10.400><c> is</c><00:20:10.640><c> awesome.</c><00:20:12.000><c> But</c><00:20:12.559><c> this</c><00:20:13.440><c> brings</c>

20:13.750 --> 20:13.760 align:start position:0%
Oh man that is awesome. But this brings
 

20:13.760 --> 20:18.070 align:start position:0%
Oh man that is awesome. But this brings
up<00:20:14.000><c> an</c><00:20:14.240><c> annoying</c><00:20:15.120><c> problem.</c>

20:18.070 --> 20:18.080 align:start position:0%
up an annoying problem.
 

20:18.080 --> 20:20.390 align:start position:0%
up an annoying problem.
See,<00:20:18.400><c> we</c><00:20:18.559><c> derived</c><00:20:18.960><c> this</c><00:20:19.120><c> operator</c><00:20:19.600><c> for</c><00:20:20.000><c> a</c>

20:20.390 --> 20:20.400 align:start position:0%
See, we derived this operator for a
 

20:20.400 --> 20:23.669 align:start position:0%
See, we derived this operator for a
special<00:20:20.880><c> case,</c><00:20:21.440><c> a</c><00:20:21.760><c> pure</c><00:20:22.160><c> sine</c><00:20:22.480><c> wave,</c><00:20:22.880><c> right?</c>

20:23.669 --> 20:23.679 align:start position:0%
special case, a pure sine wave, right?
 

20:23.679 --> 20:26.230 align:start position:0%
special case, a pure sine wave, right?
But<00:20:23.840><c> it</c><00:20:24.080><c> turns</c><00:20:24.320><c> out</c><00:20:24.720><c> the</c><00:20:25.039><c> operator</c><00:20:25.600><c> works</c><00:20:25.919><c> in</c>

20:26.230 --> 20:26.240 align:start position:0%
But it turns out the operator works in
 

20:26.240 --> 20:28.149 align:start position:0%
But it turns out the operator works in
general<00:20:26.559><c> for</c><00:20:26.799><c> any</c><00:20:27.039><c> shape.</c><00:20:27.520><c> Now,</c><00:20:27.760><c> that</c><00:20:27.919><c> is</c>

20:28.149 --> 20:28.159 align:start position:0%
general for any shape. Now, that is
 

20:28.159 --> 20:29.669 align:start position:0%
general for any shape. Now, that is
awesome.<00:20:28.480><c> I'm</c><00:20:28.720><c> not</c><00:20:28.880><c> complaining</c><00:20:29.360><c> because</c>

20:29.669 --> 20:29.679 align:start position:0%
awesome. I'm not complaining because
 

20:29.679 --> 20:32.310 align:start position:0%
awesome. I'm not complaining because
this<00:20:29.919><c> means</c><00:20:30.400><c> we</c><00:20:30.640><c> can</c><00:20:30.799><c> do</c><00:20:30.960><c> the</c><00:20:31.120><c> same</c><00:20:31.360><c> thing</c><00:20:31.600><c> for</c>

20:32.310 --> 20:32.320 align:start position:0%
this means we can do the same thing for
 

20:32.320 --> 20:34.710 align:start position:0%
this means we can do the same thing for
building<00:20:32.720><c> the</c><00:20:32.960><c> energy</c><00:20:33.360><c> operator</c><00:20:34.159><c> and</c><00:20:34.400><c> then</c><00:20:34.559><c> we</c>

20:34.710 --> 20:34.720 align:start position:0%
building the energy operator and then we
 

20:34.720 --> 20:36.630 align:start position:0%
building the energy operator and then we
can<00:20:34.880><c> finish</c><00:20:35.120><c> the</c><00:20:35.280><c> storing</c><00:20:35.919><c> equation</c><00:20:36.320><c> and</c>

20:36.630 --> 20:36.640 align:start position:0%
can finish the storing equation and
 

20:36.640 --> 20:39.669 align:start position:0%
can finish the storing equation and
fulfill<00:20:37.039><c> our</c><00:20:37.360><c> destiny.</c><00:20:38.720><c> But</c><00:20:38.960><c> I</c><00:20:39.120><c> won't</c><00:20:39.360><c> be</c><00:20:39.440><c> able</c>

20:39.669 --> 20:39.679 align:start position:0%
fulfill our destiny. But I won't be able
 

20:39.679 --> 20:42.870 align:start position:0%
fulfill our destiny. But I won't be able
to<00:20:39.840><c> sleep</c><00:20:40.080><c> at</c><00:20:40.400><c> night</c><00:20:41.360><c> because</c><00:20:42.320><c> although</c><00:20:42.720><c> we</c>

20:42.870 --> 20:42.880 align:start position:0%
to sleep at night because although we
 

20:42.880 --> 20:44.470 align:start position:0%
to sleep at night because although we
have<00:20:43.039><c> some</c><00:20:43.280><c> intuition</c><00:20:43.679><c> for</c><00:20:43.840><c> why</c><00:20:44.080><c> this</c><00:20:44.320><c> should</c>

20:44.470 --> 20:44.480 align:start position:0%
have some intuition for why this should
 

20:44.480 --> 20:46.710 align:start position:0%
have some intuition for why this should
work<00:20:44.720><c> in</c><00:20:44.880><c> general,</c><00:20:45.600><c> I</c><00:20:46.080><c> can't</c><00:20:46.400><c> actually</c>

20:46.710 --> 20:46.720 align:start position:0%
work in general, I can't actually
 

20:46.720 --> 20:50.230 align:start position:0%
work in general, I can't actually
convince<00:20:47.200><c> myself</c><00:20:48.320><c> mathematically</c><00:20:49.760><c> why</c>

20:50.230 --> 20:50.240 align:start position:0%
convince myself mathematically why
 

20:50.240 --> 20:53.430 align:start position:0%
convince myself mathematically why
something<00:20:50.640><c> we</c><00:20:50.880><c> derived</c><00:20:51.360><c> for</c><00:20:51.679><c> one</c><00:20:52.480><c> specific</c>

20:53.430 --> 20:53.440 align:start position:0%
something we derived for one specific
 

20:53.440 --> 20:58.230 align:start position:0%
something we derived for one specific
special<00:20:54.000><c> case</c><00:20:55.360><c> perfectly</c><00:20:55.919><c> works</c><00:20:56.240><c> in</c><00:20:56.480><c> general.</c>

20:58.230 --> 20:58.240 align:start position:0%
special case perfectly works in general.
 

20:58.240 --> 21:00.390 align:start position:0%
special case perfectly works in general.
I<00:20:58.480><c> was</c><00:20:58.640><c> stuck</c><00:20:58.799><c> here</c><00:20:58.960><c> for</c><00:20:59.200><c> a</c><00:20:59.360><c> while</c><00:20:59.679><c> until</c><00:21:00.080><c> I</c><00:21:00.240><c> met</c>

21:00.390 --> 21:00.400 align:start position:0%
I was stuck here for a while until I met
 

21:00.400 --> 21:03.750 align:start position:0%
I was stuck here for a while until I met
a<00:21:00.559><c> man</c><00:21:00.960><c> who</c><00:21:01.280><c> was</c><00:21:01.600><c> so</c><00:21:02.000><c> obsessed</c><00:21:02.480><c> with</c><00:21:02.799><c> heat</c><00:21:03.360><c> that</c>

21:03.750 --> 21:03.760 align:start position:0%
a man who was so obsessed with heat that
 

21:03.760 --> 21:05.990 align:start position:0%
a man who was so obsessed with heat that
apparently<00:21:04.240><c> he</c><00:21:04.480><c> kept</c><00:21:04.640><c> his</c><00:21:04.880><c> room</c><00:21:05.039><c> at</c><00:21:05.440><c> blazing</c>

21:05.990 --> 21:06.000 align:start position:0%
apparently he kept his room at blazing
 

21:06.000 --> 21:08.149 align:start position:0%
apparently he kept his room at blazing
temperatures<00:21:06.880><c> and</c><00:21:07.039><c> he</c><00:21:07.280><c> sat</c><00:21:07.440><c> inside</c><00:21:07.760><c> them</c>

21:08.149 --> 21:08.159 align:start position:0%
temperatures and he sat inside them
 

21:08.159 --> 21:11.510 align:start position:0%
temperatures and he sat inside them
wrapped<00:21:08.480><c> in</c><00:21:08.640><c> a</c><00:21:08.880><c> blanket</c><00:21:09.440><c> during</c><00:21:09.919><c> hot</c><00:21:10.240><c> summers.</c>

21:11.510 --> 21:11.520 align:start position:0%
wrapped in a blanket during hot summers.
 

21:11.520 --> 21:17.750 align:start position:0%
wrapped in a blanket during hot summers.
His<00:21:12.000><c> name</c><00:21:12.720><c> John</c><00:21:13.440><c> Baptist</c><00:21:14.320><c> Joseph</c><00:21:15.360><c> Forier.</c>

21:17.750 --> 21:17.760 align:start position:0%
His name John Baptist Joseph Forier.
 

21:17.760 --> 21:20.470 align:start position:0%
His name John Baptist Joseph Forier.
Forier<00:21:18.240><c> believed</c><00:21:18.720><c> heat</c><00:21:19.120><c> had</c><00:21:19.440><c> magical</c><00:21:20.000><c> healing</c>

21:20.470 --> 21:20.480 align:start position:0%
Forier believed heat had magical healing
 

21:20.480 --> 21:22.549 align:start position:0%
Forier believed heat had magical healing
powers.<00:21:21.280><c> But</c><00:21:21.440><c> historians</c><00:21:22.000><c> think</c><00:21:22.240><c> it's</c>

21:22.549 --> 21:22.559 align:start position:0%
powers. But historians think it's
 

21:22.559 --> 21:24.070 align:start position:0%
powers. But historians think it's
probably<00:21:22.799><c> because,</c><00:21:23.200><c> you</c><00:21:23.360><c> know,</c><00:21:23.440><c> he</c><00:21:23.679><c> developed</c>

21:24.070 --> 21:24.080 align:start position:0%
probably because, you know, he developed
 

21:24.080 --> 21:26.310 align:start position:0%
probably because, you know, he developed
extreme<00:21:24.559><c> cold</c><00:21:24.880><c> sensitivity</c><00:21:25.520><c> during</c><00:21:25.840><c> his</c><00:21:26.159><c> time</c>

21:26.310 --> 21:26.320 align:start position:0%
extreme cold sensitivity during his time
 

21:26.320 --> 21:28.789 align:start position:0%
extreme cold sensitivity during his time
in<00:21:26.559><c> the</c><00:21:26.720><c> Egyptian</c><00:21:27.200><c> desert.</c><00:21:28.159><c> But</c><00:21:28.320><c> whatever</c><00:21:28.640><c> it</c>

21:28.789 --> 21:28.799 align:start position:0%
in the Egyptian desert. But whatever it
 

21:28.799 --> 21:30.470 align:start position:0%
in the Egyptian desert. But whatever it
is,<00:21:29.039><c> what's</c><00:21:29.280><c> important</c><00:21:29.520><c> for</c><00:21:29.679><c> us</c><00:21:29.840><c> is</c><00:21:30.080><c> that</c><00:21:30.240><c> he</c>

21:30.470 --> 21:30.480 align:start position:0%
is, what's important for us is that he
 

21:30.480 --> 21:32.950 align:start position:0%
is, what's important for us is that he
was<00:21:30.640><c> a</c><00:21:30.799><c> mathematical</c><00:21:31.440><c> genius.</c><00:21:32.240><c> And</c><00:21:32.480><c> so</c>

21:32.950 --> 21:32.960 align:start position:0%
was a mathematical genius. And so
 

21:32.960 --> 21:35.909 align:start position:0%
was a mathematical genius. And so
obviously<00:21:33.760><c> he</c><00:21:34.159><c> wanted</c><00:21:34.559><c> to</c><00:21:35.039><c> mathematically</c>

21:35.909 --> 21:35.919 align:start position:0%
obviously he wanted to mathematically
 

21:35.919 --> 21:39.830 align:start position:0%
obviously he wanted to mathematically
model<00:21:36.559><c> how</c><00:21:36.960><c> heat</c><00:21:37.360><c> flows.</c><00:21:38.559><c> And</c><00:21:38.799><c> he</c><00:21:39.039><c> did</c><00:21:39.200><c> that.</c>

21:39.830 --> 21:39.840 align:start position:0%
model how heat flows. And he did that.
 

21:39.840 --> 21:42.310 align:start position:0%
model how heat flows. And he did that.
And<00:21:40.159><c> while</c><00:21:40.480><c> doing</c><00:21:40.720><c> so</c><00:21:41.120><c> he</c><00:21:41.440><c> developed</c><00:21:42.000><c> an</c>

21:42.310 --> 21:42.320 align:start position:0%
And while doing so he developed an
 

21:42.320 --> 21:45.190 align:start position:0%
And while doing so he developed an
incredible<00:21:43.200><c> principle.</c><00:21:44.080><c> He</c><00:21:44.320><c> found</c><00:21:44.480><c> that</c><00:21:44.880><c> any</c>

21:45.190 --> 21:45.200 align:start position:0%
incredible principle. He found that any
 

21:45.200 --> 21:48.870 align:start position:0%
incredible principle. He found that any
wave<00:21:45.840><c> or</c><00:21:46.159><c> any</c><00:21:46.559><c> shape</c><00:21:47.039><c> at</c><00:21:47.360><c> all</c><00:21:48.159><c> can</c><00:21:48.400><c> be</c><00:21:48.559><c> written</c>

21:48.870 --> 21:48.880 align:start position:0%
wave or any shape at all can be written
 

21:48.880 --> 21:52.870 align:start position:0%
wave or any shape at all can be written
as<00:21:49.200><c> a</c><00:21:49.440><c> sum</c><00:21:50.400><c> of</c><00:21:50.960><c> ss</c><00:21:51.360><c> and</c><00:21:51.600><c> cosine</c><00:21:52.080><c> waves.</c><00:21:52.640><c> Here's</c>

21:52.870 --> 21:52.880 align:start position:0%
as a sum of ss and cosine waves. Here's
 

21:52.880 --> 21:55.669 align:start position:0%
as a sum of ss and cosine waves. Here's
what<00:21:53.039><c> I</c><00:21:53.200><c> mean.</c><00:21:53.840><c> Take</c><00:21:54.080><c> this</c><00:21:54.400><c> square</c><00:21:54.720><c> wave.</c>

21:55.669 --> 21:55.679 align:start position:0%
what I mean. Take this square wave.
 

21:55.679 --> 21:57.430 align:start position:0%
what I mean. Take this square wave.
According<00:21:56.000><c> to</c><00:21:56.159><c> Foryer,</c><00:21:56.720><c> you</c><00:21:56.960><c> can</c><00:21:57.039><c> write</c><00:21:57.280><c> this</c>

21:57.430 --> 21:57.440 align:start position:0%
According to Foryer, you can write this
 

21:57.440 --> 21:59.990 align:start position:0%
According to Foryer, you can write this
as<00:21:57.679><c> just</c><00:21:57.919><c> sums</c><00:21:58.320><c> of</c><00:21:58.480><c> ss</c><00:21:58.799><c> and</c><00:21:58.960><c> cosiness.</c><00:21:59.679><c> If</c><00:21:59.840><c> you</c>

21:59.990 --> 22:00.000 align:start position:0%
as just sums of ss and cosiness. If you
 

22:00.000 --> 22:01.830 align:start position:0%
as just sums of ss and cosiness. If you
just<00:22:00.159><c> use</c><00:22:00.400><c> one</c><00:22:00.720><c> sine</c><00:22:00.960><c> wave,</c><00:22:01.360><c> that</c><00:22:01.600><c> doesn't</c>

22:01.830 --> 22:01.840 align:start position:0%
just use one sine wave, that doesn't
 

22:01.840 --> 22:03.590 align:start position:0%
just use one sine wave, that doesn't
look<00:22:02.080><c> like</c><00:22:02.240><c> much.</c><00:22:02.640><c> But</c><00:22:02.720><c> if</c><00:22:02.880><c> you</c><00:22:02.960><c> add</c><00:22:03.120><c> a</c><00:22:03.360><c> second</c>

22:03.590 --> 22:03.600 align:start position:0%
look like much. But if you add a second
 

22:03.600 --> 22:06.070 align:start position:0%
look like much. But if you add a second
one<00:22:04.080><c> with</c><00:22:04.320><c> a</c><00:22:04.640><c> slightly</c><00:22:04.960><c> different</c><00:22:05.280><c> height</c><00:22:05.760><c> and</c>

22:06.070 --> 22:06.080 align:start position:0%
one with a slightly different height and
 

22:06.080 --> 22:09.510 align:start position:0%
one with a slightly different height and
width,<00:22:06.720><c> look,</c><00:22:07.120><c> it</c><00:22:07.440><c> gets</c><00:22:07.679><c> closer.</c><00:22:08.720><c> Add</c><00:22:08.960><c> a</c><00:22:09.200><c> third</c>

22:09.510 --> 22:09.520 align:start position:0%
width, look, it gets closer. Add a third
 

22:09.520 --> 22:11.270 align:start position:0%
width, look, it gets closer. Add a third
and<00:22:09.679><c> a</c><00:22:09.919><c> fourth</c><00:22:10.159><c> and</c><00:22:10.320><c> a</c><00:22:10.480><c> fifth</c><00:22:10.720><c> and</c><00:22:10.880><c> you</c><00:22:11.039><c> keep</c>

22:11.270 --> 22:11.280 align:start position:0%
and a fourth and a fifth and you keep
 

22:11.280 --> 22:14.789 align:start position:0%
and a fourth and a fifth and you keep
going<00:22:11.600><c> and</c><00:22:12.000><c> look</c><00:22:12.559><c> look</c><00:22:13.039><c> the</c><00:22:13.280><c> shape</c><00:22:14.000><c> converges</c>

22:14.789 --> 22:14.799 align:start position:0%
going and look look the shape converges
 

22:14.799 --> 22:18.710 align:start position:0%
going and look look the shape converges
towards<00:22:15.360><c> a</c><00:22:15.760><c> perfect</c><00:22:16.720><c> square</c><00:22:17.440><c> wave.</c><00:22:18.240><c> Now</c><00:22:18.480><c> of</c>

22:18.710 --> 22:18.720 align:start position:0%
towards a perfect square wave. Now of
 

22:18.720 --> 22:20.950 align:start position:0%
towards a perfect square wave. Now of
course<00:22:19.280><c> technically</c><00:22:19.840><c> we</c><00:22:20.159><c> need</c><00:22:20.400><c> infinitely</c>

22:20.950 --> 22:20.960 align:start position:0%
course technically we need infinitely
 

22:20.960 --> 22:23.190 align:start position:0%
course technically we need infinitely
many<00:22:21.280><c> but</c><00:22:21.600><c> look</c><00:22:22.000><c> I</c><00:22:22.159><c> mean</c><00:22:22.320><c> even</c><00:22:22.480><c> with</c><00:22:22.640><c> a</c><00:22:22.799><c> few</c><00:22:23.039><c> we</c>

22:23.190 --> 22:23.200 align:start position:0%
many but look I mean even with a few we
 

22:23.200 --> 22:26.789 align:start position:0%
many but look I mean even with a few we
can<00:22:23.440><c> get</c><00:22:23.679><c> remarkably</c><00:22:24.320><c> close</c><00:22:24.720><c> right</c>

22:26.789 --> 22:26.799 align:start position:0%
can get remarkably close right
 

22:26.799 --> 22:30.070 align:start position:0%
can get remarkably close right
here's<00:22:27.200><c> another</c><00:22:27.440><c> example</c><00:22:28.480><c> again</c><00:22:29.360><c> by</c><00:22:29.600><c> adding</c>

22:30.070 --> 22:30.080 align:start position:0%
here's another example again by adding
 

22:30.080 --> 22:31.990 align:start position:0%
here's another example again by adding
multiple<00:22:30.559><c> sine</c><00:22:30.799><c> waves</c><00:22:31.120><c> of</c><00:22:31.360><c> just</c><00:22:31.600><c> the</c><00:22:31.760><c> right</c>

22:31.990 --> 22:32.000 align:start position:0%
multiple sine waves of just the right
 

22:32.000 --> 22:34.549 align:start position:0%
multiple sine waves of just the right
width<00:22:32.320><c> and</c><00:22:32.640><c> height</c><00:22:33.360><c> we</c><00:22:33.600><c> can</c><00:22:33.760><c> construct</c><00:22:34.320><c> this</c>

22:34.549 --> 22:34.559 align:start position:0%
width and height we can construct this
 

22:34.559 --> 22:37.669 align:start position:0%
width and height we can construct this
shape<00:22:34.960><c> too.</c><00:22:35.360><c> we</c><00:22:35.600><c> can</c><00:22:35.760><c> construct</c><00:22:36.240><c> any</c><00:22:36.559><c> shape</c>

22:37.669 --> 22:37.679 align:start position:0%
shape too. we can construct any shape
 

22:37.679 --> 22:40.310 align:start position:0%
shape too. we can construct any shape
and<00:22:38.080><c> forer</c><00:22:38.480><c> showed</c><00:22:38.799><c> that</c><00:22:39.120><c> this</c><00:22:39.440><c> works</c><00:22:39.919><c> in</c>

22:40.310 --> 22:40.320 align:start position:0%
and forer showed that this works in
 

22:40.320 --> 22:43.190 align:start position:0%
and forer showed that this works in
general<00:22:41.039><c> and</c><00:22:41.280><c> the</c><00:22:41.440><c> idea</c><00:22:41.679><c> is</c><00:22:41.919><c> called</c><00:22:42.320><c> the</c><00:22:42.640><c> 4year</c>

22:43.190 --> 22:43.200 align:start position:0%
general and the idea is called the 4year
 

22:43.200 --> 22:46.149 align:start position:0%
general and the idea is called the 4year
series<00:22:44.000><c> or</c><00:22:44.240><c> more</c><00:22:44.559><c> generally</c><00:22:45.360><c> it's</c><00:22:45.760><c> called</c><00:22:45.919><c> the</c>

22:46.149 --> 22:46.159 align:start position:0%
series or more generally it's called the
 

22:46.159 --> 22:49.350 align:start position:0%
series or more generally it's called the
4year<00:22:46.799><c> transforms.</c><00:22:48.080><c> This</c><00:22:48.320><c> idea</c><00:22:48.640><c> was</c><00:22:48.960><c> so</c>

22:49.350 --> 22:49.360 align:start position:0%
4year transforms. This idea was so
 

22:49.360 --> 22:51.350 align:start position:0%
4year transforms. This idea was so
radical<00:22:49.760><c> that</c><00:22:50.080><c> even</c><00:22:50.320><c> the</c><00:22:50.480><c> top</c><00:22:50.640><c> mathematicians</c>

22:51.350 --> 22:51.360 align:start position:0%
radical that even the top mathematicians
 

22:51.360 --> 22:53.430 align:start position:0%
radical that even the top mathematicians
back<00:22:51.600><c> then</c><00:22:52.000><c> just</c><00:22:52.240><c> couldn't</c><00:22:52.640><c> believe</c><00:22:52.960><c> it</c><00:22:53.200><c> this</c>

22:53.430 --> 22:53.440 align:start position:0%
back then just couldn't believe it this
 

22:53.440 --> 22:56.950 align:start position:0%
back then just couldn't believe it this
would<00:22:53.600><c> be</c><00:22:53.760><c> true.</c><00:22:54.640><c> But</c><00:22:54.880><c> today</c><00:22:55.600><c> we</c><00:22:55.919><c> have</c><00:22:56.159><c> a</c><00:22:56.480><c> very</c>

22:56.950 --> 22:56.960 align:start position:0%
would be true. But today we have a very
 

22:56.960 --> 22:58.789 align:start position:0%
would be true. But today we have a very
elegant<00:22:57.440><c> proof</c><00:22:57.760><c> for</c><00:22:57.919><c> it.</c><00:22:58.080><c> Let</c><00:22:58.240><c> me</c><00:22:58.320><c> know</c><00:22:58.480><c> if</c><00:22:58.640><c> you</c>

22:58.789 --> 22:58.799 align:start position:0%
elegant proof for it. Let me know if you
 

22:58.799 --> 23:01.510 align:start position:0%
elegant proof for it. Let me know if you
want<00:22:58.960><c> me</c><00:22:59.120><c> to</c><00:22:59.280><c> make</c><00:22:59.360><c> a</c><00:22:59.600><c> video</c><00:22:59.760><c> on</c><00:22:59.919><c> that.</c><00:23:00.880><c> But</c><00:23:01.200><c> for</c>

23:01.510 --> 23:01.520 align:start position:0%
want me to make a video on that. But for
 

