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1870s a young boy went to his physics
professor and said hey I want to be a
physicist and the professor replied
physics is almost complete don't waste
your time fortunately that boy decided
to waste his time and get a PhD because
pretty soon we realized that physics was
broken you see classical physics
predicted that everything in this
universe including you me and donkeys
and that cup of coffee should radiate
infinite energy at any temperature it
give out Infinite Energy in all mostly
in high frequency ultraviolet and x-rays
this later became known as the
ultraviolet catastrophe it was a
catastrophic result clearly it is wrong
coffees do not give you Infinite Energy
in high frequency X-rays and um
ultraviolet rays instead they give out
finite amount of energy mostly in
infrared radiation and that's because
the actual graph looks like this today
we call this the black body radiation
curve and you can see it Peak and then
it goes back down at low temperature
that Peak is mostly in the infrared
region of course at high temperature
that Peak can go back to the visible
region and therefore at high
temperatures things can glow like your
encantas and bulb and stuff but the big
question was how do we fix this how do
we go from what the classical disastrous
result that physics was giving us to the
actual result what needs to be modified
to give us this answer nobody had any
clue nobody was able to solve this
puzzle nobody except for the man who
wasted his time pursuing a Physics
degree because he realized using trial
and error and in later what he called an
act of Despair that the only way to
solve this is to assume that energy is
quantized yes this man is MOX plunk who
accidentally saved physics and then
regretted it we have all heard of the
story multiple times how Max flong
quantized Energy and saved the day and
solved this whole catastrophe but the
big question for me was how I'm not
talking about the math part of it but
what is the logic behind it how does
quantizing energy solve this well that's
what I've been trying to understand for
the last few weeks and I think I finally
have the intuition for it and I'm super
excited to share that with you so the
goal of this video is not just to
understand this but for us to put
ourselves in MX plunk shoes pretend we
don't know anything about e equal to hu
and then ReDiscover it from scratch so
if you're ready for this let's begin in
so plon where do we start plon says
mahes the first big question we need to
ask ourselves is what really caused the
ultraviolet catastrophe I mean there are
a ton of details but at the very core
what are some main reasons for it turns
out there are three the first one is you
can simulate a black body a perfect
black body as a hole in a metallic Cube
because a hole is pretty much a perfect
absorb so it can also perfectly emit
radiation and now if you heat up that
metallic Cube then its thermal energy
will be stored in electromagnetic
standing waves each of these standing
wave represents a degree of Freedom an
independent way of storing energy but
the thing about the standing waves is
that you can set up any amount of
standing waves of shorter and shorter
wavelength there is absolutely no limit
to it which means you can have infinite
degrees of freedom the second part is
that the second point is that not only
do we have infinite degrees of freedom
but the degrees of freedom actually
increases with frequency and again this
is the reason the reason for this is
because we live in three dimensions see
in one dimension there's only one way to
set up a particular standing wave so
there's only one degree of Freedom per
frequency but in higher um dimensions
for example in two Dimensions as we can
see over here notice that if there are
you know at higher frequencies if there
are more Lopes then there are more ways
in which you can set it up and therefore
you will have more degrees of freedom at
higher frequencies so putting it these
two together what we see is that at
lower frequencies meaning High
wavelength you have less number of
degrees of freedom but then as the
frequency becomes higher and higher the
number of degrees of freedom becomes
more and more and more and the final
piece of the puzzle over there was equip
partition theorem which suggested that
at any temperature um the average energy
per degree of Freedom must be the same
so if each degree of freedom on average
gets the same amount of energy
immediately you can see the higher
frequency ones which have more degrees
of freedom they end up getting most of
the energy and since there is no limit
to it that causes that disastrous
curve now again we've talked about this
in great detail in our previous video
but again you don't have to watch that
if this summary makes sense to you we
can go forward no need to break the flow
over here but plank now the question is
how do we solve for this and plon says M
think logically we know this is wrong
that means at least one of these three
must be wrong isn't it and so PL asks
mahes which of this do you think is
wrong and I'm like it has to be the
first one I mean clearly that's the
problem there's right over there that's
the problem right and plong says well
let's see first of all can you simulate
a perfect black body as a whole in a
metallic Cube answer is yes there's
nothing wrong with that because a hole
does act like a perfect absorb so it's a
perfect emitter nothing wrong with that
now if you heat up that particular
metalic Cube then indeed the thermal
radiation will be stored in the standing
electromagnetic waves which act like a
degree of freedom and now it is true
that there is no limit to how short the
frequency can be you can set up any
amount of standing weight of shorter and
shorter and shorter frequencies there's
no limit to that which means this is
accurate indeed you get infinite degrees
of freedom and that is accurate and I'm
like okay that is very interesting what
about the next one degrees of freedom
increases with frequency well we saw it
and that is also accurate this is just
math there's nothing wrong with that so
