Physical Chemistry Third Edition

25.1 The Quantum Statistical Mechanics of a Simple Model System 1041

whereεν 1 ,εν 2 ,εν 3 , andεν 4 are harmonic oscillator energy eigenvalues corresponding

to the quantum numbersν 1 ,ν 2 ,ν 3 , andν 4. In this chapter, we will use a lowercaseε

for a molecular energy eigenvalue and a lowercaseψfor a molecular wave function.

We will use reserveEfor a system energy eigenvalue and capitalΨfor a system wave

function.

Exercise 25.1

a.Assume that the four oscillators in our model system are indistinguishable and are fermions.

Antisymmetrize the wave function of Eq. (25.1-2) by writing 24 terms with appropriate

positive and negative signs. Exchange a few pairs of coordinates to satisfy yourself that the

wave function is antisymmetric. Show that no two quantum numbers can be equal to each

other. Write the wave function as a Slater determinant.

b.Assume that the four oscillators in our system are indistinguishable and are bosons. Sym-

metrize the wave function of Eq. (25.1-2) by writing 24 terms with positive signs. Exchange

a few pairs of coordinates to satisfy yourself that the wave function is symmetric. Show that

it is possible for two or more quantum numbers to be equal to each other.

From the expression for the energy eigenvalues of a harmonic oscillator the energy

of the system is

Ehν(v 1 +v 2 +v 3 +v 4 )+ 4

(

1

2

hν

)

(25.1-4)

The final term is the zero-point energy. We change the zero of potential energy so that

the ground-state vibrational energy is equal to zero, and Eq. (25.1-4) becomes

Ehν(v 1 +v 2 +v 3 +v 4 ) (25.1-5)

We require three variables such as the energy, volume, and amount of substance to

specify the equilibrium macrostate of a fluid system. The energy of our model system

is independent of its volume, so its macrostate is specified by two variables, the energy

and the number of particles in the system. We consider the macrostate corresponding

toE 4 hνandN4. The 35 system microstates compatible with this macrostate

are listed in Table 25.1, and any one of these microstates might be occupied, given the

specified macrostate.

We have labeled the microstates with an indexkthat ranges from 1 to 35. The set

of numbersN 0 (k),N 1 (k),N 2 (k),N 3 (k), andN 4 (k) specify the number of oscillators

occupying each oscillator state for a given system microstate numberk. Such a set

of numbers is called adistribution. We denote a distribution by{N}, a symbol that

stands for the entire set ofN’s. The distribution for each of the 35 system states is

exhibited in Table 25.1. Several of the system states have the same distribution, and we

denote the number of system states corresponding to a given distribution byW({N}).

There are two distributions withW12, one withW6, one withW4, and one

withW1.

Exercise 25.2

Assuming that the four harmonic oscillators in our system are indistinguishable fermions, make

a list of the possible microstates of the system.