23:01.520 --> 23:03.510 align:start position:0%
want me to make a video on that. But for
now<00:23:01.760><c> we'll</c><00:23:02.080><c> just</c><00:23:02.240><c> accept</c><00:23:02.559><c> that.</c><00:23:02.960><c> So</c><00:23:03.200><c> according</c>

23:03.510 --> 23:03.520 align:start position:0%
now we'll just accept that. So according
 

23:03.520 --> 23:06.789 align:start position:0%
now we'll just accept that. So according
to<00:23:03.679><c> Foryer</c><00:23:04.559><c> our</c><00:23:04.960><c> matter</c><00:23:05.360><c> wave</c><00:23:05.919><c> can</c><00:23:06.159><c> be</c><00:23:06.400><c> written</c>

23:06.789 --> 23:06.799 align:start position:0%
to Foryer our matter wave can be written
 

23:06.799 --> 23:11.110 align:start position:0%
to Foryer our matter wave can be written
as<00:23:07.120><c> the</c><00:23:07.520><c> sum</c><00:23:08.000><c> of</c><00:23:08.559><c> lots</c><00:23:08.960><c> and</c><00:23:09.200><c> lots</c><00:23:09.520><c> of</c><00:23:10.159><c> pure</c><00:23:10.799><c> sine</c>

23:11.110 --> 23:11.120 align:start position:0%
as the sum of lots and lots of pure sine
 

23:11.120 --> 23:13.350 align:start position:0%
as the sum of lots and lots of pure sine
waves.<00:23:11.919><c> So</c><00:23:12.159><c> now</c><00:23:12.400><c> what</c><00:23:12.640><c> will</c><00:23:12.799><c> happen</c><00:23:12.960><c> if</c><00:23:13.200><c> we</c>

23:13.350 --> 23:13.360 align:start position:0%
waves. So now what will happen if we
 

23:13.360 --> 23:16.149 align:start position:0%
waves. So now what will happen if we
apply<00:23:13.840><c> that</c><00:23:14.240><c> operator</c><00:23:14.799><c> we</c><00:23:14.960><c> built</c><00:23:15.360><c> to</c><00:23:15.679><c> this</c>

23:16.149 --> 23:16.159 align:start position:0%
apply that operator we built to this
 

23:16.159 --> 23:19.830 align:start position:0%
apply that operator we built to this
general<00:23:16.720><c> wave?</c><00:23:17.919><c> Well</c><00:23:18.720><c> derivatives</c><00:23:19.440><c> are</c>

23:19.830 --> 23:19.840 align:start position:0%
general wave? Well derivatives are
 

23:19.840 --> 23:22.470 align:start position:0%
general wave? Well derivatives are
linear<00:23:20.799><c> meaning</c><00:23:21.120><c> derivative</c><00:23:21.520><c> of</c><00:23:21.760><c> a</c><00:23:22.000><c> plus</c><00:23:22.240><c> b</c>

23:22.470 --> 23:22.480 align:start position:0%
linear meaning derivative of a plus b
 

23:22.480 --> 23:24.470 align:start position:0%
linear meaning derivative of a plus b
equals<00:23:22.799><c> derivative</c><00:23:23.200><c> of</c><00:23:23.440><c> a</c><00:23:23.679><c> plus</c><00:23:24.080><c> derivative</c>

23:24.470 --> 23:24.480 align:start position:0%
equals derivative of a plus derivative
 

23:24.480 --> 23:28.789 align:start position:0%
equals derivative of a plus derivative
of<00:23:24.720><c> b.</c><00:23:26.080><c> That</c><00:23:26.400><c> means</c><00:23:27.120><c> this</c><00:23:27.520><c> operator</c><00:23:28.400><c> gets</c>

23:28.789 --> 23:28.799 align:start position:0%
of b. That means this operator gets
 

23:28.799 --> 23:34.149 align:start position:0%
of b. That means this operator gets
applied<00:23:30.000><c> to</c><00:23:30.480><c> every</c><00:23:31.120><c> single</c><00:23:31.840><c> component.</c>

23:34.149 --> 23:34.159 align:start position:0%
applied to every single component.
 

23:34.159 --> 23:37.750 align:start position:0%
applied to every single component.
For<00:23:34.480><c> each</c><00:23:34.880><c> component,</c><00:23:35.919><c> it</c><00:23:36.320><c> spits</c><00:23:36.799><c> out</c><00:23:37.440><c> the</c>

23:37.750 --> 23:37.760 align:start position:0%
For each component, it spits out the
 

23:37.760 --> 23:41.430 align:start position:0%
For each component, it spits out the
components<00:23:38.240><c> kinetic</c><00:23:38.640><c> energy</c><00:23:39.600><c> multiplied</c><00:23:40.720><c> by</c>

23:41.430 --> 23:41.440 align:start position:0%
components kinetic energy multiplied by
 

23:41.440 --> 23:44.470 align:start position:0%
components kinetic energy multiplied by
the<00:23:41.760><c> components</c><00:23:42.159><c> wave</c><00:23:42.480><c> function.</c><00:23:43.600><c> And</c><00:23:43.840><c> then</c>

23:44.470 --> 23:44.480 align:start position:0%
the components wave function. And then
 

23:44.480 --> 23:49.909 align:start position:0%
the components wave function. And then
it<00:23:45.039><c> adds</c><00:23:45.600><c> up</c><00:23:46.480><c> all</c><00:23:46.880><c> of</c><00:23:47.039><c> those</c><00:23:48.159><c> and</c><00:23:48.799><c> spits</c><00:23:49.520><c> this</c>

23:49.909 --> 23:49.919 align:start position:0%
it adds up all of those and spits this
 

23:49.919 --> 23:53.990 align:start position:0%
it adds up all of those and spits this
entire<00:23:50.799><c> sum</c><00:23:51.360><c> back.</c><00:23:52.320><c> Which</c><00:23:52.559><c> means</c><00:23:52.960><c> look</c><00:23:53.600><c> the</c>

23:53.990 --> 23:54.000 align:start position:0%
entire sum back. Which means look the
 

23:54.000 --> 23:57.909 align:start position:0%
entire sum back. Which means look the
operator<00:23:54.640><c> when</c><00:23:55.360><c> working</c><00:23:55.679><c> on</c><00:23:55.840><c> a</c><00:23:56.240><c> general</c><00:23:56.799><c> wave</c>

23:57.909 --> 23:57.919 align:start position:0%
operator when working on a general wave
 

23:57.919 --> 24:00.870 align:start position:0%
operator when working on a general wave
spits<00:23:58.400><c> out</c><00:23:58.880><c> the</c><00:23:59.280><c> full</c><00:23:59.919><c> mixture</c><00:24:00.480><c> of</c><00:24:00.720><c> the</c>

24:00.870 --> 24:00.880 align:start position:0%
spits out the full mixture of the
 

24:00.880 --> 24:04.789 align:start position:0%
spits out the full mixture of the
kinetic<00:24:01.280><c> energy</c><00:24:01.919><c> as</c><00:24:02.240><c> a</c><00:24:02.559><c> weighted</c><00:24:03.520><c> sum.</c><00:24:04.320><c> And</c><00:24:04.480><c> so</c>

24:04.789 --> 24:04.799 align:start position:0%
kinetic energy as a weighted sum. And so
 

24:04.799 --> 24:07.190 align:start position:0%
kinetic energy as a weighted sum. And so
for<00:24:05.280><c> helps</c><00:24:05.520><c> us</c><00:24:05.679><c> understand</c><00:24:06.159><c> why</c><00:24:06.799><c> if</c><00:24:07.039><c> an</c>

24:07.190 --> 24:07.200 align:start position:0%
for helps us understand why if an
 

24:07.200 --> 24:10.390 align:start position:0%
for helps us understand why if an
operator<00:24:07.919><c> works</c><00:24:08.320><c> for</c><00:24:08.640><c> a</c><00:24:08.880><c> sine</c><00:24:09.200><c> wave</c><00:24:09.840><c> it</c><00:24:10.159><c> should</c>

24:10.390 --> 24:10.400 align:start position:0%
operator works for a sine wave it should
 

24:10.400 --> 24:14.870 align:start position:0%
operator works for a sine wave it should
work<00:24:10.640><c> for</c><00:24:11.039><c> any</c><00:24:11.600><c> wave</c><00:24:12.240><c> in</c><00:24:12.640><c> general.</c><00:24:13.679><c> Oh</c><00:24:14.000><c> man</c><00:24:14.400><c> for</c>

24:14.870 --> 24:14.880 align:start position:0%
work for any wave in general. Oh man for
 

24:14.880 --> 24:17.909 align:start position:0%
work for any wave in general. Oh man for
you<00:24:15.039><c> beauty.</c><00:24:15.840><c> Imagine</c><00:24:16.320><c> if</c><00:24:16.559><c> forier</c><00:24:17.039><c> knew</c><00:24:17.600><c> that</c>

24:17.909 --> 24:17.919 align:start position:0%
you beauty. Imagine if forier knew that
 

24:17.919 --> 24:20.230 align:start position:0%
you beauty. Imagine if forier knew that
his<00:24:18.240><c> obsession</c><00:24:18.640><c> with</c><00:24:18.880><c> heat</c><00:24:19.600><c> would</c><00:24:19.840><c> one</c><00:24:20.080><c> day</c>

24:20.230 --> 24:20.240 align:start position:0%
his obsession with heat would one day
 

24:20.240 --> 24:22.549 align:start position:0%
his obsession with heat would one day
unlock<00:24:20.799><c> the</c><00:24:21.120><c> framework</c><00:24:21.679><c> for</c><00:24:22.000><c> just</c><00:24:22.240><c> you</c><00:24:22.400><c> know</c>

24:22.549 --> 24:22.559 align:start position:0%
unlock the framework for just you know
 

24:22.559 --> 24:25.430 align:start position:0%
unlock the framework for just you know
modeling<00:24:23.120><c> quantum</c><00:24:23.840><c> world.</c><00:24:24.559><c> Oh</c><00:24:24.799><c> my</c><00:24:24.960><c> god,</c><00:24:25.279><c> how</c>

24:25.430 --> 24:25.440 align:start position:0%
modeling quantum world. Oh my god, how
 

24:25.440 --> 24:28.870 align:start position:0%
modeling quantum world. Oh my god, how
would<00:24:25.600><c> he</c><00:24:25.760><c> be</c><00:24:25.919><c> feeling?</c><00:24:26.400><c> Oh</c><00:24:26.640><c> my</c><00:24:26.880><c> god.</c><00:24:27.840><c> Anyways,</c>

24:28.870 --> 24:28.880 align:start position:0%
would he be feeling? Oh my god. Anyways,
 

24:28.880 --> 24:30.789 align:start position:0%
would he be feeling? Oh my god. Anyways,
this<00:24:29.120><c> means</c><00:24:29.520><c> all</c><00:24:29.760><c> we</c><00:24:29.919><c> have</c><00:24:30.000><c> to</c><00:24:30.159><c> do</c><00:24:30.320><c> is</c><00:24:30.559><c> repeat</c>

24:30.789 --> 24:30.799 align:start position:0%
this means all we have to do is repeat
 

24:30.799 --> 24:33.830 align:start position:0%
this means all we have to do is repeat
the<00:24:31.120><c> same</c><00:24:31.360><c> process</c><00:24:31.840><c> for</c><00:24:32.159><c> extracting</c><00:24:32.880><c> energy</c>

24:33.830 --> 24:33.840 align:start position:0%
the same process for extracting energy
 

24:33.840 --> 24:36.230 align:start position:0%
the same process for extracting energy
and<00:24:34.159><c> we</c><00:24:34.320><c> are</c><00:24:34.480><c> done.</c><00:24:35.039><c> It</c><00:24:35.200><c> will</c><00:24:35.360><c> be</c><00:24:35.440><c> a</c><00:24:35.679><c> great</c><00:24:35.919><c> idea</c>

24:36.230 --> 24:36.240 align:start position:0%
and we are done. It will be a great idea
 

24:36.240 --> 24:38.070 align:start position:0%
and we are done. It will be a great idea
to<00:24:36.480><c> pause</c><00:24:36.720><c> the</c><00:24:36.960><c> video</c><00:24:37.120><c> over</c><00:24:37.360><c> here</c><00:24:37.520><c> and</c><00:24:37.760><c> see</c><00:24:37.919><c> if</c>

24:38.070 --> 24:38.080 align:start position:0%
to pause the video over here and see if
 

24:38.080 --> 24:39.909 align:start position:0%
to pause the video over here and see if
you<00:24:38.159><c> can</c><00:24:38.320><c> try</c><00:24:38.480><c> this</c><00:24:38.799><c> yourself.</c><00:24:39.360><c> Please</c><00:24:39.679><c> do</c>

24:39.909 --> 24:39.919 align:start position:0%
you can try this yourself. Please do
 

24:39.919 --> 24:44.470 align:start position:0%
you can try this yourself. Please do
that<00:24:40.640><c> moment</c><00:24:40.960><c> of</c><00:24:41.200><c> truth.</c>

24:44.470 --> 24:44.480 align:start position:0%
 
 

24:44.480 --> 24:48.630 align:start position:0%
 
All<00:24:44.559><c> right.</c><00:24:45.760><c> So</c><00:24:46.240><c> to</c><00:24:46.559><c> extract</c><00:24:47.120><c> energy</c><00:24:48.080><c> look</c><00:24:48.320><c> I</c>

24:48.630 --> 24:48.640 align:start position:0%
All right. So to extract energy look I
 

24:48.640 --> 24:50.789 align:start position:0%
All right. So to extract energy look I
have<00:24:48.720><c> to</c><00:24:48.880><c> differentiate</c><00:24:49.440><c> with</c><00:24:49.760><c> respect</c><00:24:50.159><c> to</c>

24:50.789 --> 24:50.799 align:start position:0%
have to differentiate with respect to
 

24:50.799 --> 24:53.990 align:start position:0%
have to differentiate with respect to
time.<00:24:52.000><c> This</c><00:24:52.240><c> time</c><00:24:52.720><c> this</c><00:24:53.039><c> term</c><00:24:53.279><c> is</c><00:24:53.440><c> a</c><00:24:53.600><c> constant.</c>

24:53.990 --> 24:54.000 align:start position:0%
time. This time this term is a constant.
 

24:54.000 --> 24:55.750 align:start position:0%
time. This time this term is a constant.
So<00:24:54.159><c> nothing</c><00:24:54.320><c> happens</c><00:24:54.640><c> to</c><00:24:54.799><c> it.</c><00:24:55.200><c> So</c><00:24:55.360><c> when</c><00:24:55.520><c> I</c>

24:55.750 --> 24:55.760 align:start position:0%
So nothing happens to it. So when I
 

24:55.760 --> 24:58.149 align:start position:0%
So nothing happens to it. So when I
differentiate<00:24:56.320><c> sign</c><00:24:56.880><c> well</c><00:24:57.120><c> again</c><00:24:57.360><c> e</c><00:24:57.679><c> over</c><00:24:57.840><c> h</c>

24:58.149 --> 24:58.159 align:start position:0%
differentiate sign well again e over h
 

24:58.159 --> 25:01.590 align:start position:0%
differentiate sign well again e over h
pops<00:24:58.480><c> out</c><00:24:58.799><c> and</c><00:24:59.120><c> sign</c><00:24:59.440><c> turns</c><00:24:59.760><c> into</c><00:25:00.000><c> cos.</c><00:25:01.200><c> We</c>

25:01.590 --> 25:01.600 align:start position:0%
pops out and sign turns into cos. We
 

25:01.600 --> 25:03.909 align:start position:0%
pops out and sign turns into cos. We
have<00:25:01.919><c> our</c><00:25:02.320><c> energy.</c><00:25:03.120><c> So</c><00:25:03.440><c> let's</c><00:25:03.679><c> just</c>

25:03.909 --> 25:03.919 align:start position:0%
have our energy. So let's just
 

25:03.919 --> 25:05.510 align:start position:0%
have our energy. So let's just
rearrange.

25:05.510 --> 25:05.520 align:start position:0%
rearrange.
 

25:05.520 --> 25:07.029 align:start position:0%
rearrange.
And

25:07.029 --> 25:07.039 align:start position:0%
And
 

25:07.039 --> 25:08.710 align:start position:0%
And
wait

25:08.710 --> 25:08.720 align:start position:0%
wait
 

25:08.720 --> 25:10.630 align:start position:0%
wait
there's<00:25:08.960><c> a</c><00:25:09.120><c> problem.</c><00:25:09.679><c> I'm</c><00:25:10.000><c> not</c><00:25:10.159><c> getting</c><00:25:10.400><c> my</c>

25:10.630 --> 25:10.640 align:start position:0%
there's a problem. I'm not getting my
 

25:10.640 --> 25:13.269 align:start position:0%
there's a problem. I'm not getting my
function<00:25:11.039><c> back</c><00:25:11.279><c> because</c><00:25:11.600><c> sine</c><00:25:12.159><c> turned</c><00:25:12.559><c> into</c>

25:13.269 --> 25:13.279 align:start position:0%
function back because sine turned into
 

25:13.279 --> 25:18.470 align:start position:0%
function back because sine turned into
cos.<00:25:14.000><c> So</c><00:25:14.159><c> I</c><00:25:14.480><c> can't</c><00:25:14.799><c> write</c><00:25:15.120><c> this</c><00:25:15.440><c> as</c><00:25:16.320><c> s.</c><00:25:17.679><c> Wait,</c>

25:18.470 --> 25:18.480 align:start position:0%
cos. So I can't write this as s. Wait,
 

25:18.480 --> 25:21.750 align:start position:0%
cos. So I can't write this as s. Wait,
why<00:25:18.720><c> did</c><00:25:18.880><c> it</c><00:25:19.039><c> work</c><00:25:19.360><c> last</c><00:25:19.679><c> time?</c><00:25:20.480><c> Oh,</c><00:25:21.279><c> last</c><00:25:21.600><c> time</c>

25:21.750 --> 25:21.760 align:start position:0%
why did it work last time? Oh, last time
 

25:21.760 --> 25:23.510 align:start position:0%
why did it work last time? Oh, last time
it<00:25:22.000><c> worked</c><00:25:22.240><c> because</c><00:25:22.480><c> we</c><00:25:22.640><c> took</c><00:25:22.960><c> a</c><00:25:23.200><c> double</c>

25:23.510 --> 25:23.520 align:start position:0%
it worked because we took a double
 

25:23.520 --> 25:25.190 align:start position:0%
it worked because we took a double
derivative,<00:25:24.159><c> right?</c><00:25:24.480><c> Because</c><00:25:24.640><c> I</c><00:25:24.880><c> wanted</c>

25:25.190 --> 25:25.200 align:start position:0%
derivative, right? Because I wanted
 

25:25.200 --> 25:27.190 align:start position:0%
derivative, right? Because I wanted
momentum<00:25:25.840><c> squared.</c><00:25:26.320><c> And</c><00:25:26.480><c> so</c><00:25:26.640><c> when</c><00:25:26.799><c> I</c><00:25:26.880><c> took</c><00:25:27.039><c> a</c>

25:27.190 --> 25:27.200 align:start position:0%
momentum squared. And so when I took a
 

25:27.200 --> 25:29.350 align:start position:0%
momentum squared. And so when I took a
second<00:25:27.360><c> derivative,</c><00:25:28.240><c> well,</c><00:25:28.640><c> s</c><00:25:28.960><c> turned</c><00:25:29.200><c> to</c>

25:29.350 --> 25:29.360 align:start position:0%
second derivative, well, s turned to
 

25:29.360 --> 25:31.510 align:start position:0%
second derivative, well, s turned to
cost<00:25:29.520><c> and</c><00:25:29.679><c> cost</c><00:25:30.000><c> turned</c><00:25:30.240><c> back</c><00:25:30.400><c> to</c><00:25:30.559><c> s.</c><00:25:31.200><c> And</c><00:25:31.360><c> I</c>

25:31.510 --> 25:31.520 align:start position:0%
cost and cost turned back to s. And I
 

25:31.520 --> 25:33.750 align:start position:0%
cost and cost turned back to s. And I
was<00:25:31.679><c> able</c><00:25:31.840><c> to</c><00:25:32.000><c> cleanly</c><00:25:32.640><c> write</c><00:25:32.880><c> this</c><00:25:33.120><c> as</c><00:25:33.279><c> s</c><00:25:33.600><c> and</c>

25:33.750 --> 25:33.760 align:start position:0%
was able to cleanly write this as s and
 

25:33.760 --> 25:37.029 align:start position:0%
was able to cleanly write this as s and
build<00:25:34.000><c> my</c><00:25:34.159><c> operator.</c><00:25:35.279><c> Oh,</c><00:25:35.760><c> but</c><00:25:36.000><c> this</c><00:25:36.240><c> time</c><00:25:36.799><c> I</c>

25:37.029 --> 25:37.039 align:start position:0%
build my operator. Oh, but this time I
 

25:37.039 --> 25:38.870 align:start position:0%
build my operator. Oh, but this time I
just<00:25:37.200><c> need</c><00:25:37.279><c> a</c><00:25:37.520><c> single</c><00:25:37.679><c> derivative</c><00:25:38.320><c> because</c><00:25:38.640><c> I</c>

25:38.870 --> 25:38.880 align:start position:0%
just need a single derivative because I
 

25:38.880 --> 25:40.390 align:start position:0%
just need a single derivative because I
already<00:25:39.120><c> found</c><00:25:39.440><c> energy.</c><00:25:40.000><c> I</c><00:25:40.159><c> don't</c><00:25:40.320><c> want</c>

25:40.390 --> 25:40.400 align:start position:0%
already found energy. I don't want
 

25:40.400 --> 25:41.990 align:start position:0%
already found energy. I don't want
energy<00:25:40.799><c> squared.</c><00:25:41.279><c> So,</c><00:25:41.360><c> I</c><00:25:41.520><c> can't</c><00:25:41.679><c> take</c><00:25:41.840><c> a</c>

25:41.990 --> 25:42.000 align:start position:0%
energy squared. So, I can't take a
 

25:42.000 --> 25:44.870 align:start position:0%
energy squared. So, I can't take a
second<00:25:42.159><c> derivative.</c><00:25:43.279><c> But</c><00:25:44.000><c> I</c><00:25:44.320><c> can't</c><00:25:44.640><c> write</c>

25:44.870 --> 25:44.880 align:start position:0%
second derivative. But I can't write
 

25:44.880 --> 25:46.710 align:start position:0%
second derivative. But I can't write
this<00:25:45.200><c> as</c><00:25:45.440><c> s</c><00:25:45.840><c> because</c><00:25:46.000><c> the</c><00:25:46.159><c> sign</c><00:25:46.400><c> has</c><00:25:46.559><c> turned</c>

25:46.710 --> 25:46.720 align:start position:0%
this as s because the sign has turned
 

25:46.720 --> 25:49.510 align:start position:0%
this as s because the sign has turned
into<00:25:46.960><c> cost.</c><00:25:47.600><c> Oh</c><00:25:47.919><c> man,</c><00:25:48.240><c> we</c><00:25:48.400><c> were</c><00:25:48.559><c> so</c><00:25:48.799><c> close.</c>

25:49.510 --> 25:49.520 align:start position:0%
into cost. Oh man, we were so close.
 

25:49.520 --> 25:52.870 align:start position:0%
into cost. Oh man, we were so close.
What?<00:25:50.000><c> What</c><00:25:50.320><c> do</c><00:25:50.480><c> we</c><00:25:50.720><c> do?</c>

25:52.870 --> 25:52.880 align:start position:0%
What? What do we do?
 

