1549380323-Statistical Mechanics Theory and Molecular Simulation

Boltzmann statistics 413

E{fnm}=

∑

m

∑

n

εnfnm, (11.2.19)

which is just a sum over all possible energies multiplied by the number ofparticles
having each energy. The formulation of the eigenvalue problem in terms of accessible
statesφn,m(x), energy levelsεn, and occupation numbers for these states and energies
is known assecond quantization. The framework of second quantization leads to a
simple and elegant procedure for constructing the partition function.

11.3 An ideal gas of distinguishable quantum particles

To illustrate the use of occupation numbers in the evaluation of the quantum partition
function, let us suppose we can ignore the symmetry of the wave function under particle
exchange. Neglect of spin statistics leads to an approximation known asBoltzmann
statistics. Boltzmann statistics are equivalent to an assumption that the particles are
distinguishable because theN-particle wave function for Boltzmann particles is just a
simple product of the functionsφnimi(xi). In this case, spin can also be neglected. The
canonical partition functionQ(N,V,T) can be expressed as a sum over the quantum
numbersn 1 ,...,nNfor each particle:

Q(N,V,T) =

∑

n 1

∑

n 2

···

∑

nN

e−βEn^1 ,...,nN

=

∑

n 1

∑

n 2

···

∑

nN

e−βεn^1 e−βεn^2 ···e−βεnN

=

(

∑

n 1

e−βεn^1

)(

∑

n 2

e−βεn^2

)

···

(

∑

nN

e−βεnN

)

=

(

∑

n

e−βεnN

)N

. (11.3.1)

In terms of occupation numbers, the partition function is

Q(N,V,T) =

∑

{f}

g({f})e−β

∑

nεnfn, (11.3.2)

whereg({f}) is a factor that tells how many distinct physical states can be represented
by a given set of occupation numbers{f}. For Boltzmann particles, exchanging the
momentum labelsniof two particles leads to a new physical state but leaves the occu-
pation numbers unchanged. Thus, the counting problem becomes one of determining
how many different waysNparticles can be placed in the physical states. This means
thatg({f}) is given simply by the combinatorial factor

g({f}) =

N!

∏

nfn!

. (11.3.3)