Statistical Mechanics · Problem 3.26

The Grand Canonical Ensemble — R. K. Pathria

Physics — Statistical Mechanics · s-dimensional-harmonic-oscillator

Problem

The energy eigenvalues of an

s

-dimensional harmonic oscillator can be written as

\varepsilon_j = (j + s/2)\hbar\omega; \quad j = 0, 1, 2, \ldots

Show that the

j

th energy level has a multiplicity

(j + s - 1)!/j!(s-1)!

. Evaluate the partition function, and the major thermodynamic properties, of a system of

N

such oscillators, and compare your results with a corresponding system of

sN

one-dimensional oscillators. Show, in particular, that the chemical potential

\mu_s = s\mu_1