Quantum Electrodynamics

Abstract

As is well known, the eigen-vibrations of a cubical cavity, with edge l and volume V=l³ whose walls are perfect reflectors, have the following properties: The components k _i of the wave vector whose magnitudes amount to 2π divided by the wave-length, have the eigenvalues

$$k_t = 2\pi \frac{{s_t }} {{2l}},\,\,\,\,s_1 ,s_2 ,s_3 = 1,2,3,...$$

where the integers s _i are to be restricted to positive numbers, since we are dealing with standing waves. The number dN of eigen-vibrations with wave vector components k _i lying between k _i and k _i + dk _i is then given by

$$dN = V \cdot 2 \cdot 8 \cdot \frac{1} {{\left( {2\pi } \right)^3 }}\,\,dk_1 \,dk_2 \,dk_3 ,$$

where the factor 2 takes care of the two polarisation directions of the wave, while the factor 8 arises because for standing waves we have to restrict ourselves to the positive octant of the k-space.