that means there's only one option left
for us which can be wrong it is the
equipartition theorem it has to be be
wrong and I'm like how can it be wrong
it's a statistical law which says on
average every degree of Freedom should
get the same amount of energy it makes
sense and plong says yeah and yet it has
to be wrong because we know the other
two are right and so the next big
question for us is that if that is wrong
need to figure out why it is wrong and
what is the new Partition theorem how do
we figure this out PL says here's the
plan let's go back to the classical
physics one more time time the same
thing that we saw earlier less degrees
of freedom more degrees of freedom over
here and by the way the numbers are just
for representation purpose they're not
mathematically accurate okay anyways now
let's look at what the actual graph was
it looks like this and let's compare the
two what's interesting is that at higher
wavelengths the classical physics
actually agrees with the experiment
that's interesting it's at lower
wavelengths you can see classical
physics starts
overestimating of course lower the
wavelength the overestimation becomes
more and more and more in fact if you
were to zoom in over here you will see
even here is an overestimate but it's
very small so we can start get we can
kind of get the picture the classical
physics is doing an overestimate equip
partition theorem is an overestimate but
at higher at higher wavelength that
overestimation is small but at lower and
lower a wavelength that overestimation
becomes bigger and bigger and bigger and
that's why the graph ends up blowing up
so since equipartition theorem is an
overestimate this means that in reality
the energy per degree of freedom on
average
must be less than e e divide by
something we don't know how we don't
know why but it has to be not divid by
something so that we can humble this
graph but more importantly over here
it's an excellent agreement which means
over here it must be pretty much e divid
by 1 point something there is a small
discrepancy so it must be very close
over here the denominators must be very
close over here to one but as I go
shorter and shorter the denominator must
increase because the discrepancy
increases for example when I come over
here the denominator might become five
so that 100 by 5 becomes 20 and there is
a significant decrease and that's how we
can fix that point and if I come even
shorter maybe the denominator becomes
even more and therefore it be the graph
becomes more and more humbled and then
finally over here when I go to even left
the denominator becomes so much big that
the product starts decreasing again and
eventually at very very short
wavelengths or very high frequencies you
have very high number of degrees of
freedom but maybe the average energy
over there is pretty much zero so the
product goes to
zero this is how if we have this kind of
a partition theorem then the graph can
be explained does that make sense and
I'm like yeah I'm starting to get some
intuition I mean over here the average
energy per degree of freedom is pretty
much the same as e not therefore the
product keeps growing just like what
classically was predicted but then the
Energy starts reducing and so the growth
is less than what is classically
predicted but eventually the energy
becomes so small that the product starts
decreasing and then eventually goes to
zero what's interesting over here is
it's kind of like what we did in our La
exams like you know we didn't get the
right output and then we went back and
changed the numbers to try and fit it
right I mean I wasn't the only one who
did that right but the big question now
is plunk how do we explain this I mean
equip partition theorem made statistical
sense to me right like every degree of
freedom on average should get the same
amount of energy but now what's the
explanation for this and plong says well
Mahesh quantization quantization is
going to give an explanation but I'm
like how how does eqution theorem gets
transformed into this particular energy
function just by introducing
quantization well PL say it's kind of
description now back to understanding how quantization Saves the Day PL how
how does quantization do anything help
you understand and flang says well let's
do look at it step by step first let's
take an example imagine you have a bunch
of tennis balls just floating around at
different heights then you will have
some of these balls with very low
potential energies and some of these
balls at very high potential energies
right and therefore there will be some
average energy we'll call that average
energy as e not what's interesting over
here is that H you'll find more balls at
lower energy and as you go higher and
higher there'll be lesser and lesser
balls it's kind of like the air
molecules for example over here the air
density is very high closer to the
ground but as you go higher up as you
know as you as the energy increases the
number of air molecules per volume
starts decreasing okay and the same
thing applies to the standing waves
inside our metallic Cube if you take any
specific wavelength it can't have any
energies right so the distribution would
be well you will have all the possible
energies you will have the lowest
energies and you will have the highest
energies remember energy of an
electromagnetic wave only depends on its
amplitude so lower energy is low
amplitude higher energy is high
amplitude but the distribution would be
you'll find a lot of these waves at very
low energy and as you in as you go up
the energy the number of these waves
will start reducing but again what's
important is that there will be some
average value and that average value is
e not now classically speaking classical
physics says the same story should hold
good for all other frequencies so if you
consider high frequency visible light
the same story The distribution is the
same everything is the same remember
energy only depends upon amplitude as
far as classical physics is concerned
has nothing to do with frequency so the
same story holds for frequency and so
even for higher frequency waves well the
average energy stays the same regardless
