This single equation helped unlock the
modern world. It gave us computer chips,
electron microscopes, atomic clocks,
GPS, high-speed internet, the list goes
on. But where did this equation come
from? Well, according to Richard
Feineman, it comes from
>> nowhere. Out of man's imagination,
struggles with the details of experiment
and all kinds of mysteries.
>> That man was Irvin Schroinger. He
derived it in 1925 while vacationing in
the Swiss Alps with his mistress.
So I have two questions. First, how can
we intuitively build this equation
ourselves from scratch? But secondly,
why is an equation with such real impact
built using an imaginary number? What's
I doing there? If you're ready, let's
find out.
It all starts in 1853 when the physicist
Anders Enstrom finds that hot hydrogen
gas gives out very specific colors of
light. We soon figured out it's not just
hydrogen. Every hot element gives out
its own signature spectrum. And suddenly
this became a powerful tool to discover
brand new elements just by looking at
their light. This is how we discovered
helium for the very first time in the
sun. So in his honor, the unit for
measuring these wavelengths was named
the Enstrom.
But nobody knew why these elements gave
out those specific colors of light. The
first clue actually came a few decades
later from a Swiss math teacher named
John Balmer. Balmer was obsessed with
numbers and patterns. And he finds a
surprisingly simple formula for the
hydrogen spectrum just by trial and
error. And the formula predicts there
should be more lines. And we actually
confirm it. But nobody had any clue what
this formula meant. Why did it work?
To answer this, we needed a good theory
of light. By now, we knew light is a
wave confirmed by the interference
pattern. And Maxwell showed that light
is basically a ripple in the
electromagnetic field produced by
accelerating charges. So wiggling
charges produce light. Wiggling it
slowly gives us low frequency light and
wiggling it faster gets us high
frequency light. And these EM waves or
light could wiggle other charges which
meant wireless communication.
It was a breakthrough in communication
technology, but it couldn't explain the
hydrogen spectrum.
See, according to Maxwell, a hot glowing
gas has billions of randomly jiggling
charges from very low frequencies to
very high frequencies, which meant they
should be giving out every color of
light,
but they didn't. So something was
horribly wrong with Maxwell's theory.
But it got worse. Pretty soon we
discovered that atoms had a positive
nuclear core. So we thought the negative
electrons must be orbiting this nucleus,
making the atoms stable. But if they did
that, they would be constantly
accelerating and accelerating charges
radiate EM waves. So they would lose
energy and collapse. So now we couldn't
even explain why atoms were stable.
Physics seemed to be in crisis. But a
few years later, everything changed.
If you shine UV light on zinc, for
example, electrons come out. That makes
sense. Electrons receive energy from the
EM waves. But if you shine a brighter
visible light, no electrons came out.
That didn't make any sense. I mean,
you're pumping in more energy. So, we
would expect electrons to come out with
more energy. So, why didn't they why did
the color matter and not the brightness?
To explain this, Albert Einstein
proposed something radical. What if
light doesn't deliver energy
continuously but in discrete chunks
called photons? And what if the energy
of each photon depends only on its
frequency? Then a bright visible light
will deliver a lot of photons per
second, but each one's too weak to budge
an electron. It's like throwing ping
pong balls at a bowling ball. But on the
other hand, a dim UV light will deliver
fewer photons per second, but each one
is strong enough to knock it off like a
single cannonball.
So this explained the photoelectric
mystery beautifully and Einstein won the
Nobel Prize for it. And this constant is
you probably know the planks constant.
But a few years later, a Danish
physicist named Neils Boore pushed this
idea even further.
Bore wondered if light is absorbed in
chunks, it must also be emitted in
chunks, right? Which meant electrons had
to transition from a higher energy level
straight to a lower energy one to
release these energy chunks. So he
postulated that electrons orbit the
nucleus only at some special energy
levels, nowhere in between. And in these
special orbits, he said they don't
radiate energy. It seemed like he was
just making stuff up. But look what
happens now. When you heat up a gas,
electrons jump to a higher allowed
level. But when they fall back down,
they release the energy difference as a
photon of a specific frequency. This
explained the hydrogen spectrum. And the
photon's energy is basically the energy
lost by the electron. From this he
derived an expression for the frequency
and got the Balmer's formula. So his
postulate explained the hydrogen
spectrum, the Balmer's formula and the
atomic stability all at once. This was
huge.
