where ˜ = 2 /(2IkB ), and σ is the symmetry factor. The summation is over

the symmetry-allowed values of the quantum number J: for a homonuclear

diatomic molecule σ = 2 because J can be either even or odd, depending

on the symmetry of the wave function on interchange of nuclei, and on the

symmetry of the spin state of the nuclei. The high-temperature limit for the

linear rotator, valid for most molecules at ordinary temperatures, is

2IkB T

qrot = . (17.81)

σ2

For a general non-linear molecule rotating in three dimensions, with moment

of inertia tensor16 I, the high-temperature limit for the partition function

is given by

(2kB T )3/2

qrot = π det(I). (17.82)

σ3

In contrast to the rotational partition function, the vibrational partition

16 For the de¬nition of the inertia tensor see (15.52) on page 409.

472 Review of statistical mechanics

function can in general not be approximated by its classical high-tempera-

ture limit. Low-temperature molecular vibrations can be approximated by

a set of independent harmonic oscillators (normal modes with frequency νi )

and the vibrational partition function is a product of the p.f. of each normal

mode. A harmonic oscillator with frequency ν has equidistant energy levels

(if the minimum of the potential well is taken as zero energy):

µn = (n + 1 )hν, n = 0, 1, 2, . . . , (17.83)

2

and the partition function is

exp(’ 1 ξ)

= 1 [sinh(ξ/2)]’1 ,

2

qho = (17.84)

1 ’ exp(’ξ) 2

where ξ = hν/kB T . In the low-temperature limit only the ¬rst level is oc-

cupied and q tends to exp(’ 1 ξ) (or one if the lowest level is taken as zero

2

energy); in the high-temperature (classical) limit q tends to kB T /hν. Fig-

ure 17.4 compares the quantum-statistical partition function, (free) energy

and entropy with the classical limit: although the di¬erence in Q is not

large, the di¬erence in S is dramatic. The classical entropy tends to ’∞

as T ’ 0, which is a worrying result! For temperatures above hν/kB the

classical limit is a good approximation.

17.6 The classical approximation

A full quantum calculation of the partition function of a multidimensional

system is in general impossible, also if the Hamiltonian can be accurately

speci¬ed. But, as quantum dynamics for “heavy” particles can be approxi-

mated by classical mechanics, quantum statistics can be approximated by a

classical version of statistical mechanics. In the previous section we consid-

ered the classical limit for an ideal gas of (quantum) molecules, and found

q N /N ! for the classical or Boltzmann limit of the partition function of N

indistinguishable molecules (see (17.74) on page 470). We also found the

¬rst-order correction for either Fermi“Dirac or Bose“Einstein particles in

terms of a virial coe¬cient proportional to h3 (Eq. (17.68) on page 469).

But these equations are only valid in the ideal gas case when the interaction

between the particles can be ignored. In this section we shall consider a

system of interacting particles and try to expand the partition function in

powers of . We expect the zero-order term to be the classical limit, and

we expect at least a third-order term to distinguish between FD and BE

statistics.

The approach to the classical limit of the quantum partition function

17.6 The classical approximation 473

2.5

Q

2

1.5

1

0.5

0 0.5 1 1.5 2

A,U/hν

2

U

1

0

A

’1

’2

0 0.5 1 1.5 2

S/k

2

1

0

’1

’2

0 0.5 1 1.5 2

Temperature kT/hν

Figure 17.4 Partition function Q, Helmholtz free energy A, energy U and entropy S

for the harmonic oscillator as a function of temperature, for the quantum oscillator

(drawn lines) and the classical oscillator (dashed lines). Temperature is expressed

as kB T /hν, energies are in units of hν and entropy is in units of kB (ν being the

oscillator frequency).

474 Review of statistical mechanics

was e¬ectively solved in the early 1930s. The original expansion in powers

of was done by Wigner (1932), but without considering the symmetry

properties that distinguish FD and BE statistics. The latter was solved

separately by Uhlenbeck and Gropper (1932). Kirkwood (1933) gave a lucid

combined derivation that found its way into most textbooks on quantum

statistics. We shall not copy the derivation, but only give Kirkwood™s line

of reasoning and the main result.

We start with the expression of the quantum partition function (17.47)

for a system of N identical particles:17

Q = tr exp(’βH), (17.85)

where H is the Hamiltonian matrix in an arbitrary orthonormal complete set

of basis functions. The basis functions must have the symmetry imposed by

the particle characters, such as speci¬ed in (17.50). One convenient choice

of basis function is the product of the wave functions for a single particle in

a cubic periodic box with edge a and volume V , see (17.61). For this choice

of plane wave functions

1

ψk e’β H(r) ψk dr,

— ˆ

Q= (17.86)

N!

k

with

2π

k = {k1 , k2 , . . . , kN }, kj = nj , nj ∈ Z3 , (17.87)

a

r = {r 1 , r 2 , . . . , r N }, r i ∈ box, (17.88)

1

(“1)P eiP[ j kj ·r j ] .

ψk = √ V ’N/2 (17.89)

N! P

The permutation P permutes indexes of identical particles, such as exchang-

ing r i and r j , but it does not touch the indexing of k. The sign (“1)P is

negative when an odd number of fermion pairs are interchanged, and positive

otherwise. The factor 1/N ! in the partition function needs some discussion.