25:52.880 --> 25:54.630 align:start position:0%
What? What do we do?
Let's<00:25:53.120><c> just</c><00:25:53.279><c> think</c><00:25:53.440><c> mathematically.</c><00:25:54.400><c> For</c>

25:54.630 --> 25:54.640 align:start position:0%
Let's just think mathematically. For
 

25:54.640 --> 25:57.909 align:start position:0%
Let's just think mathematically. For
this<00:25:54.799><c> to</c><00:25:54.960><c> work,</c><00:25:55.520><c> to</c><00:25:55.760><c> get</c><00:25:56.000><c> our</c><00:25:56.480><c> s</c><00:25:56.960><c> back,</c><00:25:57.679><c> this</c>

25:57.909 --> 25:57.919 align:start position:0%
this to work, to get our s back, this
 

25:57.919 --> 26:00.390 align:start position:0%
this to work, to get our s back, this
function<00:25:58.240><c> of</c><00:25:58.400><c> time</c><00:25:58.799><c> has</c><00:25:59.120><c> to</c><00:25:59.279><c> be</c><00:25:59.520><c> such</c><00:25:59.760><c> that</c><00:26:00.000><c> its</c>

26:00.390 --> 26:00.400 align:start position:0%
function of time has to be such that its
 

26:00.400 --> 26:03.669 align:start position:0%
function of time has to be such that its
first<00:26:00.720><c> derivative</c><00:26:01.440><c> must</c><00:26:01.679><c> be</c><00:26:01.840><c> itself.</c><00:26:02.960><c> But</c><00:26:03.200><c> a</c><00:26:03.360><c> s</c>

26:03.669 --> 26:03.679 align:start position:0%
first derivative must be itself. But a s
 

26:03.679 --> 26:05.909 align:start position:0%
first derivative must be itself. But a s
or<00:26:03.840><c> a</c><00:26:04.000><c> cosine</c><00:26:04.400><c> or</c><00:26:04.720><c> any</c><00:26:05.039><c> periodic</c><00:26:05.440><c> function</c><00:26:05.679><c> for</c>

26:05.909 --> 26:05.919 align:start position:0%
or a cosine or any periodic function for
 

26:05.919 --> 26:08.710 align:start position:0%
or a cosine or any periodic function for
that<00:26:06.080><c> matter,</c><00:26:06.480><c> none</c><00:26:06.720><c> of</c><00:26:07.039><c> them</c><00:26:07.440><c> give</c><00:26:07.600><c> us</c><00:26:07.840><c> that.</c>

26:08.710 --> 26:08.720 align:start position:0%
that matter, none of them give us that.
 

26:08.720 --> 26:11.909 align:start position:0%
that matter, none of them give us that.
There's<00:26:09.039><c> only</c><00:26:09.440><c> one</c><00:26:09.840><c> function</c><00:26:10.159><c> in</c><00:26:10.480><c> this</c><00:26:10.720><c> entire</c>

26:11.909 --> 26:11.919 align:start position:0%
There's only one function in this entire
 

26:11.919 --> 26:15.350 align:start position:0%
There's only one function in this entire
multiverse<00:26:13.360><c> with</c><00:26:13.600><c> that</c><00:26:13.840><c> property.</c><00:26:14.960><c> It's</c><00:26:15.120><c> the</c>

26:15.350 --> 26:15.360 align:start position:0%
multiverse with that property. It's the
 

26:15.360 --> 26:17.430 align:start position:0%
multiverse with that property. It's the
exponential<00:26:16.000><c> function.</c><00:26:16.640><c> I</c><00:26:16.880><c> mean,</c><00:26:17.200><c> think</c>

26:17.430 --> 26:17.440 align:start position:0%
exponential function. I mean, think
 

26:17.440 --> 26:20.710 align:start position:0%
exponential function. I mean, think
about<00:26:17.600><c> it</c><00:26:17.919><c> for</c><00:26:18.159><c> a</c><00:26:18.400><c> second.</c><00:26:19.360><c> Imagine</c><00:26:19.679><c> that</c><00:26:20.400><c> this</c>

26:20.710 --> 26:20.720 align:start position:0%
about it for a second. Imagine that this
 

26:20.720 --> 26:24.230 align:start position:0%
about it for a second. Imagine that this
function<00:26:21.200><c> was</c><00:26:21.679><c> exponential</c><00:26:22.720><c> in</c><00:26:23.200><c> time.</c><00:26:24.000><c> Now,</c>

26:24.230 --> 26:24.240 align:start position:0%
function was exponential in time. Now,
 

26:24.240 --> 26:25.990 align:start position:0%
function was exponential in time. Now,
if<00:26:24.480><c> we</c><00:26:24.720><c> differentiate</c><00:26:25.120><c> it</c><00:26:25.279><c> with</c><00:26:25.440><c> respect</c><00:26:25.760><c> to</c>

26:25.990 --> 26:26.000 align:start position:0%
if we differentiate it with respect to
 

26:26.000 --> 26:29.990 align:start position:0%
if we differentiate it with respect to
time,<00:26:26.640><c> again,</c><00:26:26.960><c> e</c><00:26:27.279><c> over</c><00:26:27.440><c> h</c><00:26:27.760><c> bar</c><00:26:28.480><c> pops</c><00:26:28.880><c> out.</c><00:26:29.600><c> And</c>

26:29.990 --> 26:30.000 align:start position:0%
time, again, e over h bar pops out. And
 

26:30.000 --> 26:31.110 align:start position:0%
time, again, e over h bar pops out. And
this<00:26:30.159><c> time</c><00:26:30.320><c> because</c><00:26:30.559><c> the</c><00:26:30.720><c> derivative</c><00:26:31.039><c> of</c>

26:31.110 --> 26:31.120 align:start position:0%
this time because the derivative of
 

26:31.120 --> 26:34.870 align:start position:0%
this time because the derivative of
exponential<00:26:31.600><c> is</c><00:26:31.840><c> itself.</c><00:26:32.960><c> Look,</c><00:26:33.600><c> we</c><00:26:34.000><c> get</c><00:26:34.559><c> our</c>

26:34.870 --> 26:34.880 align:start position:0%
exponential is itself. Look, we get our
 

26:34.880 --> 26:38.070 align:start position:0%
exponential is itself. Look, we get our
function<00:26:35.600><c> s</c><00:26:36.400><c> back.</c><00:26:37.039><c> We</c><00:26:37.279><c> can</c><00:26:37.360><c> now</c><00:26:37.600><c> rearrange</c>

26:38.070 --> 26:38.080 align:start position:0%
function s back. We can now rearrange
 

26:38.080 --> 26:40.310 align:start position:0%
function s back. We can now rearrange
and<00:26:38.320><c> build</c><00:26:38.559><c> our</c><00:26:38.880><c> operator</c><00:26:39.360><c> just</c><00:26:39.600><c> like</c><00:26:39.760><c> before.</c>

26:40.310 --> 26:40.320 align:start position:0%
and build our operator just like before.
 

26:40.320 --> 26:42.870 align:start position:0%
and build our operator just like before.
And<00:26:40.559><c> the</c><00:26:40.720><c> math</c><00:26:41.120><c> would</c><00:26:41.360><c> work</c><00:26:41.600><c> out.</c><00:26:42.400><c> But</c><00:26:42.640><c> the</c>

26:42.870 --> 26:42.880 align:start position:0%
And the math would work out. But the
 

26:42.880 --> 26:46.230 align:start position:0%
And the math would work out. But the
problem<00:26:43.200><c> is</c><00:26:44.159><c> this</c><00:26:44.559><c> is</c><00:26:44.799><c> not</c><00:26:45.120><c> a</c><00:26:45.360><c> wave.</c><00:26:45.760><c> For</c><00:26:46.080><c> a</c>

26:46.230 --> 26:46.240 align:start position:0%
problem is this is not a wave. For a
 

26:46.240 --> 26:48.470 align:start position:0%
problem is this is not a wave. For a
wave,<00:26:46.640><c> the</c><00:26:46.880><c> amplitude</c><00:26:47.520><c> needs</c><00:26:47.840><c> to</c><00:26:48.000><c> be</c><00:26:48.240><c> some</c>

26:48.470 --> 26:48.480 align:start position:0%
wave, the amplitude needs to be some
 

26:48.480 --> 26:51.269 align:start position:0%
wave, the amplitude needs to be some
kind<00:26:48.640><c> of</c><00:26:48.720><c> a</c><00:26:48.880><c> periodic</c><00:26:49.440><c> function</c><00:26:49.840><c> of</c><00:26:50.240><c> time.</c><00:26:51.120><c> But</c>

26:51.269 --> 26:51.279 align:start position:0%
kind of a periodic function of time. But
 

26:51.279 --> 26:53.269 align:start position:0%
kind of a periodic function of time. But
exponentials<00:26:51.919><c> are</c><00:26:52.080><c> not</c><00:26:52.240><c> periodic</c><00:26:52.720><c> function.</c>

26:53.269 --> 26:53.279 align:start position:0%
exponentials are not periodic function.
 

26:53.279 --> 26:56.230 align:start position:0%
exponentials are not periodic function.
Exponentials<00:26:54.000><c> will</c><00:26:54.240><c> just</c><00:26:54.640><c> keep</c><00:26:54.960><c> blowing</c><00:26:55.360><c> up</c>

26:56.230 --> 26:56.240 align:start position:0%
Exponentials will just keep blowing up
 

26:56.240 --> 26:58.149 align:start position:0%
Exponentials will just keep blowing up
forever,

26:58.149 --> 26:58.159 align:start position:0%
forever,
 

26:58.159 --> 27:01.029 align:start position:0%
forever,
right?

27:01.029 --> 27:01.039 align:start position:0%
 
 

27:01.039 --> 27:03.430 align:start position:0%
 
Or<00:27:01.360><c> if</c><00:27:01.520><c> you</c><00:27:01.760><c> if</c><00:27:01.919><c> you</c><00:27:02.080><c> put</c><00:27:02.240><c> a</c><00:27:02.559><c> negative</c><00:27:02.960><c> side</c><00:27:03.279><c> for</c>

27:03.430 --> 27:03.440 align:start position:0%
Or if you if you put a negative side for
 

27:03.440 --> 27:05.510 align:start position:0%
Or if you if you put a negative side for
example<00:27:04.080><c> then</c><00:27:04.320><c> yeah</c><00:27:04.640><c> then</c><00:27:04.799><c> the</c><00:27:05.039><c> exponential</c>

27:05.510 --> 27:05.520 align:start position:0%
example then yeah then the exponential
 

27:05.520 --> 27:09.430 align:start position:0%
example then yeah then the exponential
would<00:27:05.760><c> just</c><00:27:05.919><c> keep</c><00:27:06.400><c> decaying</c><00:27:07.679><c> forever.</c>

27:09.430 --> 27:09.440 align:start position:0%
would just keep decaying forever.
 

27:09.440 --> 27:14.710 align:start position:0%
would just keep decaying forever.
Whatever<00:27:09.840><c> it</c><00:27:10.080><c> is</c><00:27:11.200><c> this</c><00:27:11.679><c> is</c><00:27:12.000><c> not</c><00:27:12.559><c> a</c><00:27:12.880><c> wave.</c>

27:14.710 --> 27:14.720 align:start position:0%
Whatever it is this is not a wave.
 

27:14.720 --> 27:17.190 align:start position:0%
Whatever it is this is not a wave.
So<00:27:14.880><c> we</c><00:27:15.120><c> have</c><00:27:15.200><c> a</c><00:27:15.440><c> problem.</c><00:27:16.159><c> I</c><00:27:16.400><c> mean</c><00:27:16.640><c> for</c><00:27:16.880><c> the</c>

27:17.190 --> 27:17.200 align:start position:0%
So we have a problem. I mean for the
 

27:17.200 --> 27:20.070 align:start position:0%
So we have a problem. I mean for the
physics<00:27:17.600><c> to</c><00:27:17.840><c> work</c><00:27:18.559><c> we</c><00:27:18.880><c> need</c><00:27:19.039><c> this</c><00:27:19.360><c> function</c><00:27:19.760><c> to</c>

27:20.070 --> 27:20.080 align:start position:0%
physics to work we need this function to
 

27:20.080 --> 27:23.029 align:start position:0%
physics to work we need this function to
be<00:27:20.240><c> periodic.</c><00:27:21.520><c> But</c><00:27:21.919><c> when</c><00:27:22.159><c> you</c><00:27:22.320><c> use</c><00:27:22.480><c> a</c><00:27:22.640><c> periodic</c>

27:23.029 --> 27:23.039 align:start position:0%
be periodic. But when you use a periodic
 

27:23.039 --> 27:24.710 align:start position:0%
be periodic. But when you use a periodic
function<00:27:23.360><c> the</c><00:27:23.520><c> math</c><00:27:23.760><c> breaks</c><00:27:24.000><c> down</c><00:27:24.159><c> because</c><00:27:24.480><c> I</c>

27:24.710 --> 27:24.720 align:start position:0%
function the math breaks down because I
 

27:24.720 --> 27:26.630 align:start position:0%
function the math breaks down because I
cannot<00:27:25.120><c> extract</c><00:27:25.600><c> energy</c><00:27:25.919><c> and</c><00:27:26.159><c> build</c><00:27:26.400><c> an</c>

27:26.630 --> 27:26.640 align:start position:0%
cannot extract energy and build an
 

27:26.640 --> 27:29.029 align:start position:0%
cannot extract energy and build an
operator.<00:27:27.440><c> For</c><00:27:27.679><c> that</c><00:27:27.919><c> to</c><00:27:28.080><c> happen,</c><00:27:28.480><c> I</c><00:27:28.799><c> need</c>

27:29.029 --> 27:29.039 align:start position:0%
operator. For that to happen, I need
 

27:29.039 --> 27:31.269 align:start position:0%
operator. For that to happen, I need
that<00:27:29.279><c> function</c><00:27:29.600><c> to</c><00:27:29.760><c> be</c><00:27:30.000><c> exponential.</c><00:27:31.200><c> But</c>

27:31.269 --> 27:31.279 align:start position:0%
that function to be exponential. But
 

27:31.279 --> 27:32.789 align:start position:0%
that function to be exponential. But
then<00:27:31.520><c> the</c><00:27:31.679><c> math</c><00:27:31.919><c> works,</c><00:27:32.159><c> but</c><00:27:32.320><c> the</c><00:27:32.559><c> physics</c>

27:32.789 --> 27:32.799 align:start position:0%
then the math works, but the physics
 

27:32.799 --> 27:36.549 align:start position:0%
then the math works, but the physics
doesn't<00:27:33.120><c> work</c><00:27:33.440><c> because</c><00:27:34.240><c> it's</c><00:27:34.559><c> not</c><00:27:34.799><c> a</c><00:27:35.039><c> wave.</c><00:27:36.320><c> So</c>

27:36.549 --> 27:36.559 align:start position:0%
doesn't work because it's not a wave. So
 

27:36.559 --> 27:39.510 align:start position:0%
doesn't work because it's not a wave. So
to<00:27:36.880><c> make</c><00:27:37.360><c> both</c><00:27:37.760><c> of</c><00:27:37.919><c> them</c><00:27:38.240><c> work,</c><00:27:38.799><c> we</c><00:27:39.039><c> somehow</c>

27:39.510 --> 27:39.520 align:start position:0%
to make both of them work, we somehow
 

27:39.520 --> 27:42.950 align:start position:0%
to make both of them work, we somehow
need<00:27:39.840><c> an</c><00:27:40.240><c> exponential</c><00:27:41.039><c> function</c><00:27:41.360><c> of</c><00:27:41.600><c> time</c>

27:42.950 --> 27:42.960 align:start position:0%
need an exponential function of time
 

27:42.960 --> 27:46.230 align:start position:0%
need an exponential function of time
that's<00:27:43.600><c> also</c><00:27:44.400><c> periodic.</c>

27:46.230 --> 27:46.240 align:start position:0%
that's also periodic.
 

27:46.240 --> 27:49.110 align:start position:0%
that's also periodic.
But<00:27:46.480><c> that's</c><00:27:46.880><c> impossible.</c><00:27:48.080><c> By</c><00:27:48.480><c> definition,</c>

27:49.110 --> 27:49.120 align:start position:0%
But that's impossible. By definition,
 

27:49.120 --> 27:52.070 align:start position:0%
But that's impossible. By definition,
exponential<00:27:49.760><c> functions</c><00:27:50.400><c> are</c><00:27:50.799><c> exponential.</c>

27:52.070 --> 27:52.080 align:start position:0%
exponential functions are exponential.
 

27:52.080 --> 27:53.990 align:start position:0%
exponential functions are exponential.
So,<00:27:52.400><c> how</c><00:27:52.640><c> in</c><00:27:52.799><c> the</c><00:27:52.960><c> world</c><00:27:53.279><c> can</c><00:27:53.440><c> we</c><00:27:53.679><c> build</c><00:27:53.840><c> a</c>

27:53.990 --> 27:54.000 align:start position:0%
So, how in the world can we build a
 

27:54.000 --> 27:56.549 align:start position:0%
So, how in the world can we build a
function<00:27:54.320><c> that's</c><00:27:54.640><c> both</c><00:27:55.039><c> exponential</c><00:27:56.080><c> and</c>

27:56.549 --> 27:56.559 align:start position:0%
function that's both exponential and
 

27:56.559 --> 27:58.710 align:start position:0%
function that's both exponential and
periodic?<00:27:57.600><c> Well,</c><00:27:57.760><c> it's</c><00:27:58.000><c> kind</c><00:27:58.159><c> of</c><00:27:58.240><c> like</c><00:27:58.399><c> how</c><00:27:58.559><c> we</c>

27:56.559 --> 27:59.589 align:start position:0%
subscription. So, happy learning. Back
to<00:29:28.480><c> the</c><00:29:28.720><c> video.</c><00:29:29.760><c> Okay,</c><00:29:30.000><c> so</c><00:29:30.240><c> coming</c><00:29:30.559><c> back,</c><00:29:31.200><c> how</c>

27:59.195 --> 27:59.205 align:start position:0%
to the video. Okay, so coming back, how
 

27:59.205 --> 28:01.275 align:start position:0%
to the video. Okay, so coming back, how
do<00:29:31.520><c> we</c><00:29:31.760><c> create</c><00:29:32.000><c> a</c><00:29:32.240><c> function</c><00:29:32.799><c> that's</c><00:29:33.120><c> both</c>

28:01.275 --> 28:01.285 align:start position:0%
do we create a function that's both
 

28:01.285 --> 28:03.835 align:start position:0%
do we create a function that's both
exponential<00:29:34.000><c> and</c><00:29:34.320><c> periodic?</c><00:29:35.360><c> The</c><00:29:35.600><c> answer</c>

28:03.835 --> 28:03.845 align:start position:0%
exponential and periodic? The answer
 

28:03.845 --> 28:06.475 align:start position:0%
exponential and periodic? The answer
actually<00:29:36.480><c> comes</c><00:29:36.880><c> from</c><00:29:37.279><c> a</c><00:29:37.679><c> bookkeeper</c><00:29:38.240><c> and</c><00:29:38.480><c> an</c>

28:06.475 --> 28:06.485 align:start position:0%
actually comes from a bookkeeper and an
 

28:06.485 --> 28:09.595 align:start position:0%
actually comes from a bookkeeper and an
accountant<00:29:39.039><c> from</c><00:29:39.279><c> Paris.</c><00:29:39.919><c> His</c><00:29:40.399><c> name</c><00:29:41.200><c> John</c>

28:09.595 --> 28:09.605 align:start position:0%
accountant from Paris. His name John
 

28:09.605 --> 28:12.795 align:start position:0%
accountant from Paris. His name John
Robert<00:29:42.320><c> Arand.</c><00:29:43.120><c> Argan</c><00:29:43.600><c> asks</c><00:29:43.840><c> Mahesh,</c><00:29:44.559><c> why</c><00:29:44.799><c> do</c>

28:12.795 --> 28:12.805 align:start position:0%
Robert Arand. Argan asks Mahesh, why do
 

28:12.805 --> 28:14.955 align:start position:0%
Robert Arand. Argan asks Mahesh, why do
exponential<00:29:45.520><c> functions</c><00:29:45.919><c> in</c><00:29:46.159><c> general</c><00:29:46.640><c> blow</c><00:29:46.960><c> up</c>

28:14.955 --> 28:14.965 align:start position:0%
exponential functions in general blow up
 

28:14.965 --> 28:17.994 align:start position:0%
exponential functions in general blow up
or<00:29:47.440><c> decay?</c><00:29:48.480><c> Well,</c><00:29:48.720><c> I</c><00:29:48.960><c> say</c><00:29:49.360><c> that's</c><00:29:49.840><c> because</c>

28:17.994 --> 28:18.004 align:start position:0%
or decay? Well, I say that's because
 

28:18.004 --> 28:19.595 align:start position:0%
or decay? Well, I say that's because
that's<00:29:50.480><c> literally</c><00:29:50.799><c> what</c><00:29:50.960><c> exponential</c><00:29:51.520><c> is.</c>

28:19.595 --> 28:19.605 align:start position:0%
that's literally what exponential is.
 

28:19.605 --> 28:21.435 align:start position:0%
that's literally what exponential is.
There's<00:29:51.919><c> no</c><00:29:52.159><c> deeper</c><00:29:52.480><c> explanation,</c><00:29:53.039><c> right?</c>

28:21.435 --> 28:21.445 align:start position:0%
There's no deeper explanation, right?
 

28:21.445 --> 28:22.795 align:start position:0%
There's no deeper explanation, right?
Well,<00:29:53.840><c> he</c><00:29:54.000><c> says,</c><00:29:54.240><c> think</c><00:29:54.480><c> about</c><00:29:54.640><c> it</c>

28:22.795 --> 28:22.805 align:start position:0%
Well, he says, think about it
 

28:22.805 --> 28:24.795 align:start position:0%
Well, he says, think about it
physically.<00:29:55.919><c> The</c><00:29:56.159><c> derivative</c><00:29:56.640><c> of</c><00:29:56.799><c> an</c>

28:24.795 --> 28:24.805 align:start position:0%
physically. The derivative of an
 

28:24.805 --> 28:29.195 align:start position:0%
physically. The derivative of an
exponential<00:29:58.080><c> is</c><00:29:58.880><c> proportional</c><00:29:59.840><c> to</c><00:30:00.240><c> itself,</c>

28:29.195 --> 28:29.205 align:start position:0%
exponential is proportional to itself,
 

28:29.205 --> 28:32.874 align:start position:0%
exponential is proportional to itself,
right?<00:30:02.320><c> So,</c><00:30:02.720><c> think</c><00:30:02.960><c> of</c><00:30:03.200><c> this</c><00:30:03.520><c> as</c><00:30:04.080><c> velocity.</c>

28:32.874 --> 28:32.884 align:start position:0%
right? So, think of this as velocity.
 

28:32.884 --> 28:35.035 align:start position:0%
right? So, think of this as velocity.
The<00:30:05.279><c> velocity</c><00:30:05.679><c> in</c><00:30:05.919><c> this</c><00:30:06.080><c> particular</c><00:30:06.399><c> case</c><00:30:06.640><c> is</c>

28:35.035 --> 28:35.045 align:start position:0%
The velocity in this particular case is
 

28:35.045 --> 28:38.234 align:start position:0%
The velocity in this particular case is
proportional<00:30:07.840><c> to</c><00:30:08.320><c> the</c><00:30:08.559><c> position</c><00:30:09.520><c> and</c><00:30:09.919><c> it's</c><00:30:10.159><c> in</c>

28:38.234 --> 28:38.244 align:start position:0%
proportional to the position and it's in
 

28:38.244 --> 28:40.795 align:start position:0%
proportional to the position and it's in
the<00:30:10.720><c> same</c><00:30:11.039><c> direction</c><00:30:11.520><c> as</c><00:30:11.760><c> the</c><00:30:12.000><c> position.</c><00:30:12.559><c> So</c><00:30:12.720><c> a</c>

28:40.795 --> 28:40.805 align:start position:0%
the same direction as the position. So a
 

28:40.805 --> 28:43.754 align:start position:0%
the same direction as the position. So a
little<00:30:13.120><c> time</c><00:30:13.279><c> later</c><00:30:13.600><c> the</c><00:30:13.840><c> position</c><00:30:14.559><c> grows</c><00:30:15.600><c> but</c>

28:43.754 --> 28:43.764 align:start position:0%
little time later the position grows but
 

28:43.764 --> 28:45.835 align:start position:0%
little time later the position grows but
the<00:30:16.159><c> velocity</c><00:30:16.559><c> grows</c><00:30:16.799><c> as</c><00:30:17.039><c> well.</c><00:30:17.440><c> So</c><00:30:17.679><c> now</c><00:30:17.840><c> the</c>

28:45.835 --> 28:45.845 align:start position:0%
the velocity grows as well. So now the
 

28:45.845 --> 28:48.234 align:start position:0%
the velocity grows as well. So now the
position<00:30:18.320><c> grows</c><00:30:18.559><c> even</c><00:30:18.799><c> faster.</c><00:30:19.520><c> The</c><00:30:19.919><c> velocity</c>

28:48.234 --> 28:48.244 align:start position:0%
position grows even faster. The velocity
 

28:48.244 --> 28:50.394 align:start position:0%
position grows even faster. The velocity
grows<00:30:20.720><c> even</c><00:30:21.039><c> faster</c><00:30:21.600><c> and</c><00:30:21.840><c> that's</c><00:30:22.080><c> how</c><00:30:22.320><c> we</c><00:30:22.480><c> get</c>

28:50.394 --> 28:50.404 align:start position:0%
grows even faster and that's how we get
 

28:50.404 --> 28:53.195 align:start position:0%
grows even faster and that's how we get
a<00:30:22.720><c> runaway</c><00:30:23.120><c> effect.</c><00:30:23.440><c> The</c><00:30:23.679><c> whole</c><00:30:23.840><c> thing</c><00:30:24.240><c> blows.</c>

28:53.195 --> 28:53.205 align:start position:0%
a runaway effect. The whole thing blows.
 

