.

T

•

•

φk , µk , nk

••

•

Figure 17.2 Single-particle quantum states k, with wave function φk , energy µk and

occupation number nk . For fermions nk is restricted to 0 or 1. The shown double

occupation of the third level is only possible for bosons.

total wave function is an (anti)symmetrized sum of product states

1

Ψi (r 1 , r 2 , . . .) = √ (“1)P P [φk1 (r 1 )φk2 (r 2 ) . . .]. (17.50)

N! P

Here the sum is over all possible N ! permutations of the N particles, and

(“1)P is a shorthand notation for ’1 in case of an odd number of permuta-

tions of two fermions (the upper sign) and +1 in case of bosons (lower sign).

√

The factor 1/ N ! is needed to normalize the total wave function again.

It is clear that the total wave function vanishes if two fermions occupy the

same single-particle wave function φk . Therefore the number of particles nk

occupying wave function k is restricted to 0 or 1, while no such restriction

exists for bosons:

nk = 0, 1 (fermions)., (17.51)

nk = 0, 1, 2, . . . (bosons). (17.52)

A N -particle wave function is characterized by the set of occupation numbers

n = {n1 , n2 , . . . , nk , . . .} with the restriction

N= nk . (17.53)

k

The energy En is given by

En = n k µk . (17.54)

k

All possible states with all possible numbers of particles are generated by

all possible sets n of numbers subject to the condition (17.51) for fermions.

17.5 Fermi“Dirac, Bose“Einstein and Boltzmann statistics 467

Thus the grand partition function equals

∞

e’βEn

»N

Ξ=

n

N =0

nk ’β

···» nk µk

= e

k k

n1 n2

nk

· · · Πk »e’βµk

=

n1 n2

n1 n2

»e’βµ1 »e’βµ2 ···

=

n1 n2

nk

»e’βµk

= Πk . (17.55)

nk

where each sum over nk runs over the allowed values. For fermions only the

values 0 or 1 are allowed, yielding the Fermi“Dirac statistics:

ΞFD = Πk 1 + »e’βµk . (17.56)

For bosons all values of nk are allowed, yielding the Bose“Einstein statistics:

’1

ΞBE = Πk 1 + »e’βµk + »2 e’2βµk + . . . = Πk 1 ’ »e’βµk . (17.57)

Equations (17.56) and (17.57) can be combined as

±1

’βµk

= Πk 1 ± »e

ΞFD (17.58)

BE

with thermodynamic relation

ln 1 ± »e’βµk

βpV = ln Ξ = ± (17.59)

k

and occupancy numbers given by

exp[’β(µk ’ μ/NA )]

» exp(’βµk )

nk = = (17.60)

1 ± » exp(’βµk ) 1 ± exp[’β(µk ’ μ/NA )]

(upper sign: FD; lower sign: BE). Figure 17.3 shows that fermions will

¬ll low-lying energy levels approximately until the thermodynamic potential

per particle (which is called the Fermi level) is reached; one example is

electrons in metals (see exercises). Bosons tend to accumulate on the lowest

levels; the thermodynamic potential of bosons is always lower than the lowest

level. This Bose condensation phenomenon is only apparent at very low

temperatures.

468 Review of statistical mechanics

nk

2

1.5

BE

1

B

FD

0.5

0

’4 ’2 0 2 4

β(µk ’ μ/NAv)

Figure 17.3 Occupation number nk of kth single-particle quantum state, in an ideal

quantum gas, as function of the energy level µk above the thermodynamic potential

μ, for fermions (FD) and bosons (BE). The dashed line indicates the classical

approximation (Boltzmann statistics).

17.5.3 The Boltzmann limit

In gases at high temperature or low density the number of available quantum

states considerably exceeds the number of particles. In a periodic box of

dimensions a — a — a; V = a3 , the functions

2π

φk = V ’1/2 exp(ikr), n, n ∈ Z3 ,

k= (17.61)

a

are eigenfunctions of the kinetic energy operator, with eigenvalues

2 k2

µk =

. (17.62)

2m

This implies (see Exercise 17.3) that the number of single-particle transla-

tional quantum states between µ and µ + dµ is given by

√ m3/2 V 1/2