Later in an interview, he says, "As soon
as I saw Balmer's formula, the whole
thing was immediately clear to me." He
was probably showing off, but there were
many unanswered questions here. I mean,
why were electrons restricted to those
specific orbits? and why don't they
radiate while they are there? Boore
basically said, "Trust me." But a French
PhD student named Louis De Bruy came up
with an answer. He pushed Bor's idea to
the limit. Light moves as a wave but
interacts with stuff like a particle,
right? Dual nature. So he wondered, what
if matter behaves the same way? What if
matter 2 interacts like a particle but
moves as a wave? Then electrons inside
an atom could form standing waves. And
just like a guitar string can only
vibrate with say three loops or four
loops or five but nothing in between.
Electron waves can only vibrate with
three or four or five loops but nothing
in between.
This explained why electrons can only
exist at those specific distances from
the nucleus as Boore said. And it also
explains why they don't radiate while
sitting at those energy levels because
they're not orbiting particles. They're
not accelerating. They're stationary
waves now. They only radiate a photon
when they transition from a higher to a
lower level.
This was so radical. Matter behaving as
waves. And that's not all. For his PhD
thesis, he derived an expression for its
wavelength using special relativity. He
had found wavelength of matter. It seems
so bizarre. His examiners only approved
it after confirming it with Einstein
himself. Finally in the following year a
professor at the University of Zurich
gave a seminar on this very topic and
after the talk a colleague in the
audience asked if matter is a wave where
is the wave equation. The professor was
Irvin Schroinger and he took that
question seriously. So he went on a
vacation to the Swiss Alps over
Christmas and came back with the wave
equation. But how did he do it?
I went through Shinger's original
derivation and I couldn't understand a
thing. In fact, he himself called it
unintelligible later on. So, I looked up
Fineman's derivation and that was also
quite mathy. But after a lot of
searching, I found a modern version
which seemed pretty intuitive. It starts
by asking what rule should matter waves
obey in general. The most fundamental
one we know is energy conservation,
right? Total energy equals kinetic
energy plus potential energy. Okay, that
makes sense. Um, kinetic energy is half
mv squared. So if you multiply the top
and the bottom by m, we can write it as
momentum squared over 2m. And I'm like,
perfect. This is all starting to make
sense. And in the final step, it says to
make it quantum, replace energy with
energy operator. Wait, what? kinetic
energy with kinetic energy operator.
What? What's going on? And they act on
the wave function sigh, giving us the
Schroinger's equation. What just
happened?
After calming down a bit, I realized I
just had to answer two questions. First
of all, why do we need these operators?
And second of all, how do I build them
myself intuitively? So, let's start with
the first question. We know E equals HF.
And we also know P= H / lambda deoy. Why
not just substitute them directly into
this equation, right? That was what I
was thinking. Well, the answer is
actually right in front of us. You see,
that would only work for an infinitely
long pure sine wave because look, only
then it would have one single definite
veil length and frequency.
But a general wave doesn't have a single
wavelength or frequency. It can have a
whole range all mixed together.
Now for ordinary waves, that's not a big
deal. But for matter waves, think about
what it means. This means the particle
doesn't even have a single value of
momentum or energy. At this point, we
don't even know how to interpret that. I
mean, what does it even mean for an
electron to not have a definite energy?
Well, let's keep that question aside.
We'll come back to it later. But for
now, what's important is because mow
waves don't have a single value of
energy or momentum, we can't substitute
directly. Instead, we need something
that extracts the entire mixture.
That's what these operators do. The
energy operator over here pulls the
entire range of energy hiding in the
wave. Similarly, the kinetic energy
operator pulls the entire range of
kinetic energy. That's why in quantum
mechanics we always talk about
operators, right? Because matter waves
don't have single values for energy or
momentum or whatever. They have a whole
range. Okay, so first question answered.
On to the second one and the most
important one. How do I build these
operators myself? Here's a powerful
problem solving principle. When you're
trying to solve a hard problem, first
see if you can create a simpler version
of that problem and try to solve that.
