# Deriving Planck's radiation law from microscopic considerations?

In the usual derivation of Planck's radiation law, the energies or frequencies $\omega$ of the oscillators depend on the measurements $L$ of the black body. The model is such that the only characteristic energy is given by these oscillator excitations and in terms of the temperature $T$. Also, the oscillation amplitude vanishes on the walls. Specifically, the structure of the atom walls doesn't enter the computation. After all, that's pretty much the definition of a black body.

My question:

Are there computations (maybe QED + statistical physics?) for more realistic systems, which might model the interaction of photons with the wall atoms and which give results such that one can see the limiting case to an ideal black body?

The question came up when I wondered about the theory behind emission and absorptions of photons on the walls of a black body.

A blackbody is not a blackbox

for an illuminating account of the derivation of the Planck spectrum without enclosing the field in a box. If you cant get the published version, see the arxiv version.

EDIT (25 March 2012)

Planck's Radiation Law: A Many Body Theory Perspective

discusses blackbody radiation from a many-body viewpoint. Note that they also consider interactions among the photons and electrons, and still show Planck's law is valid. This might perhaps be considered as arising from the interactions of the photons with the electrons in the walls, if you like. This paper is, of course, also referred to in the first paper I mentioned.

• +1 for an extremely interesting citation that I had not previously seen. I recommend it to anyone who is interested enough in the Question to be here. Illuminating, indeed. – Peter Morgan Mar 8 '12 at 14:35
• +1 Thanks. He argues that the density of the wave in the medium is $(1-e^{\alpha\ l})\frac{\epsilon}{\alpha}$ and so exponentially goes against the ratio emission/absorption, which by Kirchhoffs law is universal and independent of any box geometry. He uses field equilibrium, so the field is just there. A problem I still have is that it doesn't lead me to understand why the field, with all its universal frequencies, gets there. If I heat up a body of some sort of atoms (specific spectrum), how does the Planck spectrum for the field with continous $\omega$ come about? Field-self-interaction? – Nikolaj-K Mar 8 '12 at 15:16
• There are a nice set of papers which discuss how the continuous Planck spectrum arises from discrete atomic spectra. – Vijay Murthy Mar 8 '12 at 15:45
• Thanks again. To give an abstract to the last paper: To model the transition to the continuous spectrum from energy gaps, they include a Doppler broadening due to motion of the atoms, given by Maxwell-Boltzmann and smear out the frequencies. – Nikolaj-K Mar 8 '12 at 16:24

An old paper that I've enjoyed immensely for its early and unique take on the relationship between the black-body radiation and the Maxwell velocity distribution of molecules is this one:

A. Einstein, On the Quantum Theory of Radiation. Physikalische Zeitschrift 18, 121 (1917).

It is unfortunately not online anywhere that I could find, but you can find it in English in this book (which can also be a bit hard to find):

D. ter Haar, The Old Quantum Theory. Pergamon Press, 1967. See pp. 167-183.

It annoys me to no end that the copy of this book that I got from Amazon was stamped "DISCARDED - SUNY Geneseo". This particular paper has some remarkable (and still valid!) insights by Einstein that were mostly overlooked in the subsequent rush to the new quantum theory.

My view is that old papers by the Greats have value and hidden nuggets even now! Just read how Feynman redefined how all of modern particle physics is done by coming across an old, mostly ignored paper by Dirac on applying Lagrangian methods to quantum theory.