Physical Chemistry Third Edition

1040 25 Equilibrium Statistical Mechanics. I. The Probability Distribution for Molecular States

25.1 The Quantum Statistical Mechanics of

a Simple Model System

In Part III of this textbook, we discussed the quantum-mechanical states of single

atoms and molecules. We now focus on systems containing many molecules. There

are two principal types of states that we need to discuss: (1) the mechanical states

(microscopic statesormicrostates) of the entire system, and (2) the thermodynamic

states (macroscopic statesormacrostates) of the system.

Macroscopic states involve variables that pertain to the entire system, such as the

pressureP, the temperatureT, and the volumeV. For a fluid system of one substance

and one phase, the equilibrium macrostate is specified by only three variables, such as

P,T, andV. If we assume that classical mechanics is an adequate approximation, the

microstate of such a system is specified by the position and velocity of every particle

in the system. If quantum mechanics must be used for a dilute gas, there are several

quantum numbers required to specify the state of each molecule in the system. This

is a very large number of independent variables or a very large number of quantum

numbers. Statistical mechanics is the theory that relates the small amount of information

in the macrostates and the large amount of information in the microstates.

A Simple Model System

In order to introduce the principles of statistical mechanics we discuss amodel system,

which means a nonexistent system that is simpler to analyze than a real system. Our

model system consists of four harmonic oscillators, all with the same frequencyν.

The harmonic oscillators in the model system can be distinguished from each other

and do not exert forces on each other. This is of course not a large system contain-

ing many molecules, but will serve to introduce some of the concepts of statistical

mechanics.

The Hamiltonian operator of the system is a sum of four harmonic oscillator

Hamiltonians:

Ĥsystem(1,2,3,4)Ĥ(1)+̂H(2)+Ĥ(3)+Ĥ(4) (25.1-1)

Ĥ(1) is the Hamiltonian operator of harmonic oscillator number 1, and so forth, as

shown in Eq. (15.4-1). The numbers in the parentheses are abbreviations forx 1 ,x 2 ,

x 3 , andx 4 , the coordinates of the four oscillators. There are no terms involving two

or more coordinates in the same term because the oscillators do not interact with each

other.

The Hamiltonian operator in Eq. (25.1-1) corresponds to a time-independent

Schrödinger equation that can be solved by separation of variables, using a trial solu-

tion:

Ψ(1,2,3,4)ψ 1 (1)ψ 2 (2)ψ 3 (3)ψ 4 (4) (25.1-2)

When the solution is carried out, the factorsψ 1 (1),ψ 2 (2),ψ 3 (3), andψ 4 (4) are harmonic

oscillator energy eigenfunctions with quantum numbersν 1 ,ν 2 ,ν 3 , andν 4. Since the

oscillators are distinguishable from each other, the system wave function does not have

to be symmetrized or antisymmetrized, and there are no restrictions on the values of the

quantum numbers. The system energy eigenvalues corresponding to Eq. (25.1-2) are

Eεν 1 +εν 2 +εν 3 +εν 4 (25.1-3)