So in our case, the hard problem is to
build operators that work in general.
The simpler version would be to try and
build these operators for the simplest
wave possible,
a sine wave.
So maybe if I can build an operator for
a sine wave, I can then use that
intuition to generalize it. First of
all, this could be a traveling wave or
you know it could be a standing wave
like you know de Bruy described. I like
standing waves. The math feels slightly
more intuitive. So let's go with that.
Let's draw a couple of axes. You have
x-axis and you have s which is you know
shinger's own notation. So the first
question is what is the equation for s?
Well, let's pause the animation. We can
write s= sinx. I mean we could also
write s= cos x. It's just a matter of
where we put the origin. But let's stick
to sinx. That's the simplest equation,
right? But guess what? S's height swings
between +1 and minus1. I want our height
to be slightly more general. Let's call
it a. So how do we do that? Well, we
scale this by a. That's the amplitude.
Now I can control the amplitude.
Perfect. But sign always resets after 2
pi. Which means right now our wavelength
is locked exactly at 2 pi. I don't want
that. I want to be able to have any
wavelength. I want lambda. So what do we
do? Well, we multiply x by 2 pi over
lambda. I mean, think about it. Now when
x equals lambda the lambda cancels out
and the argument becomes 2 pi and the
wave resets. So now lambda has become
our wavelength. But this isn't a wave
yet. It's a frozen picture.
A standing wave means the amplitude
itself changes over time periodically.
So a itself needs to be some periodic
function of time. Again we'll choose the
simplest function s but just like before
s has a period of 2 pi. So right now our
you know time period is locked at 2 pi.
So we want the period to be let's say
capital t. So we use the same trick as
before. We multiply this by 2 pi over
capital t. And we are done. We just need
a bit of cleaning up over here. 1 / t is
frequency. And physicists hate writing 2
pi over and over again. So we will
define 2 pi f as a new variable omega
and similarly 2 pi over lambda as a new
variable kappa. We're only doing this so
that we don't have to write two pies
over and over again. Okay. If we
substitute it, boom, we have built the
equation for our standing wave. But this
is still a generic wave. There's nothing
quantum about it.
To make it quantum, we bring in Einstein
and De Bruy equation. Now, since this is
a pure sine wave of one specific
frequency and wavelength, we can
directly substitute over here. But
before we do that, we have to write this
in terms of omega and kappa as well. So,
let's quickly do that. To do that, we'll
just multiply and divide by 2 pi
everywhere. And now look at what we get.
H over 2 pi, we'll call that as h bar.
We call this the reduced plank constant.
And 2 pi f is omega. And 2 pi over
lambda. Well, we have kappa. And this
cleaning up actually makes things much
more beautiful. I mean, think about it.
What exactly is omega over here? Omega
basically tells us number of waves per
second, right? So we can call it the
temporal frequency. What about kappa?
Well, kappa tells us the number of waves
per meter. Look at this. per meter. So
that is the spatial frequency
which means for matter waves the
temporal frequency encodes the total
energy. It lives in the time domain and
the spatial frequency encodes the
momentum. It lives in the space domain.
When I saw this, a light bulb went off.
I mean, think about it. You probably
know that in special relativity, space
and time are not two separate things.
They're just shadows of the underlying
four-dimensional spaceime, right? So, we
could guess that energy and momentum
aren't separate things. Maybe they are
just two components,
shadows of something much deeper, a
four-dimensional object. And that's
exactly what we have in special
relativity. It's called four momentum.
So, in relativity, we don't think of
energy and momentum separately. We just
think of them as two components of the
underlying
object called for momentum. I know this
is a deg uh this is a tangent but oh my
god like that connection is beautiful.
Anyways we substitute now for omega and
kappa and boom we have built our very
first mowave equation. But remember what
our actual goal was here was to build
energy and kinetic energy operators. So
let's start with kinetic energy. Since
it has momentum squared in it, our
question would be how do we pull
momentum out of this equation? We can
differentiate it with respect to x,
right? Well, actually we need to do
partial derivative. Then this part
becomes a constant. And now look, p or h
pops out in front and s turns to cos.
But we don't want just momentum. We want
momentum squared because our goal is to
get kinetic energy. So what do we do?
