Max Planck
Borrowing from Ludwig

The question was how to get such a simple equation from the then-theory of the day. The trouble was you couldn't. You had to come up with something new.

Actually, Max had some help. One of his friends was Ludwig Boltzmann who like Max was a professor. Ludwig had derived a similar shaped equation by simply figuring out how many ways molecules could shift their energy around. Although the derivation took some tricks and a few steps (which you can see in a new window if you click here), it was nevertheless pretty much limited to what is considered high school mathematics.

Ludwig found that if you had a total of N molecules, the probability that any given molecule, n_i, would have a particular energy, ε_i, is given by:

P(n_i) = Ne^-kε_i/T

... where T is the temperature.

Now if you make a graph of this equation, you'll notice that it doesn't have the right shape. That's because the actual distribution of the energy of the molecules is the equation multiplied by the energy level itself:

D(P) = ε_iNe^-kε_i/T

Plot this out and the curve looks OK.

Max realized there was a trick that would work. You just assume the light beams inside the box were acting as "oscillators" that could not assume any energy value. If the energy was restricted to discrete multiples of the wavelength - actually the frequency - then this would produce similar mathematics as having only a finite number of molecules.

So with no other reason except that it would work, Max assumed that the energy of radiation was quantized:

E = nhν

... where n = 1, 2, 3, ... on up.

E is the energy, ν is the frequency of the radiation and h is a constant now commensensically called Planck's constant.

One thing working in Max's favor is that Ludwig's equation applied to anything that had discrete energies. It didn't need to be molecules but "oscillators" would work fine. So Max could start off with Ludwig's basic formula:

n_i = Ne^-hν_i/_T

The hope, of course, is that as you got to higher and higher energies, there would be fewer and fewer quanta with those energies. So your equation would start off at zero, rise to a maximum and then drop off.

Now a distinction between Max's equation and Ludwig's is that Max's N is not the total number of quanta. It is the total number of oscillators in the lowest energy state. So Max is not arbitrarily limiting the number of packets as you can for molecules.

Up to this point, we've have engaged in fairly subtle reasoning. Fortunately, from now on out, it's pretty much middle school algebra.

The total energy of all the oscillators, then, is simply made by summing up all the number and their energy levels.

NTotal	=	N0 + N1 + N2 + N3 ... + Nn + ...
	=	N0 + N0e-hν + N0e-2hν + N0e-3hν + ... + N0e-nhν + ...
	=	N0(1 + e-hν + e-2hν + e-3hν + ... + e-nhν + ...)

Now to make things simpler you can now write the equation as:

=

N0(1 + x + x2 + x3 + ... + xn + ...)

... where x = e^-hν

Now, we can do the same thing with total energy of all the quanta as:

ETotal	=	E1 + E2 + E3 ... + En + ...
	=	N0hν + N0hνe-hν + 2N0hνe-2hν + 3N0hνe-3hν + ... + nN0hνe-nhν + ...
	=	N0hν(e-hν + 2e-2hν + 3e-3hν + ... + ne-nhν + ...)

... which if you substitute x = e^-hν you get:

ETotal

=

N0hν(x + 2x2 + 3x3 + ... + nxn + ...)

At this point, we'll remind ourselves that Max's equation does not a priori limit the number of quanta as Ludwig did when talking about molecules. On the other hand if you get to a high enough energy level, the exponential term (x = e^-kT/h) gets so small that it cuts off the number of quanta we have (remember that you can only have discrete number of quanta and anything less than 1 means you have no quanta).

So we will ultimately have a finite total number of quanta. And since we are looking for the average energy, , we divide the total energy by the number of quanta.

<E>		=		ETotal NTotal
		=		N0hν(x 2x2 + 3x3 + ... + nxn + ...) N0(1 + x2 + x3 + ... + xn + ...)
		=		hν(x + 2x2 + 3x3 + ... + nxn + ...) 1 + x2 + x3 + ... + xn + ...

We'll skip the details, but as x tends toward infinity, the sums in the numerator, S_n, and in the denominator, S_d, both converge.

Sn

=

x (1 - x)2

Sd

=

1 (1 - x)

So you make the substitutions:

<E>		=		hνSn Sd
		=		hνx/(1 - x)2 1/(1 - x)

... and canceling out (1 - x), we get:

<E>

=

hνx 1 - x

... and if you multiply both the numerator and denominator by ¹/_x we have:

<E>

=

hν 1/x - 1

... which if you remember that x = e^-hν/_T we now end up with:

<E>

=

hν ehν/T - 1

... which if you plot against ν and take the usual value of h as 6.62606983 x 10^-34 joules-seconds, gives you the right equation.

Max, we remember, thought he was simply coming up with a mathematical trick and that light really wasn't really bundled up in little packets. Today, though, the reality of his quanta - now called photons - is about as undisputable as you can get.

References

"Derivation of Planck's Radiation Law", The Curious Astronomer.

Part 1: https://thecuriousastronomer.wordpress.com/2015/08/06/derivation-of-plancks-radiation-law-part-1/

Part 2: https://thecuriousastronomer.wordpress.com/2015/08/18/derivation-of-plancks-radiation-law-part-2/

Part 3: https://thecuriousastronomer.wordpress.com/2015/09/03/derivation-of-the-planck-radiation-law-part-3/

Part 4: https://thecuriousastronomer.wordpress.com/2016/01/07/derivation-of-plancks-radiation-law-part-4-final-part/

The derivation here is essentially that given above and is probably the best derivation of Max's equation.