Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physic8 311

(b) There is no difference between a two-dimensional oscillator and two
independent one-dimensional oscillators, then the partition function is

eaz/2 eay/2

z=-.-

eaz - 1

When w, = wy, a, = ay = a, we have

eau - 1 *

ea

2=

(en - 1)2 *

(c) 1) We calculate the partition function

z = / exp[-(cx2 - gz3)/kT]dx.

(Note that the kinetic energy term has not been included in the expression,

this is done by adding - in the heat capacity later.) The non-harmonic

term (exp(gs3/kT) - 11 is a small quantity in the region of motion. Using

Taylor’s expansion retaining only the lowest order terms, we get

k

2

The mean value of the potential energy is

The heat capacity per oscillator is

2) In the first-order approximation, the mean value of the position x

of the oscillator is