28:53.205 --> 28:56.394 align:start position:0%
a runaway effect. The whole thing blows.
If<00:30:25.520><c> you</c><00:30:25.760><c> had</c><00:30:26.080><c> a</c><00:30:26.480><c> negative</c><00:30:27.200><c> exponent</c><00:30:28.240><c> then</c><00:30:28.399><c> you</c>

28:56.394 --> 28:56.404 align:start position:0%
If you had a negative exponent then you
 

28:56.404 --> 28:57.994 align:start position:0%
If you had a negative exponent then you
would<00:30:28.720><c> have</c><00:30:28.799><c> the</c><00:30:29.039><c> same</c><00:30:29.200><c> effect</c><00:30:29.600><c> except</c><00:30:30.000><c> now</c>

28:57.994 --> 28:58.004 align:start position:0%
would have the same effect except now
 

28:58.004 --> 29:00.075 align:start position:0%
would have the same effect except now
the<00:30:30.320><c> velocity</c><00:30:30.720><c> would</c><00:30:30.960><c> be</c><00:30:31.039><c> in</c><00:30:31.279><c> the</c><00:30:31.520><c> opposite</c>

29:00.075 --> 29:00.085 align:start position:0%
the velocity would be in the opposite
 

29:00.085 --> 29:02.155 align:start position:0%
the velocity would be in the opposite
direction<00:30:32.640><c> because</c><00:30:32.880><c> of</c><00:30:33.039><c> the</c><00:30:33.200><c> negative</c><00:30:33.520><c> sign.</c>

29:02.155 --> 29:02.165 align:start position:0%
direction because of the negative sign.
 

29:02.165 --> 29:04.475 align:start position:0%
direction because of the negative sign.
So<00:30:34.640><c> now</c><00:30:34.880><c> a</c><00:30:35.039><c> little</c><00:30:35.200><c> time</c><00:30:35.440><c> later</c><00:30:35.760><c> the</c><00:30:35.919><c> position</c>

29:04.475 --> 29:04.485 align:start position:0%
So now a little time later the position
 

29:04.485 --> 29:06.795 align:start position:0%
So now a little time later the position
reduces<00:30:37.760><c> because</c><00:30:37.919><c> of</c><00:30:38.080><c> that</c><00:30:38.320><c> the</c><00:30:38.559><c> velocity</c>

29:06.795 --> 29:06.805 align:start position:0%
reduces because of that the velocity
 

29:06.805 --> 29:09.275 align:start position:0%
reduces because of that the velocity
also<00:30:39.200><c> reduces.</c><00:30:39.840><c> So</c><00:30:40.000><c> the</c><00:30:40.399><c> position</c><00:30:40.799><c> reduces</c>

29:09.275 --> 29:09.285 align:start position:0%
also reduces. So the position reduces
 

29:09.285 --> 29:11.514 align:start position:0%
also reduces. So the position reduces
slower<00:30:42.000><c> the</c><00:30:42.159><c> velocity</c><00:30:42.559><c> reduces</c><00:30:42.960><c> even</c><00:30:43.200><c> slower</c>

29:11.514 --> 29:11.524 align:start position:0%
slower the velocity reduces even slower
 

29:11.524 --> 29:15.595 align:start position:0%
slower the velocity reduces even slower
and<00:30:43.919><c> you</c><00:30:44.159><c> get</c><00:30:44.320><c> a</c><00:30:44.640><c> decay.</c><00:30:45.840><c> But</c><00:30:46.000><c> Argan</c><00:30:46.480><c> asks</c><00:30:47.520><c> what</c>

29:15.595 --> 29:15.605 align:start position:0%
and you get a decay. But Argan asks what
 

29:15.605 --> 29:20.795 align:start position:0%
and you get a decay. But Argan asks what
if<00:30:48.559><c> somehow</c><00:30:49.760><c> we</c><00:30:50.080><c> could</c><00:30:50.399><c> make</c><00:30:51.279><c> the</c><00:30:51.600><c> velocity</c>

29:20.795 --> 29:20.805 align:start position:0%
if somehow we could make the velocity
 

29:20.805 --> 29:22.874 align:start position:0%
if somehow we could make the velocity
perpendicular

29:22.874 --> 29:22.884 align:start position:0%
perpendicular
 

29:22.884 --> 29:26.874 align:start position:0%
perpendicular
to<00:30:56.000><c> the</c><00:30:56.320><c> position.</c>

29:26.874 --> 29:26.884 align:start position:0%
to the position.
 

29:26.884 --> 29:30.874 align:start position:0%
to the position.
This<00:30:59.360><c> time</c><00:31:00.240><c> a</c><00:31:00.480><c> little</c><00:31:00.720><c> later,</c><00:31:01.520><c> the</c><00:31:01.919><c> position</c>

29:30.874 --> 29:30.884 align:start position:0%
This time a little later, the position
 

29:30.884 --> 29:33.195 align:start position:0%
This time a little later, the position
neither<00:31:03.440><c> grows</c><00:31:03.760><c> nor</c><00:31:04.080><c> shrinks,</c><00:31:04.399><c> but</c><00:31:04.720><c> it</c><00:31:04.960><c> just</c>

29:33.195 --> 29:33.205 align:start position:0%
neither grows nor shrinks, but it just
 

29:33.205 --> 29:35.835 align:start position:0%
neither grows nor shrinks, but it just
shifts<00:31:05.840><c> sideways,</c><00:31:07.039><c> which</c><00:31:07.279><c> means</c><00:31:07.679><c> the</c>

29:35.835 --> 29:35.845 align:start position:0%
shifts sideways, which means the
 

29:35.845 --> 29:38.394 align:start position:0%
shifts sideways, which means the
velocity<00:31:08.880><c> magnitude</c><00:31:09.520><c> stays</c><00:31:09.919><c> the</c><00:31:10.159><c> same</c><00:31:10.240><c> as</c>

29:38.394 --> 29:38.404 align:start position:0%
velocity magnitude stays the same as
 

29:38.404 --> 29:40.394 align:start position:0%
velocity magnitude stays the same as
well.<00:31:11.360><c> And</c><00:31:11.600><c> it</c><00:31:11.840><c> continues</c><00:31:12.240><c> to</c><00:31:12.399><c> stay</c>

29:40.394 --> 29:40.404 align:start position:0%
well. And it continues to stay
 

29:40.404 --> 29:42.555 align:start position:0%
well. And it continues to stay
perpendicular,<00:31:13.600><c> meaning</c><00:31:13.840><c> it</c><00:31:14.159><c> keeps</c><00:31:14.399><c> pushing</c>

29:42.555 --> 29:42.565 align:start position:0%
perpendicular, meaning it keeps pushing
 

29:42.565 --> 29:45.595 align:start position:0%
perpendicular, meaning it keeps pushing
it<00:31:15.200><c> sidewards</c><00:31:16.000><c> forever.</c>

29:45.595 --> 29:45.605 align:start position:0%
it sidewards forever.
 

29:45.605 --> 29:49.754 align:start position:0%
it sidewards forever.
That<00:31:18.559><c> is</c><00:31:18.720><c> a</c><00:31:19.120><c> uniform</c><00:31:19.840><c> circular</c><00:31:20.399><c> motion.</c>

29:49.754 --> 29:49.764 align:start position:0%
That is a uniform circular motion.
 

29:49.764 --> 29:53.915 align:start position:0%
That is a uniform circular motion.
That<00:31:22.559><c> will</c><00:31:22.880><c> make</c><00:31:23.200><c> our</c><00:31:23.600><c> exponential</c><00:31:24.720><c> periodic.</c>

29:53.915 --> 29:53.925 align:start position:0%
That will make our exponential periodic.
 

29:53.925 --> 29:56.315 align:start position:0%
That will make our exponential periodic.
I<00:31:26.320><c> mean,</c><00:31:26.480><c> sure,</c><00:31:26.799><c> it's</c><00:31:27.039><c> not</c><00:31:27.440><c> oscillating,</c><00:31:28.159><c> but</c>

29:56.315 --> 29:56.325 align:start position:0%
I mean, sure, it's not oscillating, but
 

29:56.325 --> 29:58.394 align:start position:0%
I mean, sure, it's not oscillating, but
it<00:31:28.720><c> that's</c><00:31:29.039><c> okay.</c><00:31:29.440><c> It's</c><00:31:29.760><c> periodic.</c><00:31:30.399><c> That's</c>

29:58.394 --> 29:58.404 align:start position:0%
it that's okay. It's periodic. That's
 

29:58.404 --> 30:01.195 align:start position:0%
it that's okay. It's periodic. That's
what<00:31:30.720><c> I</c><00:31:30.880><c> want.</c><00:31:32.240><c> But</c><00:31:32.399><c> Argan,</c><00:31:32.880><c> how</c><00:31:33.039><c> do</c><00:31:33.120><c> we</c>

30:01.195 --> 30:01.205 align:start position:0%
what I want. But Argan, how do we
 

30:01.205 --> 30:04.315 align:start position:0%
what I want. But Argan, how do we
control<00:31:33.760><c> the</c><00:31:34.159><c> direction</c><00:31:34.880><c> of</c><00:31:35.440><c> this</c><00:31:36.320><c> um</c>

30:04.315 --> 30:04.325 align:start position:0%
control the direction of this um
 

30:04.325 --> 30:05.595 align:start position:0%
control the direction of this um
velocity?<00:31:36.960><c> And</c><00:31:37.120><c> how</c><00:31:37.279><c> do</c><00:31:37.360><c> we</c><00:31:37.519><c> make</c><00:31:37.600><c> it</c>

30:05.595 --> 30:05.605 align:start position:0%
velocity? And how do we make it
 

30:05.605 --> 30:08.155 align:start position:0%
velocity? And how do we make it
perpendicular?<00:31:39.039><c> Well,</c><00:31:39.200><c> Argan</c><00:31:39.600><c> says,</c><00:31:40.080><c> think</c>

30:08.155 --> 30:08.165 align:start position:0%
perpendicular? Well, Argan says, think
 

30:08.165 --> 30:10.155 align:start position:0%
perpendicular? Well, Argan says, think
about<00:31:40.559><c> this.</c><00:31:41.279><c> When</c><00:31:41.519><c> we</c><00:31:41.679><c> multiply</c><00:31:42.080><c> the</c>

30:10.155 --> 30:10.165 align:start position:0%
about this. When we multiply the
 

30:10.165 --> 30:12.955 align:start position:0%
about this. When we multiply the
exponent<00:31:42.720><c> by</c><00:31:43.039><c> one,</c><00:31:43.760><c> the</c><00:31:44.000><c> velocity</c><00:31:44.559><c> points</c><00:31:44.960><c> in</c>

30:12.955 --> 30:12.965 align:start position:0%
exponent by one, the velocity points in
 

30:12.965 --> 30:15.754 align:start position:0%
exponent by one, the velocity points in
the<00:31:45.360><c> same</c><00:31:45.519><c> direction</c><00:31:45.840><c> as</c><00:31:46.080><c> position.</c><00:31:47.279><c> That's</c><00:31:47.679><c> a</c>

30:15.754 --> 30:15.764 align:start position:0%
the same direction as position. That's a
 

30:15.764 --> 30:18.394 align:start position:0%
the same direction as position. That's a
0°<00:31:48.559><c> rotation</c><00:31:48.799><c> of</c><00:31:48.960><c> the</c><00:31:49.120><c> velocity.</c><00:31:50.159><c> When</c><00:31:50.399><c> we</c>

30:18.394 --> 30:18.404 align:start position:0%
0° rotation of the velocity. When we
 

30:18.404 --> 30:20.715 align:start position:0%
0° rotation of the velocity. When we
multiply<00:31:50.960><c> by</c><00:31:51.519><c> -1,</c><00:31:52.080><c> it</c><00:31:52.320><c> points</c><00:31:52.559><c> in</c><00:31:52.720><c> the</c>

30:20.715 --> 30:20.725 align:start position:0%
multiply by -1, it points in the
 

30:20.725 --> 30:22.394 align:start position:0%
multiply by -1, it points in the
opposite<00:31:53.200><c> direction.</c><00:31:53.679><c> So</c><00:31:53.919><c> this</c><00:31:54.159><c> rotates</c>

30:22.394 --> 30:22.404 align:start position:0%
opposite direction. So this rotates
 

30:22.404 --> 30:26.234 align:start position:0%
opposite direction. So this rotates
velocity<00:31:54.960><c> by</c><00:31:55.600><c> 180°</c>

30:26.234 --> 30:26.244 align:start position:0%
velocity by 180°
 

30:26.244 --> 30:28.555 align:start position:0%
velocity by 180°
which<00:31:58.720><c> means</c><00:31:59.279><c> we</c><00:31:59.600><c> need</c><00:31:59.760><c> to</c><00:31:59.919><c> multiply</c><00:32:00.480><c> the</c>

30:28.555 --> 30:28.565 align:start position:0%
which means we need to multiply the
 

30:28.565 --> 30:31.354 align:start position:0%
which means we need to multiply the
exponent<00:32:01.200><c> with</c><00:32:01.440><c> some</c><00:32:01.760><c> number</c><00:32:02.640><c> that</c><00:32:03.039><c> rotates</c>

30:31.354 --> 30:31.364 align:start position:0%
exponent with some number that rotates
 

30:31.364 --> 30:33.915 align:start position:0%
exponent with some number that rotates
the<00:32:03.760><c> velocity</c><00:32:04.159><c> by</c><00:32:04.559><c> 90°.</c><00:32:05.440><c> How</c><00:32:05.600><c> do</c><00:32:05.760><c> we</c><00:32:05.919><c> find</c>

30:33.915 --> 30:33.925 align:start position:0%
the velocity by 90°. How do we find
 

30:33.925 --> 30:36.634 align:start position:0%
the velocity by 90°. How do we find
that?<00:32:06.799><c> Well</c><00:32:07.360><c> say</c><00:32:07.440><c> is</c><00:32:07.600><c> simple.</c><00:32:08.399><c> Whatever</c>

30:36.634 --> 30:36.644 align:start position:0%
that? Well say is simple. Whatever
 

30:36.644 --> 30:38.715 align:start position:0%
that? Well say is simple. Whatever
number<00:32:09.039><c> this</c><00:32:09.360><c> is</c><00:32:09.760><c> if</c><00:32:10.000><c> you</c><00:32:10.159><c> multiply</c><00:32:10.559><c> it</c><00:32:10.720><c> by</c>

30:38.715 --> 30:38.725 align:start position:0%
number this is if you multiply it by
 

30:38.725 --> 30:41.514 align:start position:0%
number this is if you multiply it by
itself<00:32:11.519><c> one</c><00:32:11.840><c> more</c><00:32:12.159><c> time</c><00:32:13.039><c> well</c><00:32:13.360><c> it</c><00:32:13.519><c> would</c>

30:41.514 --> 30:41.524 align:start position:0%
itself one more time well it would
 

30:41.524 --> 30:47.114 align:start position:0%
itself one more time well it would
rotate<00:32:14.080><c> again</c><00:32:14.559><c> by</c><00:32:14.880><c> 90°</c><00:32:16.320><c> meaning</c><00:32:17.519><c> 180°</c>

30:47.114 --> 30:47.124 align:start position:0%
rotate again by 90° meaning 180°
 

30:47.124 --> 30:49.354 align:start position:0%
rotate again by 90° meaning 180°
but<00:32:19.519><c> we</c><00:32:19.760><c> already</c><00:32:20.159><c> know</c><00:32:20.559><c> that</c><00:32:20.799><c> is</c><00:32:21.039><c> negative</c>

30:49.354 --> 30:49.364 align:start position:0%
but we already know that is negative
 

30:49.364 --> 30:51.915 align:start position:0%
but we already know that is negative
one.<00:32:22.080><c> So</c><00:32:22.320><c> in</c><00:32:22.559><c> other</c><00:32:22.720><c> words,</c><00:32:23.440><c> whatever</c><00:32:23.760><c> this</c>

30:51.915 --> 30:51.925 align:start position:0%
one. So in other words, whatever this
 

30:51.925 --> 30:54.555 align:start position:0%
one. So in other words, whatever this
number<00:32:24.320><c> is,</c><00:32:25.120><c> it</c><00:32:25.440><c> multiplied</c><00:32:25.919><c> by</c><00:32:26.159><c> itself</c>

30:54.555 --> 30:54.565 align:start position:0%
number is, it multiplied by itself
 

30:54.565 --> 30:58.475 align:start position:0%
number is, it multiplied by itself
should<00:32:26.960><c> give</c><00:32:27.120><c> me</c><00:32:27.360><c> -1</c><00:32:28.799><c> or</c><00:32:29.440><c> the</c><00:32:29.840><c> square</c><00:32:30.240><c> of</c><00:32:30.480><c> that</c>

30:58.475 --> 30:58.485 align:start position:0%
should give me -1 or the square of that
 

30:58.485 --> 31:01.675 align:start position:0%
should give me -1 or the square of that
number<00:32:30.960><c> should</c><00:32:31.200><c> give</c><00:32:31.360><c> me</c><00:32:31.679><c> -1</c><00:32:32.960><c> or</c><00:32:33.200><c> that</c><00:32:33.519><c> number</c>

31:01.675 --> 31:01.685 align:start position:0%
number should give me -1 or that number
 

31:01.685 --> 31:04.715 align:start position:0%
number should give me -1 or that number
which<00:32:34.080><c> rotates</c><00:32:34.480><c> by</c><00:32:34.640><c> 90°</c><00:32:35.440><c> is</c><00:32:35.760><c> the</c><00:32:36.159><c> square</c><00:32:36.559><c> root</c>

31:04.715 --> 31:04.725 align:start position:0%
which rotates by 90° is the square root
 

31:04.725 --> 31:07.035 align:start position:0%
which rotates by 90° is the square root
of<00:32:37.519><c> -1.</c>

31:07.035 --> 31:07.045 align:start position:0%
of -1.
 

31:07.045 --> 31:14.315 align:start position:0%
of -1.
The<00:32:39.600><c> I</c><00:32:40.720><c> has</c><00:32:41.200><c> entered</c><00:32:42.320><c> the</c><00:32:42.720><c> room.</c><00:32:43.919><c> Oh,</c>

31:14.315 --> 31:14.325 align:start position:0%
The I has entered the room. Oh,
 

31:14.325 --> 31:17.915 align:start position:0%
The I has entered the room. Oh,
that's<00:32:47.600><c> that's</c><00:32:48.000><c> where</c><00:32:48.159><c> it</c><00:32:48.559><c> ah</c><00:32:49.519><c> that's</c><00:32:49.840><c> where</c>

31:17.915 --> 31:17.925 align:start position:0%
that's that's where it ah that's where
 

31:17.925 --> 31:20.715 align:start position:0%
that's that's where it ah that's where
it<00:32:50.240><c> comes</c><00:32:50.480><c> from.</c>

31:20.715 --> 31:20.725 align:start position:0%
it comes from.
 

31:20.725 --> 31:22.715 align:start position:0%
it comes from.
I'm<00:32:53.200><c> sorry.</c><00:32:53.519><c> Let's</c><00:32:53.679><c> calm</c><00:32:53.919><c> down.</c><00:32:54.399><c> But</c><00:32:54.559><c> what</c><00:32:54.720><c> is</c>

31:22.715 --> 31:22.725 align:start position:0%
I'm sorry. Let's calm down. But what is
 

31:22.725 --> 31:25.195 align:start position:0%
I'm sorry. Let's calm down. But what is
the<00:32:54.960><c> I</c><00:32:55.279><c> doing</c><00:32:55.440><c> over</c><00:32:55.679><c> here?</c><00:32:56.320><c> The</c><00:32:56.640><c> I</c><00:32:56.960><c> in</c><00:32:57.200><c> the</c>

31:25.195 --> 31:25.205 align:start position:0%
the I doing over here? The I in the
 

31:25.205 --> 31:28.075 align:start position:0%
the I doing over here? The I in the
exponent<00:32:57.840><c> produces</c><00:32:58.240><c> in</c><00:32:58.480><c> 90°</c><00:32:59.039><c> rotation</c><00:32:59.919><c> making</c>

31:28.075 --> 31:28.085 align:start position:0%
exponent produces in 90° rotation making
 

31:28.085 --> 31:31.035 align:start position:0%
exponent produces in 90° rotation making
our<00:33:00.480><c> exponential</c><00:33:01.039><c> function</c><00:33:01.440><c> spin</c><00:33:02.080><c> making</c><00:33:02.559><c> it</c>

31:31.035 --> 31:31.045 align:start position:0%
our exponential function spin making it
 

31:31.045 --> 31:33.114 align:start position:0%
our exponential function spin making it
periodic.<00:33:04.000><c> Now,</c><00:33:04.240><c> of</c><00:33:04.399><c> course,</c><00:33:04.559><c> this</c><00:33:04.799><c> rotation</c>

31:33.114 --> 31:33.124 align:start position:0%
periodic. Now, of course, this rotation
 

31:33.124 --> 31:36.155 align:start position:0%
periodic. Now, of course, this rotation
is<00:33:05.519><c> not</c><00:33:05.679><c> in</c><00:33:06.080><c> real</c><00:33:06.480><c> space.</c><00:33:06.960><c> This</c><00:33:07.200><c> is</c><00:33:07.360><c> a</c><00:33:07.679><c> complex</c>

31:36.155 --> 31:36.165 align:start position:0%
is not in real space. This is a complex
 

31:36.165 --> 31:38.315 align:start position:0%
is not in real space. This is a complex
plane.<00:33:09.039><c> So,</c><00:33:09.279><c> this</c><00:33:09.440><c> is</c><00:33:09.519><c> the</c><00:33:09.679><c> real</c><00:33:09.919><c> axis</c><00:33:10.240><c> and</c>

31:38.315 --> 31:38.325 align:start position:0%
plane. So, this is the real axis and
 

31:38.325 --> 31:41.035 align:start position:0%
plane. So, this is the real axis and
this<00:33:10.640><c> is</c><00:33:10.799><c> the</c><00:33:11.440><c> um</c><00:33:11.600><c> imaginary</c><00:33:12.159><c> axis.</c><00:33:12.960><c> Now</c>

31:41.035 --> 31:41.045 align:start position:0%
this is the um imaginary axis. Now
 

31:41.045 --> 31:43.114 align:start position:0%
this is the um imaginary axis. Now
because<00:33:13.440><c> of</c><00:33:13.679><c> Argan's</c><00:33:14.399><c> beautiful</c><00:33:14.799><c> geometric</c>

31:43.114 --> 31:43.124 align:start position:0%
because of Argan's beautiful geometric
 

31:43.124 --> 31:45.595 align:start position:0%
because of Argan's beautiful geometric
insight<00:33:16.080><c> into</c><00:33:16.399><c> these</c><00:33:16.559><c> complex</c><00:33:16.960><c> exponentials,</c>

31:45.595 --> 31:45.605 align:start position:0%
insight into these complex exponentials,
 

31:45.605 --> 31:48.155 align:start position:0%
insight into these complex exponentials,
we<00:33:18.080><c> call</c><00:33:18.240><c> this</c><00:33:19.039><c> complex</c><00:33:19.440><c> plane</c><00:33:19.679><c> the</c><00:33:19.919><c> Argon</c>

31:48.155 --> 31:48.165 align:start position:0%
we call this complex plane the Argon
 

31:48.165 --> 31:53.275 align:start position:0%
we call this complex plane the Argon
diagram.<00:33:21.200><c> So</c><00:33:22.480><c> just</c><00:33:22.720><c> to</c><00:33:23.360><c> recap,</c><00:33:24.399><c> we</c><00:33:24.960><c> started</c>

31:53.275 --> 31:53.285 align:start position:0%
diagram. So just to recap, we started
 

31:53.285 --> 31:56.394 align:start position:0%
diagram. So just to recap, we started
with<00:33:25.919><c> microwaves</c><00:33:26.720><c> oscillating</c><00:33:27.360><c> up</c><00:33:27.600><c> and</c><00:33:27.840><c> down</c>

31:56.394 --> 31:56.404 align:start position:0%
with microwaves oscillating up and down
 

31:56.404 --> 31:59.435 align:start position:0%
with microwaves oscillating up and down
and<00:33:28.880><c> we</c><00:33:29.200><c> found</c><00:33:29.360><c> that</c><00:33:30.000><c> the</c><00:33:30.320><c> math</c><00:33:30.880><c> didn't</c><00:33:31.279><c> work.</c>

31:59.435 --> 31:59.445 align:start position:0%
and we found that the math didn't work.
 