Well, we differentiate again. On the
left hand side, we get a second
derivative. And over here, P / H pops
out one more time. And cost becomes
negative sign. And if you look
carefully, look, this is our original
function s. So we have got sigh back.
And if we rearrange,
we have built our very first operator,
the momentum squared operator.
Look, when you do this operation on S,
you get momentum squared time S. So
momentum squared has been extracted. So
this is the momentum squared operator.
For the last step, I need kinetic
energy, right? So kinetic energy is just
momentum square by 2 m. So let me just
divide by 2 m.
And this is the kinetic energy. So I
have found how to extract kinetic
energy. So this must be the kinetic
energy operator.
Wow. We've built it all by ourselves.
And if we compare it to the actual
Schroinger's equation, oh my god, it's
exactly the same thing.
Whoa.
All right, let's calm down. But what
does it actually say?
Well, second derivative is basically
curvature, right? So this says if your
wave has more curvature, then it will
have more kinetic energy. And that makes
sense because we already saw that you
know from Droy's equation that shorter
wavelength means more momentum means
more kinetic energy. Shorter wavelength
means more curvature. But our
understanding has upgraded because the
idea of wavelength only works for pure
sine waves right but the idea of
curvature can be defined
at every single point. So it is a
general way of thinking about it.
Kinetic energy is encoded in the
curvature.
Oh man that is awesome. But this brings
up an annoying problem.
See, we derived this operator for a
special case, a pure sine wave, right?
But it turns out the operator works in
general for any shape. Now, that is
awesome. I'm not complaining because
this means we can do the same thing for
building the energy operator and then we
can finish the storing equation and
fulfill our destiny. But I won't be able
to sleep at night because although we
have some intuition for why this should
work in general, I can't actually
convince myself mathematically why
something we derived for one specific
special case perfectly works in general.
I was stuck here for a while until I met
a man who was so obsessed with heat that
apparently he kept his room at blazing
temperatures and he sat inside them
wrapped in a blanket during hot summers.
His name John Baptist Joseph Forier.
Forier believed heat had magical healing
powers. But historians think it's
probably because, you know, he developed
extreme cold sensitivity during his time
in the Egyptian desert. But whatever it
is, what's important for us is that he
was a mathematical genius. And so
obviously he wanted to mathematically
model how heat flows. And he did that.
And while doing so he developed an
incredible principle. He found that any
wave or any shape at all can be written
as a sum of ss and cosine waves. Here's
what I mean. Take this square wave.
According to Foryer, you can write this
as just sums of ss and cosiness. If you
just use one sine wave, that doesn't
look like much. But if you add a second
one with a slightly different height and
width, look, it gets closer. Add a third
and a fourth and a fifth and you keep
going and look look the shape converges
towards a perfect square wave. Now of
course technically we need infinitely
many but look I mean even with a few we
can get remarkably close right
here's another example again by adding
multiple sine waves of just the right
width and height we can construct this
shape too. we can construct any shape
and forer showed that this works in
general and the idea is called the 4year
series or more generally it's called the
4year transforms. This idea was so
radical that even the top mathematicians
back then just couldn't believe it this
would be true. But today we have a very
elegant proof for it. Let me know if you
want me to make a video on that. But for
now we'll just accept that. So according
to Foryer our matter wave can be written
as the sum of lots and lots of pure sine
waves. So now what will happen if we
apply that operator we built to this
general wave? Well derivatives are
linear meaning derivative of a plus b
equals derivative of a plus derivative
of b. That means this operator gets
applied to every single component.
For each component, it spits out the
components kinetic energy multiplied by
the components wave function. And then
it adds up all of those and spits this
entire sum back. Which means look the
operator when working on a general wave
spits out the full mixture of the
kinetic energy as a weighted sum. And so
for helps us understand why if an
operator works for a sine wave it should
work for any wave in general. Oh man for
you beauty. Imagine if forier knew that
his obsession with heat would one day
unlock the framework for just you know
modeling quantum world. Oh my god, how
would he be feeling? Oh my god. Anyways,
this means all we have to do is repeat
the same process for extracting energy
and we are done. It will be a great idea
to pause the video over here and see if
you can try this yourself. Please do
that moment of truth.
