Derivation of Planck's law and the shape of the cavity

[center o bass]

When Planck's law is derive a cubical cavity is often used (for example in: http://disciplinas.stoa.usp.br/pluginfile.php/48089/course/section/16461/qsp_chapter10-plank.pdf)

However, the result is applied generally. But in general, it seems like the wave lengths of the standing waves will depend on the shape of the cavity.

So I wondered: why is the result independent of the shape of the cavity?

 

[Avodyne]

That's a good question!

First of all, we can easily do a rectangular box with side lengths $L_1$, $L_2$, and $L_3$. If you work this through, you will find that the factor of $L^3$ in the formula for $dN(p)$ on page 5 gets replaced with $L_1 L_2 L_3$, which is just the volume of the rectangular box. So that's a hint that only the volume matters.

To show that this is true for more general shapes requires some higher-level math; the general result is known as Weyl's Law.

 

[dextercioby]

Weyl's Law? Do you have a reference for that? Perhaps Wien's Law.

 

[Avodyne]

http://en.wikipedia.org/wiki/Weyl_law

 

[center o bass]

That's a good question!

First of all, we can easily do a rectangular box with side lengths $L_1$, $L_2$, and $L_3$. If you work this through, you will find that the factor of $L^3$ in the formula for $dN(p)$ on page 5 gets replaced with $L_1 L_2 L_3$, which is just the volume of the rectangular box. So that's a hint that only the volume matters.

To show that this is true for more general shapes requires some higher-level math; the general result is known as Weyl's Law.

Thats interessting! Do you have a reference where Weyl's law is utilized in deriving Planck's law?

 

[Avodyne]

Weyl was motivated by the blackbody problem:

In the crucial black body radiation experiments carried out in the 1890s, which led Planck, in 1900, to the famous radiation law named after him and to the discovery of quantum theory, one measures the energy density emitted rather than the energy itself, i.e.the energy divided by the volume $V$. Thus it follows from Rayleigh’s asymptotic result $V\cdot\nu^3$, derived for a cubical geometry, that the volume factor is canceled if one considers the energy density, in accordance with the expectations using physical arguments and, very importantly, in complete agreement with the experimental findings. It was realized, however, and emphasized by Sommerfeld and Lorentz in 1910 that there arises the mathematical problem to prove that the number of sufficiently high overtones which lie between $\nu$ and $\nu+d\nu$ is independent of the shape of the enclosure and is simply proportional to its volume. It was a great achievement when Weyl proved in 1911 that, by applying the Fredholm–Hilbert theory of integral equations, the Sommerfeld–Lorentz conjecture holds! From then on, Weyl himself and many other mathematicians and physicists have studied and generalized Weyl’s law by deriving corrections or even complete expressions for the remainder term.

Arendt, W., Nittka, R., Peter, W. and Steiner, F. (2009) Weyl's Law: Spectral Properties of the Laplacian in Mathematics and Physics, in Mathematical Analysis of Evolution, Information, and Complexity (eds W. Arendt and W. P. Schleich), Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany. doi: 10.1002/9783527628025.ch1

http://media.johnwiley.com.au/product_data/excerpt/04/35274083/3527408304.pdf

Much more about the history in this paper.

 

[Jano L.]

When Planck's law is derive a cubical cavity is often used (for example in: http://disciplinas.stoa.usp.br/pluginfile.php/48089/course/section/16461/qsp_chapter10-plank.pdf)

However, the result is applied generally. But in general, it seems like the wave lengths of the standing waves will depend on the shape of the cavity.

So I wondered: why is the result independent of the shape of the cavity?

The cavity is mentioned only for historical reasons; it is the way blackbody was prepared and measured originally. But it does not really play any explicit role in the Rayleigh-Jeans type of derivation of spectrum (treating EM modes like harmonic oscillators in thermal equilibrium). This is because one can assume any cuboid of vacuum with imaginary walls inside the real cavity, expand the field inside into Fourier series and arrive at the result.

The really important assumptions behind the derivation are:

- energy interpretation of the Poynting theorem, where $\frac{1}{2}\epsilon_0E^2 + \frac{1}{2\mu_0}B^2$ is density of energy;
- some rule for assigning average energy to one Fourier mode (equipartition rule or the energy of quantum harmonic oscillator etc.).