31:59.445 --> 32:01.195 align:start position:0%
and we found that the math didn't work.
We<00:33:31.840><c> couldn't</c><00:33:32.080><c> cleanly</c><00:33:32.399><c> extract</c><00:33:32.720><c> the</c><00:33:32.960><c> energy.</c>

32:01.195 --> 32:01.205 align:start position:0%
We couldn't cleanly extract the energy.
 

32:01.205 --> 32:02.955 align:start position:0%
We couldn't cleanly extract the energy.
Now<00:33:33.600><c> to</c><00:33:33.760><c> make</c><00:33:33.840><c> the</c><00:33:34.000><c> math</c><00:33:34.320><c> work,</c><00:33:34.559><c> we</c><00:33:34.799><c> needed</c>

32:02.955 --> 32:02.965 align:start position:0%
Now to make the math work, we needed
 

32:02.965 --> 32:06.394 align:start position:0%
Now to make the math work, we needed
exponentials,<00:33:36.399><c> but</c><00:33:36.720><c> they</c><00:33:36.960><c> aren't</c><00:33:37.279><c> periodic.</c>

32:06.394 --> 32:06.404 align:start position:0%
exponentials, but they aren't periodic.
 

32:06.404 --> 32:09.195 align:start position:0%
exponentials, but they aren't periodic.
Argan<00:33:38.960><c> gave</c><00:33:39.120><c> us</c><00:33:39.279><c> a</c><00:33:39.440><c> way</c><00:33:39.600><c> out.</c><00:33:40.640><c> complex</c>

32:09.195 --> 32:09.205 align:start position:0%
Argan gave us a way out. complex
 

32:09.205 --> 32:11.514 align:start position:0%
Argan gave us a way out. complex
exponentials.<00:33:42.640><c> Now,</c><00:33:42.880><c> the</c><00:33:43.039><c> math</c><00:33:43.360><c> works</c>

32:11.514 --> 32:11.524 align:start position:0%
exponentials. Now, the math works
 

32:11.524 --> 32:13.354 align:start position:0%
exponentials. Now, the math works
because<00:33:44.000><c> I</c><00:33:44.159><c> can</c><00:33:44.320><c> cleanly</c><00:33:44.720><c> extract</c><00:33:45.120><c> energy</c>

32:13.354 --> 32:13.364 align:start position:0%
because I can cleanly extract energy
 

32:13.364 --> 32:15.114 align:start position:0%
because I can cleanly extract energy
because<00:33:45.679><c> it's</c><00:33:45.919><c> an</c><00:33:46.000><c> exponential</c><00:33:46.480><c> function.</c>

32:15.114 --> 32:15.124 align:start position:0%
because it's an exponential function.
 

32:15.124 --> 32:17.994 align:start position:0%
because it's an exponential function.
But<00:33:47.440><c> the</c><00:33:48.080><c> physics</c><00:33:48.480><c> works</c><00:33:48.799><c> too</c><00:33:49.039><c> because</c><00:33:49.919><c> the</c>

32:17.994 --> 32:18.004 align:start position:0%
But the physics works too because the
 

32:18.004 --> 32:22.234 align:start position:0%
But the physics works too because the
function<00:33:50.480><c> has</c><00:33:50.720><c> become</c><00:33:51.039><c> periodic.</c>

32:22.234 --> 32:22.244 align:start position:0%
 
 

32:22.244 --> 32:23.835 align:start position:0%
 
This<00:33:54.640><c> means</c><00:33:54.960><c> now</c><00:33:55.200><c> we</c><00:33:55.360><c> have</c><00:33:55.440><c> to</c><00:33:55.600><c> accept</c><00:33:55.840><c> that</c>

32:23.835 --> 32:23.845 align:start position:0%
This means now we have to accept that
 

32:23.845 --> 32:25.754 align:start position:0%
This means now we have to accept that
our<00:33:56.240><c> microwave</c><00:33:56.880><c> is</c><00:33:57.039><c> rotating</c><00:33:57.440><c> in</c><00:33:57.600><c> some</c><00:33:57.760><c> kind</c>

32:25.754 --> 32:25.764 align:start position:0%
our microwave is rotating in some kind
 

32:25.764 --> 32:28.394 align:start position:0%
our microwave is rotating in some kind
of<00:33:58.000><c> an</c><00:33:58.240><c> abstract</c><00:33:58.720><c> complex</c><00:33:59.279><c> plane.</c><00:34:00.159><c> But</c><00:34:00.320><c> that's</c>

32:28.394 --> 32:28.404 align:start position:0%
of an abstract complex plane. But that's
 

32:28.404 --> 32:31.835 align:start position:0%
of an abstract complex plane. But that's
okay.<00:34:01.039><c> At</c><00:34:01.039><c> least</c><00:34:01.360><c> we</c><00:34:01.840><c> found</c><00:34:02.159><c> a</c><00:34:02.399><c> way</c><00:34:02.640><c> out.</c><00:34:03.600><c> So,</c>

32:31.835 --> 32:31.845 align:start position:0%
okay. At least we found a way out. So,
 

32:31.845 --> 32:34.394 align:start position:0%
okay. At least we found a way out. So,
let's<00:34:04.559><c> run</c><00:34:04.880><c> with</c><00:34:05.120><c> it</c><00:34:05.360><c> now.</c><00:34:06.000><c> So</c><00:34:06.240><c> what</c><00:34:06.399><c> we're</c>

32:34.394 --> 32:34.404 align:start position:0%
let's run with it now. So what we're
 

32:34.404 --> 32:37.194 align:start position:0%
let's run with it now. So what we're
going<00:34:06.640><c> to</c><00:34:06.799><c> do</c><00:34:07.039><c> is</c><00:34:07.360><c> let's</c><00:34:07.840><c> use</c><00:34:08.560><c> this</c><00:34:08.960><c> complex</c>

32:37.194 --> 32:37.204 align:start position:0%
going to do is let's use this complex
 

32:37.204 --> 32:40.315 align:start position:0%
going to do is let's use this complex
exponential<00:34:09.919><c> as</c><00:34:10.159><c> our</c><00:34:10.480><c> basic</c><00:34:11.040><c> wave.</c><00:34:11.839><c> Okay.</c><00:34:12.320><c> Now</c>

32:40.315 --> 32:40.325 align:start position:0%
exponential as our basic wave. Okay. Now
 

32:40.325 --> 32:42.795 align:start position:0%
exponential as our basic wave. Okay. Now
the<00:34:12.720><c> first</c><00:34:12.879><c> question</c><00:34:13.119><c> is</c><00:34:13.679><c> does</c><00:34:13.919><c> that</c><00:34:14.480><c> change</c>

32:42.795 --> 32:42.805 align:start position:0%
the first question is does that change
 

32:42.805 --> 32:45.035 align:start position:0%
the first question is does that change
anything<00:34:15.359><c> that</c><00:34:15.599><c> we</c><00:34:15.760><c> did</c><00:34:16.000><c> so</c><00:34:16.320><c> far?</c><00:34:16.960><c> Well</c>

32:45.035 --> 32:45.045 align:start position:0%
anything that we did so far? Well
 

32:45.045 --> 32:46.635 align:start position:0%
anything that we did so far? Well
remember<00:34:17.520><c> we</c><00:34:17.760><c> are</c><00:34:17.839><c> taking</c><00:34:18.079><c> a</c><00:34:18.399><c> partial</c>

32:46.635 --> 32:46.645 align:start position:0%
remember we are taking a partial
 

32:46.645 --> 32:48.474 align:start position:0%
remember we are taking a partial
derivative<00:34:19.280><c> with</c><00:34:19.440><c> respect</c><00:34:19.760><c> to</c><00:34:19.919><c> x</c><00:34:20.159><c> and</c><00:34:20.320><c> when</c><00:34:20.480><c> we</c>

32:48.474 --> 32:48.484 align:start position:0%
derivative with respect to x and when we
 

32:48.484 --> 32:50.395 align:start position:0%
derivative with respect to x and when we
do<00:34:20.800><c> that</c><00:34:21.119><c> the</c><00:34:21.359><c> time</c><00:34:21.599><c> part</c><00:34:21.839><c> is</c><00:34:21.919><c> treated</c><00:34:22.240><c> as</c><00:34:22.399><c> a</c>

32:50.395 --> 32:50.405 align:start position:0%
do that the time part is treated as a
 

32:50.405 --> 32:52.234 align:start position:0%
do that the time part is treated as a
constant.<00:34:22.960><c> So</c><00:34:23.119><c> whether</c><00:34:23.440><c> we</c><00:34:23.599><c> have</c><00:34:23.679><c> a</c><00:34:23.839><c> sign</c><00:34:24.159><c> here</c>

32:52.234 --> 32:52.244 align:start position:0%
constant. So whether we have a sign here
 

32:52.244 --> 32:54.635 align:start position:0%
constant. So whether we have a sign here
or<00:34:25.040><c> you</c><00:34:25.119><c> know</c><00:34:25.280><c> exponential</c><00:34:25.839><c> over</c><00:34:26.000><c> here</c><00:34:26.560><c> it</c>

32:54.635 --> 32:54.645 align:start position:0%
or you know exponential over here it
 

32:54.645 --> 32:57.194 align:start position:0%
or you know exponential over here it
doesn't<00:34:27.119><c> matter.</c><00:34:28.000><c> So</c><00:34:28.480><c> everything</c><00:34:28.800><c> stays</c><00:34:29.119><c> the</c>

32:57.194 --> 32:57.204 align:start position:0%
doesn't matter. So everything stays the
 

32:57.204 --> 32:58.714 align:start position:0%
doesn't matter. So everything stays the
same<00:34:29.520><c> and</c><00:34:29.679><c> so</c><00:34:29.919><c> nothing</c><00:34:30.079><c> changes</c><00:34:30.399><c> over</c><00:34:30.639><c> here.</c>

32:58.714 --> 32:58.724 align:start position:0%
same and so nothing changes over here.
 

32:58.724 --> 33:00.155 align:start position:0%
same and so nothing changes over here.
So<00:34:31.040><c> that's</c><00:34:31.280><c> awesome.</c><00:34:31.679><c> I</c><00:34:31.919><c> don't</c><00:34:32.000><c> have</c><00:34:32.079><c> to</c><00:34:32.240><c> do</c>

33:00.155 --> 33:00.165 align:start position:0%
So that's awesome. I don't have to do
 

33:00.165 --> 33:03.275 align:start position:0%
So that's awesome. I don't have to do
any<00:34:32.480><c> more</c><00:34:32.639><c> work</c><00:34:32.879><c> over</c><00:34:33.119><c> here.</c><00:34:34.000><c> But</c><00:34:34.320><c> now</c><00:34:35.119><c> it's</c>

33:03.275 --> 33:03.285 align:start position:0%
any more work over here. But now it's
 

33:03.285 --> 33:06.875 align:start position:0%
any more work over here. But now it's
time<00:34:35.679><c> to</c><00:34:36.000><c> build</c><00:34:36.240><c> the</c><00:34:36.639><c> energy</c><00:34:37.359><c> operator.</c><00:34:38.480><c> So</c><00:34:38.800><c> if</c>

33:06.875 --> 33:06.885 align:start position:0%
time to build the energy operator. So if
 

33:06.885 --> 33:08.635 align:start position:0%
time to build the energy operator. So if
you<00:34:39.200><c> differentiate</c><00:34:39.599><c> with</c><00:34:39.760><c> respect</c><00:34:40.000><c> to</c><00:34:40.240><c> time,</c>

33:08.635 --> 33:08.645 align:start position:0%
you differentiate with respect to time,
 

33:08.645 --> 33:11.434 align:start position:0%
you differentiate with respect to time,
we<00:34:41.200><c> get</c><00:34:41.520><c> energy</c><00:34:42.000><c> extracted</c><00:34:42.720><c> and</c><00:34:42.960><c> we</c><00:34:43.200><c> get</c><00:34:43.280><c> our</c>

33:11.434 --> 33:11.444 align:start position:0%
we get energy extracted and we get our
 

33:11.444 --> 33:14.635 align:start position:0%
we get energy extracted and we get our
function<00:34:44.000><c> back.</c><00:34:44.720><c> So</c><00:34:44.800><c> I</c><00:34:45.040><c> get</c><00:34:45.200><c> my</c><00:34:45.440><c> S</c><00:34:45.839><c> cleanly.</c><00:34:46.639><c> So</c>

33:14.635 --> 33:14.645 align:start position:0%
function back. So I get my S cleanly. So
 

33:14.645 --> 33:17.595 align:start position:0%
function back. So I get my S cleanly. So
it's<00:34:47.040><c> time</c><00:34:47.200><c> to</c><00:34:47.440><c> rearrange</c><00:34:48.800><c> and</c><00:34:49.119><c> isolate</c>

33:17.595 --> 33:17.605 align:start position:0%
it's time to rearrange and isolate
 

33:17.605 --> 33:19.674 align:start position:0%
it's time to rearrange and isolate
energy<00:34:50.159><c> over</c><00:34:50.480><c> here.</c><00:34:50.960><c> And</c><00:34:51.119><c> if</c><00:34:51.280><c> I</c><00:34:51.440><c> multiply</c>

33:19.674 --> 33:19.684 align:start position:0%
energy over here. And if I multiply
 

33:19.684 --> 33:23.914 align:start position:0%
energy over here. And if I multiply
numerator<00:34:52.159><c> and</c><00:34:52.320><c> denominator</c><00:34:52.800><c> by</c><00:34:53.119><c> I,</c><00:34:54.159><c> we</c><00:34:55.200><c> have</c>

33:23.914 --> 33:23.924 align:start position:0%
numerator and denominator by I, we have
 

33:23.924 --> 33:27.275 align:start position:0%
numerator and denominator by I, we have
done<00:34:56.320><c> it.</c><00:34:57.040><c> We</c><00:34:57.359><c> have</c><00:34:57.680><c> cleanly</c><00:34:58.560><c> extracted</c>

33:27.275 --> 33:27.285 align:start position:0%
done it. We have cleanly extracted
 

33:27.285 --> 33:34.795 align:start position:0%
done it. We have cleanly extracted
energy.<00:35:00.720><c> This</c><00:35:01.599><c> is</c><00:35:02.480><c> the</c><00:35:02.880><c> energy</c><00:35:03.839><c> operator.</c>

33:34.795 --> 33:34.805 align:start position:0%
 
 

33:34.805 --> 33:36.635 align:start position:0%
 
But<00:35:07.119><c> wait,</c><00:35:07.520><c> when</c><00:35:07.760><c> I</c><00:35:07.920><c> saw</c><00:35:08.000><c> this,</c><00:35:08.240><c> I</c><00:35:08.400><c> was</c><00:35:08.560><c> like,</c>

33:36.635 --> 33:36.645 align:start position:0%
But wait, when I saw this, I was like,
 

33:36.645 --> 33:38.155 align:start position:0%
But wait, when I saw this, I was like,
wait<00:35:08.960><c> a</c><00:35:09.119><c> second,</c><00:35:09.359><c> why</c><00:35:09.440><c> is</c><00:35:09.599><c> it</c><00:35:09.760><c> a</c><00:35:09.920><c> negative</c>

33:38.155 --> 33:38.165 align:start position:0%
wait a second, why is it a negative
 

33:38.165 --> 33:40.155 align:start position:0%
wait a second, why is it a negative
sign?<00:35:10.720><c> I</c><00:35:10.960><c> mean,</c><00:35:11.200><c> I</c><00:35:11.440><c> know</c><00:35:11.520><c> that</c><00:35:11.760><c> the</c><00:35:12.000><c> original</c>

33:40.155 --> 33:40.165 align:start position:0%
sign? I mean, I know that the original
 

33:40.165 --> 33:41.835 align:start position:0%
sign? I mean, I know that the original
Shortinger's<00:35:12.880><c> equation,</c><00:35:13.280><c> as</c><00:35:13.440><c> we</c><00:35:13.599><c> will</c><00:35:13.760><c> see,</c>

33:41.835 --> 33:41.845 align:start position:0%
Shortinger's equation, as we will see,
 

33:41.845 --> 33:43.755 align:start position:0%
Shortinger's equation, as we will see,
doesn't<00:35:14.160><c> have</c><00:35:14.320><c> a</c><00:35:14.480><c> negative</c><00:35:14.800><c> sign.</c><00:35:15.599><c> Well,</c><00:35:15.760><c> it</c>

33:43.755 --> 33:43.765 align:start position:0%
doesn't have a negative sign. Well, it
 

33:43.765 --> 33:45.995 align:start position:0%
doesn't have a negative sign. Well, it
turns<00:35:16.079><c> out</c><00:35:16.240><c> it's</c><00:35:16.400><c> a</c><00:35:16.640><c> small</c><00:35:16.960><c> convention</c><00:35:17.440><c> thing.</c>

33:45.995 --> 33:46.005 align:start position:0%
turns out it's a small convention thing.
 

33:46.005 --> 33:48.715 align:start position:0%
turns out it's a small convention thing.
We<00:35:18.640><c> chose</c><00:35:19.119><c> our</c><00:35:19.440><c> matter</c><00:35:19.839><c> wave</c><00:35:20.160><c> spinning</c><00:35:20.560><c> in</c><00:35:20.720><c> the</c>

33:48.715 --> 33:48.725 align:start position:0%
We chose our matter wave spinning in the
 

33:48.725 --> 33:51.115 align:start position:0%
We chose our matter wave spinning in the
clockwise<00:35:21.359><c> direction</c><00:35:21.680><c> as</c><00:35:21.920><c> seen</c><00:35:22.160><c> from</c><00:35:22.480><c> here.</c>

33:51.115 --> 33:51.125 align:start position:0%
clockwise direction as seen from here.
 

33:51.125 --> 33:52.475 align:start position:0%
clockwise direction as seen from here.
Well,<00:35:23.440><c> it</c><00:35:23.599><c> turns</c><00:35:23.760><c> out</c><00:35:23.920><c> physicists</c><00:35:24.320><c> like</c><00:35:24.480><c> to</c>

33:52.475 --> 33:52.485 align:start position:0%
Well, it turns out physicists like to
 

33:52.485 --> 33:54.555 align:start position:0%
Well, it turns out physicists like to
choose<00:35:24.960><c> anticlockwise</c><00:35:25.680><c> or</c><00:35:25.920><c> counterclockwise</c>

33:54.555 --> 33:54.565 align:start position:0%
choose anticlockwise or counterclockwise
 

33:54.565 --> 33:56.795 align:start position:0%
choose anticlockwise or counterclockwise
as<00:35:26.960><c> the</c><00:35:27.200><c> convention.</c><00:35:27.839><c> So</c><00:35:28.079><c> physicists</c><00:35:28.560><c> love</c><00:35:28.800><c> to</c>

33:56.795 --> 33:56.805 align:start position:0%
as the convention. So physicists love to
 

33:56.805 --> 33:58.715 align:start position:0%
as the convention. So physicists love to
choose<00:35:29.200><c> the</c><00:35:29.680><c> negative</c><00:35:30.079><c> sign</c><00:35:30.320><c> over</c><00:35:30.560><c> here</c><00:35:30.720><c> for</c>

33:58.715 --> 33:58.725 align:start position:0%
choose the negative sign over here for
 

33:58.725 --> 34:00.715 align:start position:0%
choose the negative sign over here for
their<00:35:31.040><c> exponentials</c><00:35:31.680><c> over</c><00:35:31.839><c> here.</c><00:35:32.400><c> So</c><00:35:32.640><c> this</c>

34:00.715 --> 34:00.725 align:start position:0%
their exponentials over here. So this
 

34:00.725 --> 34:02.715 align:start position:0%
their exponentials over here. So this
would<00:35:33.119><c> also</c><00:35:33.359><c> be</c><00:35:33.599><c> negative</c><00:35:34.240><c> and</c><00:35:34.480><c> so</c><00:35:34.640><c> you'll</c>

34:02.715 --> 34:02.725 align:start position:0%
would also be negative and so you'll
 

34:02.725 --> 34:04.235 align:start position:0%
would also be negative and so you'll
have<00:35:34.960><c> a</c><00:35:35.119><c> negative</c><00:35:35.440><c> popping</c><00:35:35.680><c> out</c><00:35:35.920><c> that</c><00:35:36.079><c> cancels</c>

34:04.235 --> 34:04.245 align:start position:0%
have a negative popping out that cancels
 

34:04.245 --> 34:05.674 align:start position:0%
have a negative popping out that cancels
with<00:35:36.640><c> this</c><00:35:36.800><c> one.</c><00:35:36.960><c> And</c><00:35:37.119><c> so</c><00:35:37.280><c> our</c><00:35:37.520><c> energy</c>

34:05.674 --> 34:05.684 align:start position:0%
with this one. And so our energy
 

34:05.684 --> 34:07.194 align:start position:0%
with this one. And so our energy
operator<00:35:38.320><c> wouldn't</c><00:35:38.640><c> have</c><00:35:38.720><c> a</c><00:35:38.880><c> negative</c><00:35:39.119><c> sign</c>

34:07.194 --> 34:07.204 align:start position:0%
operator wouldn't have a negative sign
 

34:07.204 --> 34:09.035 align:start position:0%
operator wouldn't have a negative sign
over<00:35:39.599><c> here.</c><00:35:40.000><c> That's</c><00:35:40.240><c> just</c><00:35:40.400><c> a</c><00:35:40.560><c> matter</c><00:35:40.800><c> of</c>

34:09.035 --> 34:09.045 align:start position:0%
over here. That's just a matter of
 

34:09.045 --> 34:13.035 align:start position:0%
over here. That's just a matter of
convention.<00:35:42.320><c> Anyways,</c><00:35:43.359><c> thanks</c><00:35:43.680><c> to</c><00:35:43.839><c> forier</c><00:35:44.960><c> I</c>

34:13.035 --> 34:13.045 align:start position:0%
convention. Anyways, thanks to forier I
 

34:13.045 --> 34:15.914 align:start position:0%
convention. Anyways, thanks to forier I
know<00:35:45.359><c> that</c><00:35:45.680><c> this</c><00:35:45.920><c> operator</c><00:35:46.480><c> works</c><00:35:46.800><c> in</c><00:35:47.119><c> general</c>

34:15.914 --> 34:15.924 align:start position:0%
know that this operator works in general
 

34:15.924 --> 34:18.395 align:start position:0%
know that this operator works in general
which<00:35:48.320><c> means</c><00:35:48.720><c> we</c><00:35:48.960><c> can</c><00:35:49.200><c> now</c><00:35:49.920><c> build</c><00:35:50.320><c> our</c>

34:18.395 --> 34:18.405 align:start position:0%
which means we can now build our
 

34:18.405 --> 34:20.875 align:start position:0%
which means we can now build our
shinger's<00:35:51.280><c> equation.</c><00:35:52.240><c> So</c><00:35:52.400><c> we</c><00:35:52.560><c> start</c><00:35:52.800><c> by</c>

34:20.875 --> 34:20.885 align:start position:0%
shinger's equation. So we start by
 

34:20.885 --> 34:22.715 align:start position:0%
shinger's equation. So we start by
energy<00:35:53.359><c> conservation.</c><00:35:54.000><c> Total</c><00:35:54.240><c> energy</c><00:35:54.560><c> equals</c>

34:22.715 --> 34:22.725 align:start position:0%
energy conservation. Total energy equals
 

34:22.725 --> 34:24.875 align:start position:0%
energy conservation. Total energy equals
kinetic<00:35:55.280><c> plus</c><00:35:55.599><c> potential.</c><00:35:56.560><c> And</c><00:35:56.720><c> then</c><00:35:56.880><c> for</c>

34:24.875 --> 34:24.885 align:start position:0%
kinetic plus potential. And then for
 

34:24.885 --> 34:27.194 align:start position:0%
kinetic plus potential. And then for
total<00:35:57.440><c> energy</c><00:35:57.920><c> we</c><00:35:58.240><c> substitute</c><00:35:58.800><c> the</c><00:35:58.960><c> energy</c>

34:27.194 --> 34:27.204 align:start position:0%
total energy we substitute the energy
 

34:27.204 --> 34:29.035 align:start position:0%
total energy we substitute the energy
operator.<00:36:00.079><c> For</c><00:36:00.240><c> the</c><00:36:00.400><c> kinetic</c><00:36:00.720><c> energy</c><00:36:00.960><c> we</c>

34:29.035 --> 34:29.045 align:start position:0%
operator. For the kinetic energy we
 

34:29.045 --> 34:30.954 align:start position:0%
operator. For the kinetic energy we
substitute<00:36:01.599><c> the</c><00:36:01.680><c> kinetic</c><00:36:02.000><c> energy</c><00:36:02.320><c> operator</c>

34:30.954 --> 34:30.964 align:start position:0%
substitute the kinetic energy operator
 

34:30.964 --> 34:32.795 align:start position:0%
substitute the kinetic energy operator
and<00:36:03.280><c> they're</c><00:36:03.599><c> work</c><00:36:03.839><c> and</c><00:36:04.000><c> they're</c><00:36:04.400><c> operating</c>

34:32.795 --> 34:32.805 align:start position:0%
and they're work and they're operating
 

34:32.805 --> 34:36.954 align:start position:0%
and they're work and they're operating
on<00:36:05.440><c> the</c><00:36:05.680><c> wave</c><00:36:05.920><c> function</c><00:36:06.400><c> s</c><00:36:07.599><c> and</c><00:36:07.839><c> we</c><00:36:08.240><c> have</c><00:36:08.880><c> our</c>

34:36.954 --> 34:36.964 align:start position:0%
on the wave function s and we have our
 

34:36.964 --> 34:42.555 align:start position:0%
on the wave function s and we have our
stroinger's<00:36:10.000><c> equation</c><00:36:10.560><c> in</c><00:36:10.800><c> its</c><00:36:11.280><c> full</c><00:36:12.160><c> glory.</c>

34:42.555 --> 34:42.565 align:start position:0%
stroinger's equation in its full glory.
 