All right. So to extract energy look I
have to differentiate with respect to
time. This time this term is a constant.
So nothing happens to it. So when I
differentiate sign well again e over h
pops out and sign turns into cos. We
have our energy. So let's just
rearrange.
And
wait
there's a problem. I'm not getting my
function back because sine turned into
cos. So I can't write this as s. Wait,
why did it work last time? Oh, last time
it worked because we took a double
derivative, right? Because I wanted
momentum squared. And so when I took a
second derivative, well, s turned to
cost and cost turned back to s. And I
was able to cleanly write this as s and
build my operator. Oh, but this time I
just need a single derivative because I
already found energy. I don't want
energy squared. So, I can't take a
second derivative. But I can't write
this as s because the sign has turned
into cost. Oh man, we were so close.
What? What do we do?
Let's just think mathematically. For
this to work, to get our s back, this
function of time has to be such that its
first derivative must be itself. But a s
or a cosine or any periodic function for
that matter, none of them give us that.
There's only one function in this entire
multiverse with that property. It's the
exponential function. I mean, think
about it for a second. Imagine that this
function was exponential in time. Now,
if we differentiate it with respect to
time, again, e over h bar pops out. And
this time because the derivative of
exponential is itself. Look, we get our
function s back. We can now rearrange
and build our operator just like before.
And the math would work out. But the
problem is this is not a wave. For a
wave, the amplitude needs to be some
kind of a periodic function of time. But
exponentials are not periodic function.
Exponentials will just keep blowing up
forever,
right?
Or if you if you put a negative side for
example then yeah then the exponential
would just keep decaying forever.
Whatever it is this is not a wave.
So we have a problem. I mean for the
physics to work we need this function to
be periodic. But when you use a periodic
function the math breaks down because I
cannot extract energy and build an
operator. For that to happen, I need
that function to be exponential. But
then the math works, but the physics
doesn't work because it's not a wave. So
to make both of them work, we somehow
need an exponential function of time
that's also periodic.
But that's impossible. By definition,
exponential functions are exponential.
So, how in the world can we build a
function that's both exponential and
periodic? Well, it's kind of like how we
subscription. So, happy learning. Back to the video. Okay, so coming back, how
do we create a function that's both
exponential and periodic? The answer
actually comes from a bookkeeper and an
accountant from Paris. His name John
Robert Arand. Argan asks Mahesh, why do
exponential functions in general blow up
or decay? Well, I say that's because
that's literally what exponential is.
There's no deeper explanation, right?
Well, he says, think about it
physically. The derivative of an
exponential is proportional to itself,
right? So, think of this as velocity.
The velocity in this particular case is
proportional to the position and it's in
the same direction as the position. So a
little time later the position grows but
the velocity grows as well. So now the
position grows even faster. The velocity
grows even faster and that's how we get
a runaway effect. The whole thing blows.
If you had a negative exponent then you
would have the same effect except now
the velocity would be in the opposite
direction because of the negative sign.
So now a little time later the position
reduces because of that the velocity
also reduces. So the position reduces
slower the velocity reduces even slower
and you get a decay. But Argan asks what
if somehow we could make the velocity
perpendicular
to the position.
This time a little later, the position
neither grows nor shrinks, but it just
shifts sideways, which means the
velocity magnitude stays the same as
well. And it continues to stay
perpendicular, meaning it keeps pushing
it sidewards forever.
That is a uniform circular motion.
That will make our exponential periodic.
I mean, sure, it's not oscillating, but
it that's okay. It's periodic. That's
what I want. But Argan, how do we
control the direction of this um
velocity? And how do we make it
perpendicular? Well, Argan says, think
about this. When we multiply the
exponent by one, the velocity points in
the same direction as position. That's a
0° rotation of the velocity. When we
multiply by -1, it points in the
opposite direction. So this rotates
velocity by 180°
which means we need to multiply the
exponent with some number that rotates
the velocity by 90°. How do we find
that? Well say is simple. Whatever
number this is if you multiply it by
itself one more time well it would
rotate again by 90° meaning 180°
but we already know that is negative
one. So in other words, whatever this
number is, it multiplied by itself
should give me -1 or the square of that
number should give me -1 or that number
which rotates by 90° is the square root
of -1.
