Problems and Solutions on Thermodynamics and Statistical Mechanics

310 Problems €4 Solutiona on Thermodynamics d Statistical Mechanica

V kT

Nv

potential for the three-dimensional gas, - = -, and in that for the two-

A1

Nn

dimensional gas, - = -. Since the two chemical potentials have the same

value, one obtains

2130

A simple harmonic one-dimensional oscillator has energy levels En =

(n + 1/2)Aw, where w is the characteristic oscillator (angular) frequency

andn=0,1,2, ....

(a) Suppose the oscillator is in thermal contact with a heat reservoir

at temperature T, with - << 1. Find the mean energy of the oscillator as

a function of the temperature T.

kT

AW

(b) For a two-dimensional oscillator, n = n, + ny, where

n, = 0,1,2,... and ny = 0,1,2,... , what is the partition function for

this case for any value of temperature? Reduce it to the degenerate case

w, = wy.

(c) If a one-dimensional classical anharmonic oscillator has potential

energy V(z) = cx2 - gz3, where gx3 << cz2, at equilibrium temperature T,

carry out the calculations as far as you can and give expressions as functions

of temperature for

1) the heat capacity per oscillator and

2) the mean value of the position z of the oscillator.

(UC, Berkeley)

tw

(a) Putting a = - = Awp, one has

kT

Solution: