Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physics 257

2080
In a perfect gas of electrons, the mean number of particles occupying
a single-particle quantum state of energy E, is:

1

N; =

.XP[(Ei - p)/kT] + 1 *

(a) Obtain a formula which could be used to determine p in terms of

(b) Show that the expression above reduces to the Maxwell-Boltzmann
distribution in the limit nX3 << 1, where X is the thermal de Broglie wave-
length.

the particle density n and various constants.

(c) Sketch Ni versus E; for T = 0 K and for T = p/5 K. Label

(UC, Berkeley)

significant points along both axes.

Solution:

(a) The particle number density is

As

x

This formula can be used to determine /I.

(b) When nX3 << 1, we must have in the above integral

It follows that

i.e., it reduces to the Boltzmann distribution.