The I has entered the room. Oh,
that's that's where it ah that's where
it comes from.
I'm sorry. Let's calm down. But what is
the I doing over here? The I in the
exponent produces in 90° rotation making
our exponential function spin making it
periodic. Now, of course, this rotation
is not in real space. This is a complex
plane. So, this is the real axis and
this is the um imaginary axis. Now
because of Argan's beautiful geometric
insight into these complex exponentials,
we call this complex plane the Argon
diagram. So just to recap, we started
with microwaves oscillating up and down
and we found that the math didn't work.
We couldn't cleanly extract the energy.
Now to make the math work, we needed
exponentials, but they aren't periodic.
Argan gave us a way out. complex
exponentials. Now, the math works
because I can cleanly extract energy
because it's an exponential function.
But the physics works too because the
function has become periodic.
This means now we have to accept that
our microwave is rotating in some kind
of an abstract complex plane. But that's
okay. At least we found a way out. So,
let's run with it now. So what we're
going to do is let's use this complex
exponential as our basic wave. Okay. Now
the first question is does that change
anything that we did so far? Well
remember we are taking a partial
derivative with respect to x and when we
do that the time part is treated as a
constant. So whether we have a sign here
or you know exponential over here it
doesn't matter. So everything stays the
same and so nothing changes over here.
So that's awesome. I don't have to do
any more work over here. But now it's
time to build the energy operator. So if
you differentiate with respect to time,
we get energy extracted and we get our
function back. So I get my S cleanly. So
it's time to rearrange and isolate
energy over here. And if I multiply
numerator and denominator by I, we have
done it. We have cleanly extracted
energy. This is the energy operator.
But wait, when I saw this, I was like,
wait a second, why is it a negative
sign? I mean, I know that the original
Shortinger's equation, as we will see,
doesn't have a negative sign. Well, it
turns out it's a small convention thing.
We chose our matter wave spinning in the
clockwise direction as seen from here.
Well, it turns out physicists like to
choose anticlockwise or counterclockwise
as the convention. So physicists love to
choose the negative sign over here for
their exponentials over here. So this
would also be negative and so you'll
have a negative popping out that cancels
with this one. And so our energy
operator wouldn't have a negative sign
over here. That's just a matter of
convention. Anyways, thanks to forier I
know that this operator works in general
which means we can now build our
shinger's equation. So we start by
energy conservation. Total energy equals
kinetic plus potential. And then for
total energy we substitute the energy
operator. For the kinetic energy we
substitute the kinetic energy operator
and they're work and they're operating
on the wave function s and we have our
stroinger's equation in its full glory.
So can we now answer our original
question intuitively? Why is there an i
over here? Well because our wave
function itself is complex. Why should
it be complex? because it needs both an
exponential and a periodic function. Why
does it need to be an exponential
function? Well, because that's the only
way I can extract energy cleanly. I can
build an energy operator because I need
the first derivative of my function to
be itself. Right? That doesn't that make
sense? Okay. Now, here's another
question. What would happen if I were to
consider the same equation without the
i? Now, the solutions would be real
exponentials in time. They will no
longer rotate which means they would
just blow up or decay forever. Would it
represent anything physical?
Yes.
Say the vertical axis was temperature
and the horizontal axis represented a
rod which means we now have a
temperature distribution.
And as time passes the hot regions cool
down and the cool regions heat up. So
the graph shrinks. But as a temperature
gets closer to each other, it gets
slower over time. Meaning we get exactly
a decay.
In other words, this now represents the
heat equation. How temperature changes
over time. But of course, it can have
different constants and we would expect
it to not have any potential energy
term. So we can now intuitively see why
the Schroinger's equation looks so
similar to the heat equation except for
the I. It's the eye that turns an
exponential into a spinning exponential
periodic.
At this point I'm really satisfied with
the Schroinger equation where it comes
from and why there is an I. And I think
I have a really good intuition behind
it. But a part of me still feels that
it's all still mathematical. It kind of
feels like, you know, there might be
some kind of a mathematical trick that
we just haven't thought of yet using
which we can get rid of that I and
that's exactly what Shinger was trying
to do for months after publishing his
equation. And when he couldn't, he wrote
a letter to Henrik Loren saying what is
unpleasant here and indeed directly to
be objected to is the use of complex
numbers. Sigh is surely fundamentally a
real function. So mathematically it
makes sense why the eye must be there.