34:42.565 --> 34:44.475 align:start position:0%
stroinger's equation in its full glory.
So<00:36:14.960><c> can</c><00:36:15.119><c> we</c><00:36:15.280><c> now</c><00:36:15.520><c> answer</c><00:36:15.920><c> our</c><00:36:16.240><c> original</c>

34:44.475 --> 34:44.485 align:start position:0%
So can we now answer our original
 

34:44.485 --> 34:47.355 align:start position:0%
So can we now answer our original
question<00:36:17.119><c> intuitively?</c><00:36:18.400><c> Why</c><00:36:18.640><c> is</c><00:36:18.800><c> there</c><00:36:18.960><c> an</c><00:36:19.200><c> i</c>

34:47.355 --> 34:47.365 align:start position:0%
question intuitively? Why is there an i
 

34:47.365 --> 34:49.194 align:start position:0%
question intuitively? Why is there an i
over<00:36:19.760><c> here?</c><00:36:20.240><c> Well</c><00:36:20.480><c> because</c><00:36:20.640><c> our</c><00:36:21.040><c> wave</c>

34:49.194 --> 34:49.204 align:start position:0%
over here? Well because our wave
 

34:49.204 --> 34:51.275 align:start position:0%
over here? Well because our wave
function<00:36:21.680><c> itself</c><00:36:22.079><c> is</c><00:36:22.320><c> complex.</c><00:36:23.040><c> Why</c><00:36:23.280><c> should</c>

34:51.275 --> 34:51.285 align:start position:0%
function itself is complex. Why should
 

34:51.285 --> 34:53.515 align:start position:0%
function itself is complex. Why should
it<00:36:23.599><c> be</c><00:36:23.760><c> complex?</c><00:36:24.480><c> because</c><00:36:24.720><c> it</c><00:36:25.040><c> needs</c><00:36:25.280><c> both</c><00:36:25.520><c> an</c>

34:53.515 --> 34:53.525 align:start position:0%
it be complex? because it needs both an
 

34:53.525 --> 34:55.914 align:start position:0%
it be complex? because it needs both an
exponential<00:36:26.240><c> and</c><00:36:26.480><c> a</c><00:36:26.640><c> periodic</c><00:36:27.119><c> function.</c><00:36:27.839><c> Why</c>

34:55.914 --> 34:55.924 align:start position:0%
exponential and a periodic function. Why
 

34:55.924 --> 34:57.275 align:start position:0%
exponential and a periodic function. Why
does<00:36:28.240><c> it</c><00:36:28.400><c> need</c><00:36:28.480><c> to</c><00:36:28.640><c> be</c><00:36:28.720><c> an</c><00:36:28.960><c> exponential</c>

34:57.275 --> 34:57.285 align:start position:0%
does it need to be an exponential
 

34:57.285 --> 34:59.115 align:start position:0%
does it need to be an exponential
function?<00:36:30.000><c> Well,</c><00:36:30.240><c> because</c><00:36:30.400><c> that's</c><00:36:30.640><c> the</c><00:36:30.960><c> only</c>

34:59.115 --> 34:59.125 align:start position:0%
function? Well, because that's the only
 

34:59.125 --> 35:01.755 align:start position:0%
function? Well, because that's the only
way<00:36:31.520><c> I</c><00:36:31.839><c> can</c><00:36:32.079><c> extract</c><00:36:32.720><c> energy</c><00:36:33.200><c> cleanly.</c><00:36:33.599><c> I</c><00:36:33.839><c> can</c>

35:01.755 --> 35:01.765 align:start position:0%
way I can extract energy cleanly. I can
 

35:01.765 --> 35:03.674 align:start position:0%
way I can extract energy cleanly. I can
build<00:36:34.079><c> an</c><00:36:34.240><c> energy</c><00:36:34.640><c> operator</c><00:36:35.119><c> because</c><00:36:35.440><c> I</c><00:36:35.680><c> need</c>

35:03.674 --> 35:03.684 align:start position:0%
build an energy operator because I need
 

35:03.684 --> 35:05.194 align:start position:0%
build an energy operator because I need
the<00:36:36.000><c> first</c><00:36:36.240><c> derivative</c><00:36:36.640><c> of</c><00:36:36.800><c> my</c><00:36:36.880><c> function</c><00:36:37.119><c> to</c>

35:05.194 --> 35:05.204 align:start position:0%
the first derivative of my function to
 

35:05.204 --> 35:08.875 align:start position:0%
the first derivative of my function to
be<00:36:37.520><c> itself.</c><00:36:38.720><c> Right?</c><00:36:39.920><c> That</c><00:36:40.400><c> doesn't</c><00:36:40.640><c> that</c><00:36:40.800><c> make</c>

35:08.875 --> 35:08.885 align:start position:0%
be itself. Right? That doesn't that make
 

35:08.885 --> 35:10.795 align:start position:0%
be itself. Right? That doesn't that make
sense?<00:36:41.760><c> Okay.</c><00:36:42.160><c> Now,</c><00:36:42.400><c> here's</c><00:36:42.720><c> another</c>

35:10.795 --> 35:10.805 align:start position:0%
sense? Okay. Now, here's another
 

35:10.805 --> 35:12.875 align:start position:0%
sense? Okay. Now, here's another
question.<00:36:43.599><c> What</c><00:36:43.839><c> would</c><00:36:44.000><c> happen</c><00:36:44.400><c> if</c><00:36:44.640><c> I</c><00:36:44.800><c> were</c><00:36:44.960><c> to</c>

35:12.875 --> 35:12.885 align:start position:0%
question. What would happen if I were to
 

35:12.885 --> 35:15.674 align:start position:0%
question. What would happen if I were to
consider<00:36:45.359><c> the</c><00:36:45.520><c> same</c><00:36:45.760><c> equation</c><00:36:46.480><c> without</c><00:36:47.440><c> the</c>

35:15.674 --> 35:15.684 align:start position:0%
consider the same equation without the
 

35:15.684 --> 35:18.795 align:start position:0%
consider the same equation without the
i?<00:36:49.040><c> Now,</c><00:36:49.280><c> the</c><00:36:49.520><c> solutions</c><00:36:49.920><c> would</c><00:36:50.160><c> be</c><00:36:50.640><c> real</c>

35:18.795 --> 35:18.805 align:start position:0%
i? Now, the solutions would be real
 

35:18.805 --> 35:20.475 align:start position:0%
i? Now, the solutions would be real
exponentials<00:36:51.520><c> in</c><00:36:51.760><c> time.</c><00:36:51.920><c> They</c><00:36:52.160><c> will</c><00:36:52.400><c> no</c>

35:20.475 --> 35:20.485 align:start position:0%
exponentials in time. They will no
 

35:20.485 --> 35:22.715 align:start position:0%
exponentials in time. They will no
longer<00:36:53.359><c> rotate</c><00:36:54.079><c> which</c><00:36:54.320><c> means</c><00:36:54.480><c> they</c><00:36:54.720><c> would</c>

35:22.715 --> 35:22.725 align:start position:0%
longer rotate which means they would
 

35:22.725 --> 35:25.194 align:start position:0%
longer rotate which means they would
just<00:36:55.040><c> blow</c><00:36:55.280><c> up</c><00:36:55.440><c> or</c><00:36:55.920><c> decay</c><00:36:56.320><c> forever.</c><00:36:57.040><c> Would</c><00:36:57.200><c> it</c>

35:25.194 --> 35:25.204 align:start position:0%
just blow up or decay forever. Would it
 

35:25.204 --> 35:27.914 align:start position:0%
just blow up or decay forever. Would it
represent<00:36:57.760><c> anything</c><00:36:58.320><c> physical?</c>

35:27.914 --> 35:27.924 align:start position:0%
represent anything physical?
 

35:27.924 --> 35:31.194 align:start position:0%
represent anything physical?
Yes.

35:31.194 --> 35:31.204 align:start position:0%
 
 

35:31.204 --> 35:33.755 align:start position:0%
 
Say<00:37:03.599><c> the</c><00:37:03.839><c> vertical</c><00:37:04.240><c> axis</c><00:37:04.720><c> was</c><00:37:05.200><c> temperature</c>

35:33.755 --> 35:33.765 align:start position:0%
Say the vertical axis was temperature
 

35:33.765 --> 35:35.995 align:start position:0%
Say the vertical axis was temperature
and<00:37:06.160><c> the</c><00:37:06.400><c> horizontal</c><00:37:06.880><c> axis</c><00:37:07.280><c> represented</c><00:37:07.920><c> a</c>

35:35.995 --> 35:36.005 align:start position:0%
and the horizontal axis represented a
 

35:36.005 --> 35:38.635 align:start position:0%
and the horizontal axis represented a
rod<00:37:08.960><c> which</c><00:37:09.200><c> means</c><00:37:09.520><c> we</c><00:37:09.839><c> now</c><00:37:10.160><c> have</c><00:37:10.400><c> a</c>

35:38.635 --> 35:38.645 align:start position:0%
rod which means we now have a
 

35:38.645 --> 35:41.275 align:start position:0%
rod which means we now have a
temperature<00:37:11.839><c> distribution.</c>

35:41.275 --> 35:41.285 align:start position:0%
temperature distribution.
 

35:41.285 --> 35:44.635 align:start position:0%
temperature distribution.
And<00:37:13.680><c> as</c><00:37:14.160><c> time</c><00:37:14.480><c> passes</c><00:37:15.200><c> the</c><00:37:15.520><c> hot</c><00:37:15.760><c> regions</c><00:37:16.480><c> cool</c>

35:44.635 --> 35:44.645 align:start position:0%
And as time passes the hot regions cool
 

35:44.645 --> 35:46.795 align:start position:0%
And as time passes the hot regions cool
down<00:37:16.960><c> and</c><00:37:17.200><c> the</c><00:37:17.359><c> cool</c><00:37:17.599><c> regions</c><00:37:18.160><c> heat</c><00:37:18.400><c> up.</c><00:37:18.880><c> So</c>

35:46.795 --> 35:46.805 align:start position:0%
down and the cool regions heat up. So
 

35:46.805 --> 35:48.875 align:start position:0%
down and the cool regions heat up. So
the<00:37:19.200><c> graph</c><00:37:19.440><c> shrinks.</c><00:37:20.079><c> But</c><00:37:20.240><c> as</c><00:37:20.400><c> a</c><00:37:20.640><c> temperature</c>

35:48.875 --> 35:48.885 align:start position:0%
the graph shrinks. But as a temperature
 

35:48.885 --> 35:50.715 align:start position:0%
the graph shrinks. But as a temperature
gets<00:37:21.280><c> closer</c><00:37:21.520><c> to</c><00:37:21.680><c> each</c><00:37:21.839><c> other,</c><00:37:22.160><c> it</c><00:37:22.400><c> gets</c>

35:50.715 --> 35:50.725 align:start position:0%
gets closer to each other, it gets
 

35:50.725 --> 35:54.795 align:start position:0%
gets closer to each other, it gets
slower<00:37:23.599><c> over</c><00:37:24.079><c> time.</c><00:37:24.960><c> Meaning</c><00:37:25.520><c> we</c><00:37:25.839><c> get</c><00:37:26.240><c> exactly</c>

35:54.795 --> 35:54.805 align:start position:0%
slower over time. Meaning we get exactly
 

35:54.805 --> 35:56.795 align:start position:0%
slower over time. Meaning we get exactly
a<00:37:27.359><c> decay.</c>

35:56.795 --> 35:56.805 align:start position:0%
a decay.
 

35:56.805 --> 36:00.315 align:start position:0%
a decay.
In<00:37:29.119><c> other</c><00:37:29.359><c> words,</c><00:37:30.320><c> this</c><00:37:30.800><c> now</c><00:37:31.440><c> represents</c><00:37:32.079><c> the</c>

36:00.315 --> 36:00.325 align:start position:0%
In other words, this now represents the
 

36:00.325 --> 36:03.115 align:start position:0%
In other words, this now represents the
heat<00:37:33.040><c> equation.</c><00:37:34.000><c> How</c><00:37:34.240><c> temperature</c><00:37:34.800><c> changes</c>

36:03.115 --> 36:03.125 align:start position:0%
heat equation. How temperature changes
 

36:03.125 --> 36:05.995 align:start position:0%
heat equation. How temperature changes
over<00:37:35.680><c> time.</c><00:37:36.880><c> But</c><00:37:37.040><c> of</c><00:37:37.200><c> course,</c><00:37:37.440><c> it</c><00:37:37.680><c> can</c><00:37:37.920><c> have</c>

36:05.995 --> 36:06.005 align:start position:0%
over time. But of course, it can have
 

36:06.005 --> 36:08.155 align:start position:0%
over time. But of course, it can have
different<00:37:38.800><c> constants</c><00:37:39.440><c> and</c><00:37:39.680><c> we</c><00:37:39.839><c> would</c><00:37:40.000><c> expect</c>

36:08.155 --> 36:08.165 align:start position:0%
different constants and we would expect
 

36:08.165 --> 36:10.555 align:start position:0%
different constants and we would expect
it<00:37:40.560><c> to</c><00:37:40.720><c> not</c><00:37:40.880><c> have</c><00:37:41.040><c> any</c><00:37:41.280><c> potential</c><00:37:42.079><c> energy</c>

36:10.555 --> 36:10.565 align:start position:0%
it to not have any potential energy
 

36:10.565 --> 36:15.115 align:start position:0%
it to not have any potential energy
term.<00:37:43.520><c> So</c><00:37:43.920><c> we</c><00:37:44.160><c> can</c><00:37:44.400><c> now</c><00:37:44.880><c> intuitively</c><00:37:45.920><c> see</c><00:37:46.800><c> why</c>

36:15.115 --> 36:15.125 align:start position:0%
term. So we can now intuitively see why
 

36:15.125 --> 36:18.075 align:start position:0%
term. So we can now intuitively see why
the<00:37:47.599><c> Schroinger's</c><00:37:48.400><c> equation</c><00:37:49.200><c> looks</c><00:37:49.760><c> so</c>

36:18.075 --> 36:18.085 align:start position:0%
the Schroinger's equation looks so
 

36:18.085 --> 36:21.515 align:start position:0%
the Schroinger's equation looks so
similar<00:37:50.800><c> to</c><00:37:51.119><c> the</c><00:37:51.359><c> heat</c><00:37:51.680><c> equation</c><00:37:52.880><c> except</c><00:37:53.359><c> for</c>

36:21.515 --> 36:21.525 align:start position:0%
similar to the heat equation except for
 

36:21.525 --> 36:24.795 align:start position:0%
similar to the heat equation except for
the<00:37:53.920><c> I.</c><00:37:54.880><c> It's</c><00:37:55.200><c> the</c><00:37:55.520><c> eye</c><00:37:55.920><c> that</c><00:37:56.320><c> turns</c><00:37:56.640><c> an</c>

36:24.795 --> 36:24.805 align:start position:0%
the I. It's the eye that turns an
 

36:24.805 --> 36:27.434 align:start position:0%
the I. It's the eye that turns an
exponential<00:37:57.680><c> into</c><00:37:58.000><c> a</c><00:37:58.240><c> spinning</c><00:37:58.960><c> exponential</c>

36:27.434 --> 36:27.444 align:start position:0%
exponential into a spinning exponential
 

36:27.444 --> 36:29.515 align:start position:0%
exponential into a spinning exponential
periodic.

36:29.515 --> 36:29.525 align:start position:0%
periodic.
 

36:29.525 --> 36:31.275 align:start position:0%
periodic.
At<00:38:01.920><c> this</c><00:38:02.079><c> point</c><00:38:02.240><c> I'm</c><00:38:02.560><c> really</c><00:38:02.880><c> satisfied</c><00:38:03.280><c> with</c>

36:31.275 --> 36:31.285 align:start position:0%
At this point I'm really satisfied with
 

36:31.285 --> 36:32.954 align:start position:0%
At this point I'm really satisfied with
the<00:38:03.599><c> Schroinger</c><00:38:04.160><c> equation</c><00:38:04.560><c> where</c><00:38:04.720><c> it</c><00:38:04.880><c> comes</c>

36:32.954 --> 36:32.964 align:start position:0%
the Schroinger equation where it comes
 

36:32.964 --> 36:34.635 align:start position:0%
the Schroinger equation where it comes
from<00:38:05.280><c> and</c><00:38:05.520><c> why</c><00:38:05.760><c> there</c><00:38:05.920><c> is</c><00:38:06.000><c> an</c><00:38:06.160><c> I.</c><00:38:06.480><c> And</c><00:38:06.560><c> I</c><00:38:06.720><c> think</c>

36:34.635 --> 36:34.645 align:start position:0%
from and why there is an I. And I think
 

36:34.645 --> 36:35.995 align:start position:0%
from and why there is an I. And I think
I<00:38:06.960><c> have</c><00:38:07.040><c> a</c><00:38:07.200><c> really</c><00:38:07.359><c> good</c><00:38:07.440><c> intuition</c><00:38:07.920><c> behind</c>

36:35.995 --> 36:36.005 align:start position:0%
I have a really good intuition behind
 

36:36.005 --> 36:38.475 align:start position:0%
I have a really good intuition behind
it.<00:38:09.119><c> But</c><00:38:09.280><c> a</c><00:38:09.440><c> part</c><00:38:09.599><c> of</c><00:38:09.680><c> me</c><00:38:09.920><c> still</c><00:38:10.079><c> feels</c><00:38:10.320><c> that</c>

36:38.475 --> 36:38.485 align:start position:0%
it. But a part of me still feels that
 

36:38.485 --> 36:40.795 align:start position:0%
it. But a part of me still feels that
it's<00:38:10.880><c> all</c><00:38:11.119><c> still</c><00:38:11.440><c> mathematical.</c><00:38:12.320><c> It</c><00:38:12.640><c> kind</c><00:38:12.800><c> of</c>

36:40.795 --> 36:40.805 align:start position:0%
it's all still mathematical. It kind of
 

36:40.805 --> 36:43.035 align:start position:0%
it's all still mathematical. It kind of
feels<00:38:13.280><c> like,</c><00:38:13.920><c> you</c><00:38:14.079><c> know,</c><00:38:14.560><c> there</c><00:38:14.800><c> might</c><00:38:14.960><c> be</c>

36:43.035 --> 36:43.045 align:start position:0%
feels like, you know, there might be
 

36:43.045 --> 36:44.875 align:start position:0%
feels like, you know, there might be
some<00:38:15.440><c> kind</c><00:38:15.599><c> of</c><00:38:15.680><c> a</c><00:38:15.839><c> mathematical</c><00:38:16.480><c> trick</c><00:38:16.720><c> that</c>

36:44.875 --> 36:44.885 align:start position:0%
some kind of a mathematical trick that
 

36:44.885 --> 36:47.755 align:start position:0%
some kind of a mathematical trick that
we<00:38:17.440><c> just</c><00:38:17.760><c> haven't</c><00:38:18.079><c> thought</c><00:38:18.320><c> of</c><00:38:18.560><c> yet</c><00:38:19.599><c> using</c>

36:47.755 --> 36:47.765 align:start position:0%
we just haven't thought of yet using
 

36:47.765 --> 36:50.875 align:start position:0%
we just haven't thought of yet using
which<00:38:20.160><c> we</c><00:38:20.400><c> can</c><00:38:20.640><c> get</c><00:38:21.119><c> rid</c><00:38:21.520><c> of</c><00:38:21.839><c> that</c><00:38:22.320><c> I</c><00:38:22.880><c> and</c>

36:50.875 --> 36:50.885 align:start position:0%
which we can get rid of that I and
 

36:50.885 --> 36:52.475 align:start position:0%
which we can get rid of that I and
that's<00:38:23.280><c> exactly</c><00:38:23.599><c> what</c><00:38:23.839><c> Shinger</c><00:38:24.400><c> was</c><00:38:24.480><c> trying</c>

36:52.475 --> 36:52.485 align:start position:0%
that's exactly what Shinger was trying
 

36:52.485 --> 36:54.715 align:start position:0%
that's exactly what Shinger was trying
to<00:38:24.800><c> do</c><00:38:24.960><c> for</c><00:38:25.280><c> months</c><00:38:25.680><c> after</c><00:38:26.000><c> publishing</c><00:38:26.560><c> his</c>

36:54.715 --> 36:54.725 align:start position:0%
to do for months after publishing his
 

36:54.725 --> 36:56.954 align:start position:0%
to do for months after publishing his
equation.<00:38:27.760><c> And</c><00:38:28.000><c> when</c><00:38:28.240><c> he</c><00:38:28.400><c> couldn't,</c><00:38:28.800><c> he</c><00:38:28.960><c> wrote</c>

36:56.954 --> 36:56.964 align:start position:0%
equation. And when he couldn't, he wrote
 

36:56.964 --> 36:59.914 align:start position:0%
equation. And when he couldn't, he wrote
a<00:38:29.359><c> letter</c><00:38:29.520><c> to</c><00:38:29.839><c> Henrik</c><00:38:30.320><c> Loren</c><00:38:31.040><c> saying</c><00:38:31.680><c> what</c><00:38:31.920><c> is</c>

36:59.914 --> 36:59.924 align:start position:0%
a letter to Henrik Loren saying what is
 

36:59.924 --> 37:02.395 align:start position:0%
a letter to Henrik Loren saying what is
unpleasant<00:38:32.720><c> here</c><00:38:33.200><c> and</c><00:38:33.520><c> indeed</c><00:38:34.000><c> directly</c><00:38:34.400><c> to</c>

37:02.395 --> 37:02.405 align:start position:0%
unpleasant here and indeed directly to
 

37:02.405 --> 37:04.395 align:start position:0%
unpleasant here and indeed directly to
be<00:38:34.720><c> objected</c><00:38:35.119><c> to</c><00:38:35.280><c> is</c><00:38:35.520><c> the</c><00:38:35.680><c> use</c><00:38:35.920><c> of</c><00:38:36.160><c> complex</c>

37:04.395 --> 37:04.405 align:start position:0%
be objected to is the use of complex
 

37:04.405 --> 37:07.275 align:start position:0%
be objected to is the use of complex
numbers.<00:38:37.440><c> Sigh</c><00:38:37.920><c> is</c><00:38:38.160><c> surely</c><00:38:38.640><c> fundamentally</c><00:38:39.200><c> a</c>

37:07.275 --> 37:07.285 align:start position:0%
numbers. Sigh is surely fundamentally a
 

37:07.285 --> 37:09.995 align:start position:0%
numbers. Sigh is surely fundamentally a
real<00:38:39.760><c> function.</c><00:38:41.040><c> So</c><00:38:41.280><c> mathematically</c><00:38:41.920><c> it</c>

37:09.995 --> 37:10.005 align:start position:0%
real function. So mathematically it
 

37:10.005 --> 37:11.995 align:start position:0%
real function. So mathematically it
makes<00:38:42.400><c> sense</c><00:38:42.560><c> why</c><00:38:42.800><c> the</c><00:38:43.040><c> eye</c><00:38:43.280><c> must</c><00:38:43.520><c> be</c><00:38:43.680><c> there.</c>

37:11.995 --> 37:12.005 align:start position:0%
makes sense why the eye must be there.
 

