## Derivation of Planck's Law

This page provides a brief derivation of Planck’s law from basic statistical principles. For more information, the reader is referred to the textbook by Rybicki and Lightman (Radiative Processes in Astrophysics, Wiley, 2004) http://books.google.com/books?id=LtdEjNABMlsC&dq=isbn:0471827592&ei=0KPFSOKvE4mIjwGQ2Oj3BA. The reader might also find interest in the historical development of early research in radiation physics as surveyed by Barr http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000028000001000042000001&idtype=cvips&gifs=yes.

## Photon gas in a box

First, consider a cubic box with each side of length L that is filled with electromagnetic (EM) radiation (a so-called ‘photon gas’) that forms standing waves whose allowable wavelengths are restricted by the size of the box. We will assume that the waves do not interact and therefore can be separated into the three orthogonal Cartesian directions such that the allowable wavelengths are:

$\lambda_i = \frac\left\{2L\right\}\left\{n_i\right\}$

where $n_i$ is an integer greater than zero, and i represents one of the three Cartesian directions—x, y, or z.

From quantum mechanics, the energy of a given mode (i.e., an allowable set $n_x, n_y, n_z$) can be expressed as

$E\left(N\right) = \left\left( N + \frac\left\{1\right\}\left\{2\right\} \right\right) \frac\left\{hc\right\}\left\{2L\right\} \sqrt\left\{n_x2 + n_y2 + n_z^2\right\}$

where h is Planck’s constant (