But what is the physical reason behind
it? Well, the breakthrough came as a
footnote actually in a paper published
by Max Bourne.
Borne says let's go back to the double
slit. We are now coming a full circle to
where we started. Awesome. This time if
we send only one photon's worth of
energy through those slits, it has to
land somewhere at one specific spot on
the screen. Right. Mah, can you tell me
where? And I say, I have no idea. And
Bon says, neither do I, but I can't tell
where it's more likely to land. The
chances of landing in these dark regions
is almost zero because no photons have
landed so far. And the brightest spot in
the center, the chances is very high
there because lots of photons have
already landed there before. So the
dimmer the regions, the lower the
chances. So even though I can't say
exactly where one photon lands, I can
talk about the probability and that
probability is proportional to the
brightness or intensity which is
proportional to the amplitude squared.
This is how Einstein's idea can be
interpreted probabilistically. And Borne
says well I just thought that we can try
and apply the same thing to mow waves.
If you replace light with a source of
electron, they too travels as waves and
then we should get the exact same
result. Which means the probability of
finding now the electron in any spot is
the square of the wave function at that
point. This is today called the borne
rule. This is what he wrote down as a
casual footnote and it gave us a way to
interpret the microwave. It's a
probability wave. And remember at the
beginning we asked a question of how
mrow waves carry mixtures of energy and
momentum values. How do you make sense
of that? Well, that mixture is really
probability distribution. So for
example, when you measure its energy or
momentum, there's some probability of
getting each particular value. And that
probability distribution is hidden in
that wave function.
So coming back to our original question,
how does the born rule give us a
physical meaning to I? Suppose our
matter wave just oscillate back and
forth on the real axis. Let's say we
found some mathematical trick to make
this work. Okay? Then the size squared
over here would also fluctuate, right?
And since size square is the probability
of finding the particle somewhere the
total area under this curve should
represent the total probability of
finding this electron anywhere in the
universe. But in this particular case
that total probability fluctuates. I
mean at some moment it can even go to
zero for example. That makes no sense
right? I mean the total probability of
finding the electron somewhere in the
universe has to be 100%. Right? But how
can it be zero? It doesn't make any
sense. The probability, for example, can
shift from place to place, but the total
should never change. So you can clearly
see a simple up and down standing wave
can't give us a probability wave, a
matter wave. There's no way to make it
work.
But a wave that is rotating in complex
plane has no such problem.
Look here at every point in the complex
plane the height of this I stays exactly
the same because it's just spinning. So
what happens to s squared? The si square
stays fixed which means the total
probability now stays fixed. Now of
course for more complicated matter waves
the probability will change with time
but the total probability will still
stay the same. However, if you were to
model matter waves using just up and
down oscilly motion in just real axis,
there's no way for that to happen. Even
the simplest wave, you can't model it.
So, it's the i that makes the wave
function spin and conserves the total
probability. It's the square root of -1,
the imaginary number that makes the
physics real.
Schroinger solved his equation for the
hydrogen atom using the 1 / r potential
function. And not only did he get the
hydrogen spectrum, he also predicted the
relative brightness of each line and
showed how adding electric or magnetic
fields splits these lines. But more
importantly, the equation predicted the
three-dimensional probability
distributions of the electrons in the
hydrogen atom, the orbitals.
In short, the equation was a complete
radical breakthrough. It became the F=
MA of quantum mechanics. As a result,
Schroinger shared the 1933 physics Nobel
Prize with Paul Durac. And today,
scientists and engineers have done
extraordinary things with it. We've
found ways to focus electron waves and
build electron microscopes powerful
enough to see individual atoms. We've
calculated approximate energy levels
inside heavier atoms like cesium for
example to build atomic clocks. And when
multiple atoms come close together, we
found that those energy levels turn into
bands. And by controlling the band gaps,
we've built ultra tiny transistor
switches that make up every single
microchip.
Our real modern world brought to you by
the imaginary number.