37:12.005 --> 37:14.555 align:start position:0%
makes sense why the eye must be there.
But<00:38:44.320><c> what</c><00:38:44.480><c> is</c><00:38:44.640><c> the</c><00:38:44.960><c> physical</c><00:38:45.680><c> reason</c><00:38:46.320><c> behind</c>

37:14.555 --> 37:14.565 align:start position:0%
But what is the physical reason behind
 

37:14.565 --> 37:18.315 align:start position:0%
But what is the physical reason behind
it?<00:38:47.680><c> Well,</c><00:38:48.079><c> the</c><00:38:48.480><c> breakthrough</c><00:38:49.040><c> came</c><00:38:49.839><c> as</c><00:38:50.079><c> a</c>

37:18.315 --> 37:18.325 align:start position:0%
it? Well, the breakthrough came as a
 

37:18.325 --> 37:20.635 align:start position:0%
it? Well, the breakthrough came as a
footnote<00:38:51.119><c> actually</c><00:38:51.680><c> in</c><00:38:51.920><c> a</c><00:38:52.079><c> paper</c><00:38:52.400><c> published</c>

37:20.635 --> 37:20.645 align:start position:0%
footnote actually in a paper published
 

37:20.645 --> 37:23.995 align:start position:0%
footnote actually in a paper published
by<00:38:53.280><c> Max</c><00:38:53.920><c> Bourne.</c>

37:23.995 --> 37:24.005 align:start position:0%
by Max Bourne.
 

37:24.005 --> 37:25.755 align:start position:0%
by Max Bourne.
Borne<00:38:56.560><c> says</c><00:38:56.960><c> let's</c><00:38:57.200><c> go</c><00:38:57.280><c> back</c><00:38:57.440><c> to</c><00:38:57.520><c> the</c><00:38:57.680><c> double</c>

37:25.755 --> 37:25.765 align:start position:0%
Borne says let's go back to the double
 

37:25.765 --> 37:28.235 align:start position:0%
Borne says let's go back to the double
slit.<00:38:58.480><c> We</c><00:38:58.720><c> are</c><00:38:58.800><c> now</c><00:38:59.040><c> coming</c><00:38:59.440><c> a</c><00:38:59.680><c> full</c><00:38:59.920><c> circle</c><00:39:00.240><c> to</c>

37:28.235 --> 37:28.245 align:start position:0%
slit. We are now coming a full circle to
 

37:28.245 --> 37:30.555 align:start position:0%
slit. We are now coming a full circle to
where<00:39:00.560><c> we</c><00:39:00.800><c> started.</c><00:39:01.280><c> Awesome.</c><00:39:02.079><c> This</c><00:39:02.240><c> time</c><00:39:02.480><c> if</c>

37:30.555 --> 37:30.565 align:start position:0%
where we started. Awesome. This time if
 

37:30.565 --> 37:32.795 align:start position:0%
where we started. Awesome. This time if
we<00:39:02.880><c> send</c><00:39:03.119><c> only</c><00:39:03.599><c> one</c><00:39:03.920><c> photon's</c><00:39:04.480><c> worth</c><00:39:04.720><c> of</c>

37:32.795 --> 37:32.805 align:start position:0%
we send only one photon's worth of
 

37:32.805 --> 37:34.795 align:start position:0%
we send only one photon's worth of
energy<00:39:05.599><c> through</c><00:39:05.839><c> those</c><00:39:06.000><c> slits,</c><00:39:06.480><c> it</c><00:39:06.720><c> has</c><00:39:06.800><c> to</c>

37:34.795 --> 37:34.805 align:start position:0%
energy through those slits, it has to
 

37:34.805 --> 37:37.355 align:start position:0%
energy through those slits, it has to
land<00:39:07.200><c> somewhere</c><00:39:07.760><c> at</c><00:39:08.160><c> one</c><00:39:08.560><c> specific</c><00:39:09.040><c> spot</c><00:39:09.359><c> on</c>

37:37.355 --> 37:37.365 align:start position:0%
land somewhere at one specific spot on
 

37:37.365 --> 37:39.275 align:start position:0%
land somewhere at one specific spot on
the<00:39:09.680><c> screen.</c><00:39:10.000><c> Right.</c><00:39:10.480><c> Mah,</c><00:39:10.880><c> can</c><00:39:11.040><c> you</c><00:39:11.200><c> tell</c><00:39:11.280><c> me</c>

37:39.275 --> 37:39.285 align:start position:0%
the screen. Right. Mah, can you tell me
 

37:39.285 --> 37:42.235 align:start position:0%
the screen. Right. Mah, can you tell me
where?<00:39:12.240><c> And</c><00:39:12.400><c> I</c><00:39:12.640><c> say,</c><00:39:12.880><c> I</c><00:39:13.119><c> have</c><00:39:13.280><c> no</c><00:39:13.440><c> idea.</c><00:39:14.240><c> And</c>

37:42.235 --> 37:42.245 align:start position:0%
where? And I say, I have no idea. And
 

37:42.245 --> 37:45.995 align:start position:0%
where? And I say, I have no idea. And
Bon<00:39:14.800><c> says,</c><00:39:15.280><c> neither</c><00:39:15.599><c> do</c><00:39:15.839><c> I,</c><00:39:16.800><c> but</c><00:39:17.359><c> I</c><00:39:17.680><c> can't</c><00:39:18.000><c> tell</c>

37:45.995 --> 37:46.005 align:start position:0%
Bon says, neither do I, but I can't tell
 

37:46.005 --> 37:48.315 align:start position:0%
Bon says, neither do I, but I can't tell
where<00:39:18.480><c> it's</c><00:39:18.720><c> more</c><00:39:19.040><c> likely</c><00:39:19.440><c> to</c><00:39:19.680><c> land.</c><00:39:20.240><c> The</c>

37:48.315 --> 37:48.325 align:start position:0%
where it's more likely to land. The
 

37:48.325 --> 37:50.875 align:start position:0%
where it's more likely to land. The
chances<00:39:20.800><c> of</c><00:39:21.040><c> landing</c><00:39:21.520><c> in</c><00:39:21.920><c> these</c><00:39:22.400><c> dark</c><00:39:22.720><c> regions</c>

37:50.875 --> 37:50.885 align:start position:0%
chances of landing in these dark regions
 

37:50.885 --> 37:52.875 align:start position:0%
chances of landing in these dark regions
is<00:39:23.280><c> almost</c><00:39:23.599><c> zero</c><00:39:24.000><c> because</c><00:39:24.240><c> no</c><00:39:24.480><c> photons</c><00:39:24.880><c> have</c>

37:52.875 --> 37:52.885 align:start position:0%
is almost zero because no photons have
 

37:52.885 --> 37:55.434 align:start position:0%
is almost zero because no photons have
landed<00:39:25.440><c> so</c><00:39:25.680><c> far.</c><00:39:26.400><c> And</c><00:39:26.640><c> the</c><00:39:26.800><c> brightest</c><00:39:27.200><c> spot</c><00:39:27.440><c> in</c>

37:55.434 --> 37:55.444 align:start position:0%
landed so far. And the brightest spot in
 

37:55.444 --> 37:57.115 align:start position:0%
landed so far. And the brightest spot in
the<00:39:27.760><c> center,</c><00:39:28.160><c> the</c><00:39:28.400><c> chances</c><00:39:28.720><c> is</c><00:39:28.880><c> very</c><00:39:29.040><c> high</c>

37:57.115 --> 37:57.125 align:start position:0%
the center, the chances is very high
 

37:57.125 --> 37:58.635 align:start position:0%
the center, the chances is very high
there<00:39:29.520><c> because</c><00:39:29.760><c> lots</c><00:39:30.079><c> of</c><00:39:30.160><c> photons</c><00:39:30.560><c> have</c>

37:58.635 --> 37:58.645 align:start position:0%
there because lots of photons have
 

37:58.645 --> 38:00.475 align:start position:0%
there because lots of photons have
already<00:39:31.040><c> landed</c><00:39:31.359><c> there</c><00:39:31.599><c> before.</c><00:39:32.240><c> So</c><00:39:32.400><c> the</c>

38:00.475 --> 38:00.485 align:start position:0%
already landed there before. So the
 

38:00.485 --> 38:02.475 align:start position:0%
already landed there before. So the
dimmer<00:39:33.040><c> the</c><00:39:33.200><c> regions,</c><00:39:33.760><c> the</c><00:39:34.000><c> lower</c><00:39:34.400><c> the</c>

38:02.475 --> 38:02.485 align:start position:0%
dimmer the regions, the lower the
 

38:02.485 --> 38:05.755 align:start position:0%
dimmer the regions, the lower the
chances.<00:39:35.839><c> So</c><00:39:36.160><c> even</c><00:39:36.480><c> though</c><00:39:36.800><c> I</c><00:39:37.280><c> can't</c><00:39:37.599><c> say</c>

38:05.755 --> 38:05.765 align:start position:0%
chances. So even though I can't say
 

38:05.765 --> 38:08.555 align:start position:0%
chances. So even though I can't say
exactly<00:39:38.400><c> where</c><00:39:38.640><c> one</c><00:39:38.880><c> photon</c><00:39:39.359><c> lands,</c><00:39:40.079><c> I</c><00:39:40.400><c> can</c>

38:08.555 --> 38:08.565 align:start position:0%
exactly where one photon lands, I can
 

38:08.565 --> 38:11.674 align:start position:0%
exactly where one photon lands, I can
talk<00:39:41.119><c> about</c><00:39:41.760><c> the</c><00:39:42.160><c> probability</c><00:39:43.119><c> and</c><00:39:43.440><c> that</c>

38:11.674 --> 38:11.684 align:start position:0%
talk about the probability and that
 

38:11.684 --> 38:14.395 align:start position:0%
talk about the probability and that
probability<00:39:44.960><c> is</c><00:39:45.359><c> proportional</c><00:39:46.000><c> to</c><00:39:46.240><c> the</c>

38:14.395 --> 38:14.405 align:start position:0%
probability is proportional to the
 

38:14.405 --> 38:17.355 align:start position:0%
probability is proportional to the
brightness<00:39:47.440><c> or</c><00:39:47.760><c> intensity</c><00:39:48.960><c> which</c><00:39:49.200><c> is</c>

38:17.355 --> 38:17.365 align:start position:0%
brightness or intensity which is
 

38:17.365 --> 38:22.954 align:start position:0%
brightness or intensity which is
proportional<00:39:50.560><c> to</c><00:39:50.800><c> the</c><00:39:50.960><c> amplitude</c><00:39:52.000><c> squared.</c>

38:22.954 --> 38:22.964 align:start position:0%
 
 

38:22.964 --> 38:24.795 align:start position:0%
 
This<00:39:55.359><c> is</c><00:39:55.520><c> how</c><00:39:55.680><c> Einstein's</c><00:39:56.240><c> idea</c><00:39:56.560><c> can</c><00:39:56.800><c> be</c>

38:24.795 --> 38:24.805 align:start position:0%
This is how Einstein's idea can be
 

38:24.805 --> 38:27.115 align:start position:0%
This is how Einstein's idea can be
interpreted<00:39:57.520><c> probabilistically.</c><00:39:58.640><c> And</c><00:39:58.800><c> Borne</c>

38:27.115 --> 38:27.125 align:start position:0%
interpreted probabilistically. And Borne
 

38:27.125 --> 38:30.075 align:start position:0%
interpreted probabilistically. And Borne
says<00:40:00.320><c> well</c><00:40:00.640><c> I</c><00:40:00.880><c> just</c><00:40:01.040><c> thought</c><00:40:01.280><c> that</c><00:40:01.760><c> we</c><00:40:02.000><c> can</c><00:40:02.160><c> try</c>

38:30.075 --> 38:30.085 align:start position:0%
says well I just thought that we can try
 

38:30.085 --> 38:32.075 align:start position:0%
says well I just thought that we can try
and<00:40:02.480><c> apply</c><00:40:02.720><c> the</c><00:40:02.880><c> same</c><00:40:03.040><c> thing</c><00:40:03.119><c> to</c><00:40:03.280><c> mow</c><00:40:03.599><c> waves.</c>

38:32.075 --> 38:32.085 align:start position:0%
and apply the same thing to mow waves.
 

38:32.085 --> 38:34.155 align:start position:0%
and apply the same thing to mow waves.
If<00:40:04.480><c> you</c><00:40:04.640><c> replace</c><00:40:05.040><c> light</c><00:40:05.359><c> with</c><00:40:05.520><c> a</c><00:40:05.760><c> source</c><00:40:06.079><c> of</c>

38:34.155 --> 38:34.165 align:start position:0%
If you replace light with a source of
 

38:34.165 --> 38:37.194 align:start position:0%
If you replace light with a source of
electron,<00:40:07.520><c> they</c><00:40:07.760><c> too</c><00:40:08.079><c> travels</c><00:40:08.480><c> as</c><00:40:08.640><c> waves</c><00:40:09.119><c> and</c>

38:37.194 --> 38:37.204 align:start position:0%
electron, they too travels as waves and
 

38:37.204 --> 38:38.954 align:start position:0%
electron, they too travels as waves and
then<00:40:09.520><c> we</c><00:40:09.760><c> should</c><00:40:10.160><c> get</c><00:40:10.320><c> the</c><00:40:10.560><c> exact</c><00:40:10.880><c> same</c>

38:38.954 --> 38:38.964 align:start position:0%
then we should get the exact same
 

38:38.964 --> 38:41.755 align:start position:0%
then we should get the exact same
result.<00:40:12.160><c> Which</c><00:40:12.320><c> means</c><00:40:12.880><c> the</c><00:40:13.200><c> probability</c><00:40:13.760><c> of</c>

38:41.755 --> 38:41.765 align:start position:0%
result. Which means the probability of
 

38:41.765 --> 38:45.914 align:start position:0%
result. Which means the probability of
finding<00:40:14.400><c> now</c><00:40:14.800><c> the</c><00:40:15.280><c> electron</c><00:40:16.000><c> in</c><00:40:16.320><c> any</c><00:40:16.560><c> spot</c><00:40:17.440><c> is</c>

38:45.914 --> 38:45.924 align:start position:0%
finding now the electron in any spot is
 

38:45.924 --> 38:50.954 align:start position:0%
finding now the electron in any spot is
the<00:40:18.640><c> square</c><00:40:19.599><c> of</c><00:40:19.839><c> the</c><00:40:20.160><c> wave</c><00:40:20.720><c> function</c><00:40:21.760><c> at</c><00:40:22.400><c> that</c>

38:50.954 --> 38:50.964 align:start position:0%
the square of the wave function at that
 

38:50.964 --> 38:54.155 align:start position:0%
the square of the wave function at that
point.<00:40:24.320><c> This</c><00:40:25.040><c> is</c><00:40:25.280><c> today</c><00:40:25.599><c> called</c><00:40:25.760><c> the</c><00:40:25.920><c> borne</c>

38:54.155 --> 38:54.165 align:start position:0%
point. This is today called the borne
 

38:54.165 --> 38:56.795 align:start position:0%
point. This is today called the borne
rule.<00:40:27.119><c> This</c><00:40:27.440><c> is</c><00:40:27.520><c> what</c><00:40:27.680><c> he</c><00:40:27.839><c> wrote</c><00:40:28.079><c> down</c><00:40:28.480><c> as</c><00:40:28.800><c> a</c>

38:56.795 --> 38:56.805 align:start position:0%
rule. This is what he wrote down as a
 

38:56.805 --> 39:01.115 align:start position:0%
rule. This is what he wrote down as a
casual<00:40:29.440><c> footnote</c><00:40:30.640><c> and</c><00:40:30.960><c> it</c><00:40:31.599><c> gave</c><00:40:31.839><c> us</c><00:40:32.079><c> a</c><00:40:32.320><c> way</c><00:40:32.720><c> to</c>

39:01.115 --> 39:01.125 align:start position:0%
casual footnote and it gave us a way to
 

39:01.125 --> 39:04.235 align:start position:0%
casual footnote and it gave us a way to
interpret<00:40:34.480><c> the</c><00:40:34.880><c> microwave.</c><00:40:35.920><c> It's</c><00:40:36.160><c> a</c>

39:04.235 --> 39:04.245 align:start position:0%
interpret the microwave. It's a
 

39:04.245 --> 39:06.155 align:start position:0%
interpret the microwave. It's a
probability<00:40:36.960><c> wave.</c><00:40:37.440><c> And</c><00:40:37.680><c> remember</c><00:40:37.920><c> at</c><00:40:38.160><c> the</c>

39:06.155 --> 39:06.165 align:start position:0%
probability wave. And remember at the
 

39:06.165 --> 39:07.835 align:start position:0%
probability wave. And remember at the
beginning<00:40:38.560><c> we</c><00:40:38.800><c> asked</c><00:40:39.040><c> a</c><00:40:39.200><c> question</c><00:40:39.440><c> of</c><00:40:39.680><c> how</c>

39:07.835 --> 39:07.845 align:start position:0%
beginning we asked a question of how
 

39:07.845 --> 39:10.315 align:start position:0%
beginning we asked a question of how
mrow<00:40:40.320><c> waves</c><00:40:40.640><c> carry</c><00:40:41.280><c> mixtures</c><00:40:41.760><c> of</c><00:40:41.920><c> energy</c><00:40:42.240><c> and</c>

39:10.315 --> 39:10.325 align:start position:0%
mrow waves carry mixtures of energy and
 

39:10.325 --> 39:11.914 align:start position:0%
mrow waves carry mixtures of energy and
momentum<00:40:42.880><c> values.</c><00:40:43.280><c> How</c><00:40:43.440><c> do</c><00:40:43.599><c> you</c><00:40:43.760><c> make</c><00:40:43.920><c> sense</c>

39:11.914 --> 39:11.924 align:start position:0%
momentum values. How do you make sense
 

39:11.924 --> 39:14.875 align:start position:0%
momentum values. How do you make sense
of<00:40:44.240><c> that?</c><00:40:44.960><c> Well,</c><00:40:45.520><c> that</c><00:40:45.839><c> mixture</c><00:40:46.240><c> is</c><00:40:46.480><c> really</c>

39:14.875 --> 39:14.885 align:start position:0%
of that? Well, that mixture is really
 

39:14.885 --> 39:16.395 align:start position:0%
of that? Well, that mixture is really
probability<00:40:47.520><c> distribution.</c><00:40:48.240><c> So</c><00:40:48.480><c> for</c>

39:16.395 --> 39:16.405 align:start position:0%
probability distribution. So for
 

39:16.405 --> 39:18.075 align:start position:0%
probability distribution. So for
example,<00:40:48.880><c> when</c><00:40:49.040><c> you</c><00:40:49.200><c> measure</c><00:40:49.359><c> its</c><00:40:49.680><c> energy</c><00:40:50.000><c> or</c>

39:18.075 --> 39:18.085 align:start position:0%
example, when you measure its energy or
 

39:18.085 --> 39:20.475 align:start position:0%
example, when you measure its energy or
momentum,<00:40:51.119><c> there's</c><00:40:51.440><c> some</c><00:40:51.680><c> probability</c><00:40:52.320><c> of</c>

39:20.475 --> 39:20.485 align:start position:0%
momentum, there's some probability of
 

39:20.485 --> 39:23.275 align:start position:0%
momentum, there's some probability of
getting<00:40:53.040><c> each</c><00:40:53.440><c> particular</c><00:40:54.240><c> value.</c><00:40:55.040><c> And</c><00:40:55.280><c> that</c>

39:23.275 --> 39:23.285 align:start position:0%
getting each particular value. And that
 

39:23.285 --> 39:25.914 align:start position:0%
getting each particular value. And that
probability<00:40:55.920><c> distribution</c><00:40:56.400><c> is</c><00:40:56.720><c> hidden</c><00:40:57.599><c> in</c>

39:25.914 --> 39:25.924 align:start position:0%
probability distribution is hidden in
 

39:25.924 --> 39:29.515 align:start position:0%
probability distribution is hidden in
that<00:40:58.400><c> wave</c><00:40:58.800><c> function.</c>

39:29.515 --> 39:29.525 align:start position:0%
that wave function.
 

39:29.525 --> 39:31.194 align:start position:0%
that wave function.
So<00:41:01.839><c> coming</c><00:41:02.079><c> back</c><00:41:02.240><c> to</c><00:41:02.400><c> our</c><00:41:02.640><c> original</c><00:41:03.040><c> question,</c>

39:31.194 --> 39:31.204 align:start position:0%
So coming back to our original question,
 

39:31.204 --> 39:33.115 align:start position:0%
So coming back to our original question,
how<00:41:03.520><c> does</c><00:41:03.680><c> the</c><00:41:03.839><c> born</c><00:41:04.240><c> rule</c><00:41:04.640><c> give</c><00:41:04.880><c> us</c><00:41:05.040><c> a</c>

39:33.115 --> 39:33.125 align:start position:0%
how does the born rule give us a
 

39:33.125 --> 39:36.075 align:start position:0%
how does the born rule give us a
physical<00:41:05.839><c> meaning</c><00:41:06.319><c> to</c><00:41:06.640><c> I?</c><00:41:07.760><c> Suppose</c><00:41:08.079><c> our</c>

39:36.075 --> 39:36.085 align:start position:0%
physical meaning to I? Suppose our
 

39:36.085 --> 39:37.755 align:start position:0%
physical meaning to I? Suppose our
matter<00:41:08.560><c> wave</c><00:41:08.880><c> just</c><00:41:09.040><c> oscillate</c><00:41:09.520><c> back</c><00:41:09.680><c> and</c>

39:37.755 --> 39:37.765 align:start position:0%
matter wave just oscillate back and
 

39:37.765 --> 39:39.835 align:start position:0%
matter wave just oscillate back and
forth<00:41:10.160><c> on</c><00:41:10.400><c> the</c><00:41:10.480><c> real</c><00:41:10.720><c> axis.</c><00:41:11.359><c> Let's</c><00:41:11.520><c> say</c><00:41:11.680><c> we</c>

39:39.835 --> 39:39.845 align:start position:0%
forth on the real axis. Let's say we
 

39:39.845 --> 39:42.075 align:start position:0%
forth on the real axis. Let's say we
found<00:41:12.319><c> some</c><00:41:12.720><c> mathematical</c><00:41:13.280><c> trick</c><00:41:13.599><c> to</c><00:41:13.920><c> make</c>

39:42.075 --> 39:42.085 align:start position:0%
found some mathematical trick to make
 

39:42.085 --> 39:46.555 align:start position:0%
found some mathematical trick to make
this<00:41:14.800><c> work.</c><00:41:15.359><c> Okay?</c><00:41:16.000><c> Then</c><00:41:16.720><c> the</c><00:41:17.040><c> size</c><00:41:17.520><c> squared</c>

39:46.555 --> 39:46.565 align:start position:0%
this work. Okay? Then the size squared
 

39:46.565 --> 39:50.475 align:start position:0%
this work. Okay? Then the size squared
over<00:41:19.040><c> here</c><00:41:19.680><c> would</c><00:41:20.079><c> also</c><00:41:20.800><c> fluctuate,</c><00:41:22.000><c> right?</c>

39:50.475 --> 39:50.485 align:start position:0%
over here would also fluctuate, right?
 

