= Solution
A cubic reflecting cavity supports <standing waves> with positive integer mode indices and <wavevector> components $k_i=\pi n_i/L$. A cell in mode space has <volume> $(\pi/L)^3$. In the continuum approximation, a spherical shell of radius $k$ and thickness $dk$ has one-eighth of its <volume> in the positive octant. There are two independent transverse <photon polarizations> for a generic <wavevector>. Hence
$$
dN=2\times\frac18\times4\pi k^2dk\left(\frac L\pi\right)^3=\frac{L^3k^2}{\pi^2}\,dk.
$$
The <photon> <energy> is $\epsilon=\hbar\omega=\hbar ck$, so $dk=d\epsilon/(\hbar c)$. Therefore
$$
\boxed{dN=\frac{L^3\epsilon^2}{\pi^2\hbar^3c^3}\,d\epsilon.}
$$
This is the leading bulk <density of states>, valid when the cavity is large compared with the relevant <wavelengths> and many modes lie in the counting interval. A finite cavity has a discrete mode spectrum, with boundary corrections beyond this approximation. Modes with vanishing indices require separate boundary counting but do not change the leading $L^3$ term. Here $\hbar$ is the <reduced Planck constant>, as printed in the original PDF; replacing it by $h$ would introduce an erroneous factor of $(2\pi)^3$.
Back to article page