of how high the frequency go go average
energy stays the same and this is your
equ partition theorem and this leads to
the disastrous ultraviolet catastrophe
but now let's introduce quantization and
see what happens let's go back to our
tennis balls over here and we'll now
introduce some levels kind of like
imagine you're introducing some planks
in between and we'll now put the
condition that the balls cannot be
hanging or floating in the air anymore
they have to be sitting on either the
ground level or the first level or the
second level or the third level nothing
in between they have to be sitting over
there this is called quantization and
this minimum level size is what we call
a quanta now the moment we do that it's
kind of like the balls will now all fall
down right so that see these balls will
be on level zero all these balls will be
at level one you don't have anything in
between and so on and so forth now when
that
happens look at the energy each ball has
lost some energy
and therefore plunk asks mahes what
happens to the overall average potential
energy well I'm like since each ball has
lost some energy the average potential
energy should be smaller than before it
should become smaller but what's
interesting is that since the quanta
size over here is pretty small I mean
you have a lot of energy levels
available the balls didn't have to fall
much they fell just a little bit as you
can see here and therefore the change
was not much so although the average
energy reduced it didn't reduce that
much it's still pretty much same as e
not but now what if we go back and
increase the quanta size let's double it
let's say for example over here what
happens now oohoo now the balls will
fall more and now look the average
energy has reduced quite a lot compared
to
before can you see that because the
balls have to fall more so if the quanta
size increases look the balls fall more
the average energy reduces more and if
you go back and increase the quantum
size even more the balls fall down even
more and look the average energy now
reduces
considerably and what if the qua size
was much larger then almost all the
balls fall down and the average energy
now almost goes to zero so what does it
all mean now plunk well plunk says and
this is the key inside over here when we
quantize it the average energy becomes
less than what classical equip partition
theorem says but what's interesting is
that if the size of the quanta is very
tiny compared to the average energy that
classical equip partition theorem
predicts if that Quant size is much
smaller than that then acup partition
theorem has pretty much no effect on it
you get the same average energy as the
classic iCal theory predicts because the
balls don't fall down much but if that
Quant size becomes bigger and starts
becoming considerably big compared to
you know the average energy that
classical physics predicts now the balls
fall down a lot and the average Energy
starts reducing bigger the size of the
quanta smaller is the average energy and
eventually the average energy just goes
to zero if the Quant size is just too
big and so PL says mes look at this this
is how quantization transforms the equip
partition theorems average energy it
reduces it and the size of the Quant
controls how much it reduces it I'm like
okay how do we put this together to
explain the graph now and pl says yeah
I'm glad you asked so what we can do
first of all calm down let's calm down
what we can do now and let's see if this
makes sense okay over here we want the
average energy to be less than
equipartition theorem what predicted
classically but not much it should be
very close to it it should be almost
approximately same as equ partion
theorem so mahes what should be the
quanta size over here it has to be small
we need to have small quanta sizes over
here so that it doesn't affect anything
but then as we get shorter and shorter
we want the you know uh the average
energy to be much smaller than what the
equip potition theorem predicted so that
means we need the quanta size to become
larger so look what we need to fix this
Gra
to fix this we need to introduce
quantization of energy and we need to
make sure that the size of that quanta
depends on the frequency at low
frequencies the quanta size should be
smaller at high frequencies that quanta
size should be bigger does that make
sense this means as the frequency
increases the size of that quanta should
increase the quanta size should be
directly related to the
frequency does this make intuitive sense
now because it's again kind of like what
we used to do in our exams not only did
we change the values but now when the
inator came we have to explain to them
why we had to change the value here to
come up with the justification please
don't tell me I was the only one who
should do that okay but this makes sense
right of course next part is purely
mathematical to figure out the exact
relationship but at least it makes sense
that the size of the Quant should be
directly related to frequency so plon
worked out the math after doing a lot of
trial and error and he figured out the
the best relationship that was actually
giving the best result was a simple one
directly proportional if the Quant size
increases directly proportional you know
increases with a direct proportionality
then you get the same shape as the graph
and then it was a matter of again doing
the math to figure out what that
proportionality constant was to get the
perfect fit and that proportionality
constant he called it the planks
constant and again what I find
fascinating about this is you can see
from the classical perspective this does
look like an act of desperation I mean
we are introducing quantization and then
we're saying that the size of that
quanta must be proportional to the
frequency should change with frequency I
mean this feels like what are we doing
over here I mean we just we just
desperately trying to fit the graph and
that's exactly what PL did and he would
be excused in thinking that this
wouldn't have any broader significance
but oh boy today we know that it did but
anyways I'm getting ahead of myself but
