Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 16.6-2

ENTROPY AND DISORDER; GENERALIZED CANONICAL FORMULATIONS

Physics — Statistical Mechanics · density-of-orbital-states-gas

Problem

For the particles of a gas

\varepsilon = p^2/2m = (\hbar^2/2m)k^2

, or

\omega = \varepsilon/\hbar = \hbar k^2/2m

. Find the density of orbital states function

D'(\omega)

. Answer:

D'(\omega) = \frac{V}{2\pi^2}k^2\bigg/\left(\frac{\hbar k}{m}\right) = \frac{m^{3/2}V}{2^{1/2}\pi^2\hbar^{3/2}}\,\omega^{1/2}