Physical Chemistry Third Edition

25.1 The Quantum Statistical Mechanics of a Simple Model System 1043

Postulate 2. All system microstates of equal energy are equally probable and enter in the

average of postulate 1 with equal weight.

A more complete version of the second postulate assumes randoma prioriphases of

the time-dependent factors in the wave functions and involves quantum-mechanical

density matrices.^1 We present only the simplest version of statistical mechanics.

Postulate 2 is equivalent to assuming that over some long period of time during

which the system remains in a given macrostate, it would spend equal time in each of

the microstates that correspond to the given macrostate. This assumption is sometimes

called theergodic hypothesis. For a system of many particles there are very many

possible microstates corresponding to a single macrostate and there is no guarantee

that all of them would be occupied in any period of time, no matter how long, or that

each of them would be occupied for the same length of time. The postulates must be

regarded as hypotheses whose use is justified only by comparing their consequences

with experiment.

The Probability Distribution of Molecular States

We now seek the probability that a randomly selected oscillator would occupy a partic-

ular vibrational state. All of the harmonic oscillators are alike, so we focus on oscillator

number 1. Of the 35 system microstates, there are 15 states withv 1 0. Since all of the

35 possible system microstates are equally probable, the probability that a randomly

chosen oscillator will havev0is

p 0 

15

35

 0. 42857 (25.1-6a)

There are 10 states for whichv 1 1, so that

p 1 

10

35

 0. 28571 (25.1-6b)

The other probabilities are

p 2 

6

35

 0. 17143 (25.1-6c)

p 3 

3

35

 0. 08571 (25.1-6d)

p 4 

1

35

 0. 02857 (25.1-6e)

Exercise 25.3

Show that the same probabilities occur if oscillator number 2 is examined.

These probabilities were obtained by averaging over system microstates, so we call

them anaverage probability distribution. When we use these probabilities to calculate

an average molecular quantity we will actually have a double average: an average over

molecular states using a probability distribution that was itself obtained by averaging

(^1) L. E. Reichl,Statistical Physics, University of Texas Press, Austin, 1980.