Blundell Thermal Physics · Problem 20.7

The partition function

Physics — Statistical Mechanics · three-harmonic-oscillators-partition-function

Problem

The energy

E

of a system of three independent harmonic oscillators is given by

E = \left(n_x + \frac{1}{2}\right)\hbar\omega + \left(n_y + \frac{1}{2}\right)\hbar\omega + \left(n_z + \frac{1}{2}\right)\hbar\omega. \tag{20.56}

Show that the partition function

Z

is given by

Z = Z_{\mathrm{SHO}}^3, \tag{20.57}

where

Z_{\mathrm{SHO}}

is the partition function of a simple harmonic oscillator given in eqn 20.3. Hence show that the Helmholtz function is given by

F = \frac{3}{2}\hbar\omega + 3k_BT\ln(1 - e^{-\beta\hbar\omega}), \tag{20.58}

and that the heat capacity tends to

3k_B

at high temperature.