Statistical Mechanics · Problem 3.30

The Grand Canonical Ensemble — R. K. Pathria

Physics — Statistical Mechanics · quantum-anharmonic-oscillator-heat-capacity

Problem

The energy levels of a quantum-mechanical, one-dimensional, anharmonic oscillator may be approximated as

\varepsilon_n = \left(n + \tfrac{1}{2}\right)\hbar\omega - x\left(n + \tfrac{1}{2}\right)^2\hbar\omega; \quad n = 0, 1, 2, \ldots

The parameter

x

, usually

\ll 1

, represents the degree of anharmonicity. Show that, to the first order in

x

and the fourth order in

u\ (\equiv \hbar\omega/kT)

, the specific heat of a system of

N

such oscillators is given by

C = Nk\left[\left(1 - \frac{1}{12}u^2 + \frac{1}{240}u^4\right) + 4x\left(\frac{1}{u} + \frac{1}{80}u^3\right)\right].

Note that the correction term increases with temperature.