We started talking about Planck's Law in the last video.
And I introduced this formula which is
the Planck's formula which relates this spectral irradiance.
And it expresses that as a function of
either the frequency or the function of the wavelength.
And, in both case, it's dependent upon
the temperature of the blackbody which features
in this exponential term right here.
So what I want to do in this video is
to give you a simple derivation of this Planck's Law.
And it turns out, you know, you don't need too much of it's not a very detailed this
derivation but very simple equation that you can derive this Planck's Law.
So let's see what, what kind of equations and math do I need?
So, the very first equation which I can write down,
thanks to Albert Einstein, is that the energy of my photon is related to the
frequency of of the light by this formula
where this energy is proportional to the frequency.
And this proportionality factor and the Planck's constant.
This is the first equation.
The second equation I can write is that the
momentum of the photon is also given by this
relationship, where it's proportional to the wave vector.
And I can write the wave vector in terms of the frequency as well.
So this wave vector can be written as a function of frequency like this as well.
So and the third equation, the only you know, I need only three equations
and the third equation I can write down, is the Heisenberg Uncertainty Principle.
That is the uncertainty
in this momentum, multiplied by the uncertainty in the, in the, in
the space, in the real space is, has to be greater than, again the Planck constant.
And you can see that the Planck constant is featuring in,
in all the three equations that I need for my Planck's Law.
So now what I can consider is I can consider this black
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images that are in the form of of this cavity overy here which has
which has the dimensions of Lx in the x direction
and Ly in the y direction and Lz in the z direction.
So now I can rewrite this this Heisenberg
Uncertainty Principle and I can rewrite for a three-dimensional
case in which I'll have a
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something like this, three uncertainty x, y, and z .
This three uncertainty in x, y, and z
multiplied by the uncertainty in location in x,
y, z, and now, this has to be greater than, greater than, greater than h cube.
So now, if I have a black body, which has this cavity of this
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dimension.
I know that the photon is essentially, it's it's it's localized, you
know, somewhere in the cavity, but I don't know its exact position.
So in this case, uncertainty in the position has to be Lx, Ly, and Lz.
If that is the case, the uncertainty in the,
in the momentum according to the Heisenberg Uncertainty Principle,
it would be essentially given by, given by the minimum would be given by
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would be given by essentially h cubed divided by this Lx,
Ly, and Lz. Also, I know that this Lx multiplied
by Ly multiplied by Lz is a volume of this cavity of
this blackbody that I had assumed. So, I can further express
this in terms of the volume of the cavity by this formula.
So now what this equation is telling me is that for
each of these states which is possible in the momentum space.
So let me draw my momentum space.
I'll have this is my three-dimensional momentum space.
So this is my, this is my, let's say Px this is my Py, and this is my P, Pz.
So what this equation is telling me is now
that each of these solutions has occupied this delta
Px amount of space here, delta Py amount of
space here, delta Pz amount of space over here.
So, each of these solutions, I can represent by, represent by this cuboid
over here where, essentially, the volume of this cuboid which is P, delta Px.
Times delta Py, times delta Pz, it
is essentially given by, given by this relationship.
So, essentially, each of these cube point has this particular volume.
So now, if I think of how many.
How many possible if I have, you know, a certain volume in my
momentum space so I can represent this volume by a form of a sphere.
How many of these, how many of these solutions can be contained in that sphere?
So now, how many states or how many states
of photons do I have within this spherical volume?
So the total number of photons that I can have, I can write
them down essentially as this volume of this sphere in the momental space.
So the volume of that sphere would be 4
by 3 multiplied by pi into the
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into the cube of the radius of this sphere.
And then the total number of states which could be contained in here is essentially
this volume divided by the volume which is occupied by each of this space.
So this is essentially divided by this.
And to calculate the number of photons I need to multiply
it, I need to multiply it with a
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factor. That is for each of these states.
How many photons can we occupy in that particular state?
So, for four times
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factor is 2. So if you have any one state, you
can occupy two photons in that particular state because you have two
perpendicular polarization of light. So, you have two of these two of these
particular photons which can occupy each of these states.
So I can take the volume in this in this equation on the, on the left-hand side.
And I get essentially the total number of, total number of
photons in this volume of the cavity. I get it and I substitute for the
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factor to be 2. So I get this formula to be 8 pi into
p cubed divided by, divided by 3 h cubed coming from over here.
So what this is, this is distributed across all the solid angle.
So if I want to find this number of photon in in a given solid angle, I could
divide it by the total solid angle which is 4 pi.
So I need to divide this by 4 pi.
And that gives me the number of photons per unit of my solid angle.
And so, essentially, I'm left with 2p cubed
divided by 3h cubed. Now I can further
substitute for momentum.
I know that my momentum in my photon, I can express this by this
formula because my momentum is related to my wave vector by using
this this particular relationship that I mentioned earlier as well.
Which was one of our three starting equations I can
substitute for the photon momentum as a function of frequency
using this relationship. So what I get now is I get
h cubed multiplied by frequency cubed
divided by h cubed divided by the speed of light cubed.
So these h cancel out.
And I essentially get this relationship my density of my photons as a
function of frequency is just simply
related by, simply related by this relationship.
So now, now I know.
So I know what the total number of photon
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for particular frequency per unit angle is given by this relationship.
So now how many, how many photons are contained between, say frequency,
between, say frequency h mu between say
frequency mu and mu plus delta mu.
So what is the density of these photons? So that would be
given by a simple derivative of this I'll
take a derivative of that in terms of frequency.
So that will be essentially given by a
derivative of this relationship with respect to frequency.
So I see that I have a third power over here.
So when I take a derivative, I'll get 3 times mu squared.
And that will cancel out with this 3 over here.
So,
this density of these photon is the function of frequency is
essentially given by this this formula. So, now how many of
now I know my density of, of photons and the function of frequency.
So, now I can write down what does a power
contain in these bodies of photons and a function on.
What does a power contain as a function of frequency?
So I can write down my spectral irradiance as
a function of, and a function of frequency and temperature.
So I expect that to be essentially give,
given by this density that I've derived over here.
Multiplied by the probability that the photon at that frequency is at that
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you know, that particular state is occupied with a photon.
And I'll multiply this by the energy of the photon, which again
is given by this relationship which was one of our starting equations.
So now, the probability is essentially, since photon is given by this
Bose-Einstein relationship which relates essentially
the probability of occupation of a photon
using this formula where mu is my frequency.
And this other term over here, this is the, this is the chemical potential.
And this is zero for, you know, photons and in general, thermal radiation.
So, my priority is essentially to simply, as a
function of my frequency, is just given by this relation.
So now I have all of the ingredients together.
Let me substitute for this substitute for this probability over here.
And then I'll substitute for this density of photons from this formula.
And that should give me essentially my formula or our for, formula for my blinds
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So let's see what do I get. I get essentially 2
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square of my
frequency term over here, and then I get this exponential term over here.
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And then it's multiplied by this energy of my individual
photon. So I get essentially instead of getting
2 over here, I get finally this
term over here. And this my, this is my
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relationship.