now we can see this eventually is what e
is equal to HF this is where it all came
from because this is how it solve the
graph okay so let's put it all together
to see if we have complete intuition for
this okay going back to those wave
distribution that we saw if we introduce
quantization then at high wavelengths or
small frequencies the quanti size is
very small if the quanti size is very
small then look there are so many energy
levels available they don't have to fall
much as a result the average energy
stays the same as what equip partition
theorem predicted but as the frequency
becomes larger wavelength becomes
smaller the quanta size becomes larger
because it's directly related to quanta
but again if the quanta is too small
then again you have a lot of energy
levels available and therefore the
average energy doesn't change much and
therefore equip potion theorem pretty
much works at high wavelengths or
smaller frequencies but now if you
reduce the wavelength a lot or increase
the frequency a lot now the Quant size
becomes significantly larger and so the
waves fall a lot more significantly and
so now the average energy reduces
significantly and again if the frequency
becomes way more very high say we go to
ultraviolet now the quanta size is very
big and now the average energy waves
fall pretty much and the average energy
pretty much dies out and goes to zero
and this is how the ultravet catastrophe
is solved so just we're making sure that
we are on the same page over here
because this is really subtle stuff okay
when when you consider continuous energy
with classical physics you get equip
partition theorem because there's no
difference between different Waves
Distribution is the same everything is
the same average energy is the same and
therefore it predicts um equipartition
theorem and you get ultrair catastrophe
but then we introduce quantization and
make sure that the quantize depends on
the frequency this is the key part that
ensures that at high frequency Quant
size are too big and so big that the
waves do not populate a lot of energy
levels and so most of their energy just
goes out the average energy for high
frequency just goes out to
zero now we could stop over here but it
would be such a shame because I still
have plankk with me over here and I have
my coffee and we now have the
opportunity to reap the benefits the
fruits of all the hard work we have done
by asking one last question to plankk
plankk why does that the peak of that
graph shift with temperature why is it
at lower temperature it's mostly in the
high wavelength and at lower at higher
temperature it starts shifting towards
lower wavelength what's going on over
here how do we explain this there are
multiple ways to answer this question
and plun says well one of my personal
favorite is to look at things this way
okay going back to the graph over here
think of it this way so at high
wavelengths the quanta sizes are very
tiny and when you have very tiny quanta
size compared to the average remember
average equ partition e not whatever
that was compared to that if the
quantize is very small then we can
ignore the quantisation effect right
it's only when the quantize becomes big
enough as I move to the left when it
becomes big enough quantization kicks in
and then the graph gets severely humbled
okay now what happens when we increase
the temperature when we increase the
temperature the average energy increases
so much more energy available so what
happens to the graph well let's think
about it step by step
well first of all the whole graph blows
up and so we're going to scale down the
whole graph so let's scale it down but
when we scale it down from this
perspective look even the Quantas over
here have become much smaller what's
happening is that
now this previously was big size Quant
size was big but now from this
perspective this Quant size is also very
small compared to the new average energy
now because so much energy is there
therefore quantization will not kick in
at this at this particular wavelength
like it did before I still have to
travel more to the left before the
quantization kicks in maybe somewhere
here now the quanti are big enough for
that quantization to kick in and now the
graph gets humed and that's why the
graph's peak shifts to the left that did
that make sense I think it weirdly Mak
sense to me the same thing happens now
at lower temperature what happens if you
lower the temperature well now now the
whole energy average energy everything
becomes much lower and so now I need to
scale up the graph to see this so let me
scale the whole thing up but when I
scale up the graph now look now look the
quantization starts kicking in very
quickly somewhere here itself the
quantization kicks in because the levels
the Quantas have become much bigger now
compared to the average energy from this
new uh level that we are looking at
things from and therefore the graph will
start getting humbled much more quickly
and so the peak will be in the high
wavelength region I again I I I think it
kind of makes sense and if it didn't
it's probably because I didn't
articulate it well um but think over it
and I think it'll probably make sense
what I find fascinating is is that this
is the reason why this coffee and you
and me and everybody over here we're all
radiating a finite amount of energy
mostly in the infrared region because
the temperatures over here is much much
lower it's only when the temperature
becomes high enough like that of the sun
that's when it starts giving out a lot
of light in the visible region and so on
and so forth so it's so beautiful but of
course plunk had no idea what he had
done he thought he was just soling he
doing a math trick he didn't think about
any broader significance to the energy
Quantas and yet he had no clue how to
explain why energy must be quantized and
why should it depend on the frequency
and stuff like that but it was Einstein
who took this idea forward and in this
video you can see how he actually
introduced the idea he took it Forward
introduced the idea of photons and
actually got quantum mechanics started
all right see you