39:50.485 --> 39:53.355 align:start position:0%
over here would also fluctuate, right?
And<00:41:22.880><c> since</c><00:41:23.280><c> size</c><00:41:23.760><c> square</c><00:41:24.319><c> is</c><00:41:24.800><c> the</c><00:41:25.119><c> probability</c>

39:53.355 --> 39:53.365 align:start position:0%
And since size square is the probability
 

39:53.365 --> 39:55.755 align:start position:0%
And since size square is the probability
of<00:41:25.760><c> finding</c><00:41:26.000><c> the</c><00:41:26.240><c> particle</c><00:41:26.640><c> somewhere</c><00:41:27.599><c> the</c>

39:55.755 --> 39:55.765 align:start position:0%
of finding the particle somewhere the
 

39:55.765 --> 39:58.795 align:start position:0%
of finding the particle somewhere the
total<00:41:28.480><c> area</c><00:41:29.119><c> under</c><00:41:29.599><c> this</c><00:41:30.000><c> curve</c><00:41:30.720><c> should</c>

39:58.795 --> 39:58.805 align:start position:0%
total area under this curve should
 

39:58.805 --> 40:00.154 align:start position:0%
total area under this curve should
represent<00:41:31.280><c> the</c><00:41:31.440><c> total</c><00:41:31.760><c> probability</c><00:41:32.079><c> of</c>

40:00.154 --> 40:00.164 align:start position:0%
represent the total probability of
 

40:00.164 --> 40:03.035 align:start position:0%
represent the total probability of
finding<00:41:33.119><c> this</c><00:41:33.440><c> electron</c><00:41:34.319><c> anywhere</c><00:41:34.800><c> in</c><00:41:34.960><c> the</c>

40:03.035 --> 40:03.045 align:start position:0%
finding this electron anywhere in the
 

40:03.045 --> 40:05.434 align:start position:0%
finding this electron anywhere in the
universe.<00:41:36.400><c> But</c><00:41:36.560><c> in</c><00:41:36.800><c> this</c><00:41:37.040><c> particular</c><00:41:37.280><c> case</c>

40:05.434 --> 40:05.444 align:start position:0%
universe. But in this particular case
 

40:05.444 --> 40:07.835 align:start position:0%
universe. But in this particular case
that<00:41:37.760><c> total</c><00:41:38.079><c> probability</c><00:41:38.880><c> fluctuates.</c><00:41:39.760><c> I</c>

40:07.835 --> 40:07.845 align:start position:0%
that total probability fluctuates. I
 

40:07.845 --> 40:10.395 align:start position:0%
that total probability fluctuates. I
mean<00:41:40.319><c> at</c><00:41:40.640><c> some</c><00:41:40.800><c> moment</c><00:41:41.440><c> it</c><00:41:41.680><c> can</c><00:41:41.839><c> even</c><00:41:42.079><c> go</c><00:41:42.240><c> to</c>

40:10.395 --> 40:10.405 align:start position:0%
mean at some moment it can even go to
 

40:10.405 --> 40:12.954 align:start position:0%
mean at some moment it can even go to
zero<00:41:43.040><c> for</c><00:41:43.200><c> example.</c><00:41:44.240><c> That</c><00:41:44.480><c> makes</c><00:41:44.720><c> no</c><00:41:44.880><c> sense</c>

40:12.954 --> 40:12.964 align:start position:0%
zero for example. That makes no sense
 

40:12.964 --> 40:14.795 align:start position:0%
zero for example. That makes no sense
right?<00:41:45.280><c> I</c><00:41:45.520><c> mean</c><00:41:45.599><c> the</c><00:41:45.760><c> total</c><00:41:46.160><c> probability</c><00:41:46.640><c> of</c>

40:14.795 --> 40:14.805 align:start position:0%
right? I mean the total probability of
 

40:14.805 --> 40:16.954 align:start position:0%
right? I mean the total probability of
finding<00:41:47.760><c> the</c><00:41:48.000><c> electron</c><00:41:48.480><c> somewhere</c><00:41:48.800><c> in</c><00:41:48.960><c> the</c>

40:16.954 --> 40:16.964 align:start position:0%
finding the electron somewhere in the
 

40:16.964 --> 40:20.635 align:start position:0%
finding the electron somewhere in the
universe<00:41:49.760><c> has</c><00:41:50.079><c> to</c><00:41:50.240><c> be</c><00:41:50.400><c> 100%.</c><00:41:51.359><c> Right?</c><00:41:52.000><c> But</c><00:41:52.560><c> how</c>

40:20.635 --> 40:20.645 align:start position:0%
universe has to be 100%. Right? But how
 

40:20.645 --> 40:22.954 align:start position:0%
universe has to be 100%. Right? But how
can<00:41:52.960><c> it</c><00:41:53.200><c> be</c><00:41:53.359><c> zero?</c><00:41:54.240><c> It</c><00:41:54.480><c> doesn't</c><00:41:54.720><c> make</c><00:41:54.880><c> any</c>

40:22.954 --> 40:22.964 align:start position:0%
can it be zero? It doesn't make any
 

40:22.964 --> 40:25.194 align:start position:0%
can it be zero? It doesn't make any
sense.<00:41:55.920><c> The</c><00:41:56.160><c> probability,</c><00:41:56.720><c> for</c><00:41:56.720><c> example,</c><00:41:57.119><c> can</c>

40:25.194 --> 40:25.204 align:start position:0%
sense. The probability, for example, can
 

40:25.204 --> 40:26.795 align:start position:0%
sense. The probability, for example, can
shift<00:41:57.520><c> from</c><00:41:57.680><c> place</c><00:41:57.920><c> to</c><00:41:58.079><c> place,</c><00:41:58.240><c> but</c><00:41:58.400><c> the</c><00:41:58.560><c> total</c>

40:26.795 --> 40:26.805 align:start position:0%
shift from place to place, but the total
 

40:26.805 --> 40:30.075 align:start position:0%
shift from place to place, but the total
should<00:41:59.200><c> never</c><00:41:59.760><c> change.</c><00:42:00.880><c> So</c><00:42:01.440><c> you</c><00:42:01.760><c> can</c><00:42:01.839><c> clearly</c>

40:30.075 --> 40:30.085 align:start position:0%
should never change. So you can clearly
 

40:30.085 --> 40:33.035 align:start position:0%
should never change. So you can clearly
see<00:42:02.400><c> a</c><00:42:02.720><c> simple</c><00:42:03.200><c> up</c><00:42:03.440><c> and</c><00:42:03.680><c> down</c><00:42:04.079><c> standing</c><00:42:04.480><c> wave</c>

40:33.035 --> 40:33.045 align:start position:0%
see a simple up and down standing wave
 

40:33.045 --> 40:35.674 align:start position:0%
see a simple up and down standing wave
can't<00:42:05.760><c> give</c><00:42:06.000><c> us</c><00:42:06.560><c> a</c><00:42:06.800><c> probability</c><00:42:07.280><c> wave,</c><00:42:07.680><c> a</c>

40:35.674 --> 40:35.684 align:start position:0%
can't give us a probability wave, a
 

40:35.684 --> 40:37.355 align:start position:0%
can't give us a probability wave, a
matter<00:42:08.160><c> wave.</c><00:42:08.560><c> There's</c><00:42:08.800><c> no</c><00:42:08.960><c> way</c><00:42:09.040><c> to</c><00:42:09.200><c> make</c><00:42:09.359><c> it</c>

40:37.355 --> 40:37.365 align:start position:0%
matter wave. There's no way to make it
 

40:37.365 --> 40:39.355 align:start position:0%
matter wave. There's no way to make it
work.

40:39.355 --> 40:39.365 align:start position:0%
work.
 

40:39.365 --> 40:42.555 align:start position:0%
work.
But<00:42:12.240><c> a</c><00:42:12.560><c> wave</c><00:42:13.119><c> that</c><00:42:13.440><c> is</c><00:42:13.680><c> rotating</c><00:42:14.079><c> in</c><00:42:14.319><c> complex</c>

40:42.555 --> 40:42.565 align:start position:0%
But a wave that is rotating in complex
 

40:42.565 --> 40:46.075 align:start position:0%
But a wave that is rotating in complex
plane<00:42:15.200><c> has</c><00:42:15.760><c> no</c><00:42:16.240><c> such</c><00:42:16.720><c> problem.</c>

40:46.075 --> 40:46.085 align:start position:0%
plane has no such problem.
 

40:46.085 --> 40:49.194 align:start position:0%
plane has no such problem.
Look<00:42:18.880><c> here</c><00:42:19.200><c> at</c><00:42:19.520><c> every</c><00:42:19.839><c> point</c><00:42:20.400><c> in</c><00:42:20.720><c> the</c><00:42:20.960><c> complex</c>

40:49.194 --> 40:49.204 align:start position:0%
Look here at every point in the complex
 

40:49.204 --> 40:52.954 align:start position:0%
Look here at every point in the complex
plane<00:42:21.760><c> the</c><00:42:22.000><c> height</c><00:42:22.400><c> of</c><00:42:22.720><c> this</c><00:42:23.040><c> I</c><00:42:23.680><c> stays</c><00:42:24.160><c> exactly</c>

40:52.954 --> 40:52.964 align:start position:0%
plane the height of this I stays exactly
 

40:52.964 --> 40:55.434 align:start position:0%
plane the height of this I stays exactly
the<00:42:25.440><c> same</c><00:42:25.839><c> because</c><00:42:26.079><c> it's</c><00:42:26.319><c> just</c><00:42:26.560><c> spinning.</c><00:42:27.359><c> So</c>

40:55.434 --> 40:55.444 align:start position:0%
the same because it's just spinning. So
 

40:55.444 --> 40:58.154 align:start position:0%
the same because it's just spinning. So
what<00:42:27.760><c> happens</c><00:42:28.000><c> to</c><00:42:28.240><c> s</c><00:42:28.560><c> squared?</c><00:42:29.440><c> The</c><00:42:29.599><c> si</c><00:42:29.920><c> square</c>

40:58.154 --> 40:58.164 align:start position:0%
what happens to s squared? The si square
 

40:58.164 --> 41:00.954 align:start position:0%
what happens to s squared? The si square
stays<00:42:30.960><c> fixed</c><00:42:32.160><c> which</c><00:42:32.400><c> means</c><00:42:32.640><c> the</c><00:42:32.880><c> total</c>

41:00.954 --> 41:00.964 align:start position:0%
stays fixed which means the total
 

41:00.964 --> 41:03.995 align:start position:0%
stays fixed which means the total
probability<00:42:33.680><c> now</c><00:42:34.000><c> stays</c><00:42:34.880><c> fixed.</c><00:42:35.760><c> Now</c><00:42:36.079><c> of</c>

41:03.995 --> 41:04.005 align:start position:0%
probability now stays fixed. Now of
 

41:04.005 --> 41:05.914 align:start position:0%
probability now stays fixed. Now of
course<00:42:36.400><c> for</c><00:42:36.640><c> more</c><00:42:36.880><c> complicated</c><00:42:37.359><c> matter</c><00:42:37.680><c> waves</c>

41:05.914 --> 41:05.924 align:start position:0%
course for more complicated matter waves
 

41:05.924 --> 41:07.995 align:start position:0%
course for more complicated matter waves
the<00:42:38.319><c> probability</c><00:42:38.960><c> will</c><00:42:39.280><c> change</c><00:42:39.520><c> with</c><00:42:39.760><c> time</c>

41:07.995 --> 41:08.005 align:start position:0%
the probability will change with time
 

41:08.005 --> 41:10.075 align:start position:0%
the probability will change with time
but<00:42:40.319><c> the</c><00:42:40.560><c> total</c><00:42:40.880><c> probability</c><00:42:41.520><c> will</c><00:42:41.839><c> still</c>

41:10.075 --> 41:10.085 align:start position:0%
but the total probability will still
 

41:10.085 --> 41:12.954 align:start position:0%
but the total probability will still
stay<00:42:42.480><c> the</c><00:42:42.800><c> same.</c><00:42:43.680><c> However,</c><00:42:44.480><c> if</c><00:42:44.720><c> you</c><00:42:44.880><c> were</c><00:42:44.960><c> to</c>

41:12.954 --> 41:12.964 align:start position:0%
stay the same. However, if you were to
 

41:12.964 --> 41:14.795 align:start position:0%
stay the same. However, if you were to
model<00:42:45.440><c> matter</c><00:42:45.839><c> waves</c><00:42:46.160><c> using</c><00:42:46.400><c> just</c><00:42:46.640><c> up</c><00:42:46.800><c> and</c>

41:14.795 --> 41:14.805 align:start position:0%
model matter waves using just up and
 

41:14.805 --> 41:17.835 align:start position:0%
model matter waves using just up and
down<00:42:47.119><c> oscilly</c><00:42:47.680><c> motion</c><00:42:48.000><c> in</c><00:42:48.240><c> just</c><00:42:48.640><c> real</c><00:42:49.040><c> axis,</c>

41:17.835 --> 41:17.845 align:start position:0%
down oscilly motion in just real axis,
 

41:17.845 --> 41:19.674 align:start position:0%
down oscilly motion in just real axis,
there's<00:42:50.400><c> no</c><00:42:50.720><c> way</c><00:42:50.880><c> for</c><00:42:51.119><c> that</c><00:42:51.280><c> to</c><00:42:51.359><c> happen.</c><00:42:51.599><c> Even</c>

41:19.674 --> 41:19.684 align:start position:0%
there's no way for that to happen. Even
 

41:19.684 --> 41:22.635 align:start position:0%
there's no way for that to happen. Even
the<00:42:52.079><c> simplest</c><00:42:52.640><c> wave,</c><00:42:53.359><c> you</c><00:42:53.599><c> can't</c><00:42:53.839><c> model</c><00:42:54.079><c> it.</c>

41:22.635 --> 41:22.645 align:start position:0%
the simplest wave, you can't model it.
 

41:22.645 --> 41:25.835 align:start position:0%
the simplest wave, you can't model it.
So,<00:42:55.680><c> it's</c><00:42:56.000><c> the</c><00:42:56.319><c> i</c><00:42:56.960><c> that</c><00:42:57.280><c> makes</c><00:42:57.520><c> the</c><00:42:57.760><c> wave</c>

41:25.835 --> 41:25.845 align:start position:0%
So, it's the i that makes the wave
 

41:25.845 --> 41:30.555 align:start position:0%
So, it's the i that makes the wave
function<00:42:58.560><c> spin</c><00:42:59.680><c> and</c><00:43:00.160><c> conserves</c><00:43:01.440><c> the</c><00:43:01.760><c> total</c>

41:30.555 --> 41:30.565 align:start position:0%
function spin and conserves the total
 

41:30.565 --> 41:34.875 align:start position:0%
function spin and conserves the total
probability.<00:43:04.160><c> It's</c><00:43:04.480><c> the</c><00:43:04.960><c> square</c><00:43:05.440><c> root</c><00:43:05.839><c> of</c><00:43:06.400><c> -1,</c>

41:34.875 --> 41:34.885 align:start position:0%
probability. It's the square root of -1,
 

41:34.885 --> 41:37.755 align:start position:0%
probability. It's the square root of -1,
the<00:43:07.359><c> imaginary</c><00:43:08.079><c> number</c><00:43:09.119><c> that</c><00:43:09.440><c> makes</c><00:43:09.680><c> the</c>

41:37.755 --> 41:37.765 align:start position:0%
the imaginary number that makes the
 

41:37.765 --> 41:41.755 align:start position:0%
the imaginary number that makes the
physics<00:43:11.280><c> real.</c>

41:41.755 --> 41:41.765 align:start position:0%
physics real.
 

41:41.765 --> 41:43.515 align:start position:0%
physics real.
Schroinger<00:43:14.480><c> solved</c><00:43:14.720><c> his</c><00:43:14.960><c> equation</c><00:43:15.280><c> for</c><00:43:15.520><c> the</c>

41:43.515 --> 41:43.525 align:start position:0%
Schroinger solved his equation for the
 

41:43.525 --> 41:46.475 align:start position:0%
Schroinger solved his equation for the
hydrogen<00:43:16.079><c> atom</c><00:43:16.400><c> using</c><00:43:16.720><c> the</c><00:43:17.200><c> 1</c><00:43:17.440><c> /</c><00:43:17.760><c> r</c><00:43:18.160><c> potential</c>

41:46.475 --> 41:46.485 align:start position:0%
hydrogen atom using the 1 / r potential
 

41:46.485 --> 41:48.395 align:start position:0%
hydrogen atom using the 1 / r potential
function.<00:43:19.440><c> And</c><00:43:19.680><c> not</c><00:43:19.920><c> only</c><00:43:20.079><c> did</c><00:43:20.240><c> he</c><00:43:20.319><c> get</c><00:43:20.400><c> the</c>

41:48.395 --> 41:48.405 align:start position:0%
function. And not only did he get the
 

41:48.405 --> 41:50.475 align:start position:0%
function. And not only did he get the
hydrogen<00:43:20.960><c> spectrum,</c><00:43:21.440><c> he</c><00:43:21.680><c> also</c><00:43:22.000><c> predicted</c><00:43:22.400><c> the</c>

41:50.475 --> 41:50.485 align:start position:0%
hydrogen spectrum, he also predicted the
 

41:50.485 --> 41:52.795 align:start position:0%
hydrogen spectrum, he also predicted the
relative<00:43:23.200><c> brightness</c><00:43:23.680><c> of</c><00:43:23.920><c> each</c><00:43:24.160><c> line</c><00:43:24.720><c> and</c>

41:52.795 --> 41:52.805 align:start position:0%
relative brightness of each line and
 

41:52.805 --> 41:54.555 align:start position:0%
relative brightness of each line and
showed<00:43:25.200><c> how</c><00:43:25.440><c> adding</c><00:43:25.760><c> electric</c><00:43:26.160><c> or</c><00:43:26.319><c> magnetic</c>

41:54.555 --> 41:54.565 align:start position:0%
showed how adding electric or magnetic
 

41:54.565 --> 41:57.434 align:start position:0%
showed how adding electric or magnetic
fields<00:43:27.280><c> splits</c><00:43:27.839><c> these</c><00:43:28.160><c> lines.</c><00:43:29.200><c> But</c><00:43:29.359><c> more</c>

41:57.434 --> 41:57.444 align:start position:0%
fields splits these lines. But more
 

41:57.444 --> 41:59.595 align:start position:0%
fields splits these lines. But more
importantly,<00:43:30.480><c> the</c><00:43:30.720><c> equation</c><00:43:31.040><c> predicted</c><00:43:31.440><c> the</c>

41:59.595 --> 41:59.605 align:start position:0%
importantly, the equation predicted the
 

41:59.605 --> 42:00.875 align:start position:0%
importantly, the equation predicted the
three-dimensional<00:43:32.480><c> probability</c>

42:00.875 --> 42:00.885 align:start position:0%
three-dimensional probability
 

42:00.885 --> 42:04.154 align:start position:0%
three-dimensional probability
distributions<00:43:34.319><c> of</c><00:43:34.640><c> the</c><00:43:34.880><c> electrons</c><00:43:35.839><c> in</c><00:43:36.160><c> the</c>

42:04.154 --> 42:04.164 align:start position:0%
distributions of the electrons in the
 

42:04.164 --> 42:07.595 align:start position:0%
distributions of the electrons in the
hydrogen<00:43:36.800><c> atom,</c><00:43:37.920><c> the</c><00:43:38.240><c> orbitals.</c>

42:07.595 --> 42:07.605 align:start position:0%
hydrogen atom, the orbitals.
 

42:07.605 --> 42:09.755 align:start position:0%
hydrogen atom, the orbitals.
In<00:43:40.000><c> short,</c><00:43:40.319><c> the</c><00:43:40.480><c> equation</c><00:43:40.720><c> was</c><00:43:40.960><c> a</c><00:43:41.119><c> complete</c>

42:09.755 --> 42:09.765 align:start position:0%
In short, the equation was a complete
 

42:09.765 --> 42:12.315 align:start position:0%
In short, the equation was a complete
radical<00:43:42.480><c> breakthrough.</c><00:43:43.200><c> It</c><00:43:43.359><c> became</c><00:43:43.680><c> the</c><00:43:43.920><c> F=</c>

42:12.315 --> 42:12.325 align:start position:0%
radical breakthrough. It became the F=
 

42:12.325 --> 42:15.035 align:start position:0%
radical breakthrough. It became the F=
MA<00:43:44.960><c> of</c><00:43:45.280><c> quantum</c><00:43:45.680><c> mechanics.</c><00:43:46.480><c> As</c><00:43:46.720><c> a</c><00:43:46.880><c> result,</c>

42:15.035 --> 42:15.045 align:start position:0%
MA of quantum mechanics. As a result,
 

42:15.045 --> 42:17.674 align:start position:0%
MA of quantum mechanics. As a result,
Schroinger<00:43:47.760><c> shared</c><00:43:48.000><c> the</c><00:43:48.160><c> 1933</c><00:43:49.119><c> physics</c><00:43:49.520><c> Nobel</c>

42:17.674 --> 42:17.684 align:start position:0%
Schroinger shared the 1933 physics Nobel
 

42:17.684 --> 42:20.075 align:start position:0%
Schroinger shared the 1933 physics Nobel
Prize<00:43:50.240><c> with</c><00:43:50.560><c> Paul</c><00:43:50.800><c> Durac.</c><00:43:51.680><c> And</c><00:43:51.839><c> today,</c>

42:20.075 --> 42:20.085 align:start position:0%
Prize with Paul Durac. And today,
 

42:20.085 --> 42:21.755 align:start position:0%
Prize with Paul Durac. And today,
scientists<00:43:52.720><c> and</c><00:43:52.960><c> engineers</c><00:43:53.359><c> have</c><00:43:53.520><c> done</c>

42:21.755 --> 42:21.765 align:start position:0%
scientists and engineers have done
 

42:21.765 --> 42:23.914 align:start position:0%
scientists and engineers have done
extraordinary<00:43:54.720><c> things</c><00:43:54.880><c> with</c><00:43:55.119><c> it.</c><00:43:55.680><c> We've</c>

42:23.914 --> 42:23.924 align:start position:0%
extraordinary things with it. We've
 

42:23.924 --> 42:26.555 align:start position:0%
extraordinary things with it. We've
found<00:43:56.400><c> ways</c><00:43:56.720><c> to</c><00:43:56.960><c> focus</c><00:43:57.440><c> electron</c><00:43:57.839><c> waves</c><00:43:58.240><c> and</c>

42:26.555 --> 42:26.565 align:start position:0%
found ways to focus electron waves and
 

42:26.565 --> 42:29.115 align:start position:0%
found ways to focus electron waves and
build<00:43:59.440><c> electron</c><00:43:59.920><c> microscopes</c><00:44:00.800><c> powerful</c>

42:29.115 --> 42:29.125 align:start position:0%
build electron microscopes powerful
 

42:29.125 --> 42:31.355 align:start position:0%
build electron microscopes powerful
enough<00:44:01.599><c> to</c><00:44:01.760><c> see</c><00:44:02.000><c> individual</c><00:44:02.400><c> atoms.</c><00:44:03.200><c> We've</c>

42:31.355 --> 42:31.365 align:start position:0%
enough to see individual atoms. We've
 

42:31.365 --> 42:33.115 align:start position:0%
enough to see individual atoms. We've
calculated<00:44:04.079><c> approximate</c><00:44:04.720><c> energy</c><00:44:05.040><c> levels</c>

42:33.115 --> 42:33.125 align:start position:0%
calculated approximate energy levels
 

42:33.125 --> 42:35.115 align:start position:0%
calculated approximate energy levels
inside<00:44:05.680><c> heavier</c><00:44:06.079><c> atoms</c><00:44:06.400><c> like</c><00:44:06.640><c> cesium</c><00:44:07.200><c> for</c>

42:35.115 --> 42:35.125 align:start position:0%
inside heavier atoms like cesium for
 

42:35.125 --> 42:38.795 align:start position:0%
inside heavier atoms like cesium for
example<00:44:07.920><c> to</c><00:44:08.160><c> build</c><00:44:08.400><c> atomic</c><00:44:08.880><c> clocks.</c><00:44:10.079><c> And</c><00:44:10.400><c> when</c>

42:38.795 --> 42:38.805 align:start position:0%
example to build atomic clocks. And when
 

42:38.805 --> 42:41.275 align:start position:0%
example to build atomic clocks. And when
multiple<00:44:11.359><c> atoms</c><00:44:11.760><c> come</c><00:44:12.079><c> close</c><00:44:12.400><c> together,</c><00:44:13.200><c> we</c>

42:41.275 --> 42:41.285 align:start position:0%
multiple atoms come close together, we
 

42:41.285 --> 42:43.115 align:start position:0%
multiple atoms come close together, we
found<00:44:13.599><c> that</c><00:44:13.760><c> those</c><00:44:14.000><c> energy</c><00:44:14.319><c> levels</c><00:44:14.560><c> turn</c><00:44:14.800><c> into</c>

42:43.115 --> 42:43.125 align:start position:0%
found that those energy levels turn into
 

42:43.125 --> 42:46.635 align:start position:0%
found that those energy levels turn into
bands.<00:44:16.240><c> And</c><00:44:16.480><c> by</c><00:44:16.800><c> controlling</c><00:44:17.599><c> the</c><00:44:17.920><c> band</c><00:44:18.240><c> gaps,</c>

42:46.635 --> 42:46.645 align:start position:0%
bands. And by controlling the band gaps,
 

42:46.645 --> 42:48.954 align:start position:0%
bands. And by controlling the band gaps,
we've<00:44:19.119><c> built</c><00:44:19.680><c> ultra</c><00:44:20.079><c> tiny</c><00:44:20.560><c> transistor</c>

42:48.954 --> 42:48.964 align:start position:0%
we've built ultra tiny transistor
 

42:48.964 --> 42:51.515 align:start position:0%
we've built ultra tiny transistor
switches<00:44:21.599><c> that</c><00:44:21.920><c> make</c><00:44:22.079><c> up</c><00:44:22.560><c> every</c><00:44:23.040><c> single</c>

42:51.515 --> 42:51.525 align:start position:0%
switches that make up every single
 

42:51.525 --> 42:53.194 align:start position:0%
switches that make up every single
microchip.

42:53.194 --> 42:53.204 align:start position:0%
microchip.
 

42:53.204 --> 42:56.954 align:start position:0%
microchip.
Our<00:44:25.839><c> real</c><00:44:26.480><c> modern</c><00:44:27.040><c> world</c><00:44:27.599><c> brought</c><00:44:27.920><c> to</c><00:44:28.160><c> you</c><00:44:28.400><c> by</c>

42:56.954 --> 42:56.964 align:start position:0%
Our real modern world brought to you by
 

42:56.964 --> 43:00.845 align:start position:0%
Our real modern world brought to you by
the<00:44:29.359><c> imaginary</c><00:44:30.000><c> number.</c>