412 Quantum ideal gases
to plus signs.^1 ) In the fermion case, the determinant leads to a wave function that is
completely antisymmetric with respect to an exchange of any two particle spin labels.
Such an exchange is equivalent to interchanging two rows of the matrixM, which has
the effect of changing the sign of the determinant. These determinants are known as
Slater determinantsafter the physicist John C. Slater (1900–1976) who introduced the
procedure.
In the preceding discussion, each individual particle was treated separately, with
total energy eigenvalues expressed as sums of single-particle eigenvalues, and over-
all wave functions given as determinants/permanents constructed from single-particle
wave functions. We will now introduce an alternative framework forsolving the quan-
tum ideal-gas problem that proves more convenient for the quantum statistical me-
chanical treatment to follow. Let us consider again the single-particle eigenfunction
and eigenvalue for a given vector of integersnand spin eigenvaluem:
φn,m(x) =1
√
V
e^2 πin·r/Lχm(s)εn=2 π^2 ̄h^2
mL^2
|n|^2. (11.2.15)We now ask: How many particles in theN-particle system are described by this wave
function and energy? Let this number befnm, which is called anoccupation number.
The occupation numberfnmtells us how many particles have an energyεnand proba-
bility amplitudeφn,m(x). Since there is an infinite number of accessible statesφn,m(x)
and associated energiesεn, there are infinitely many occupation numbers, and only a
finite subset of these can be nonzero. Indeed, the occupation numbers are subject to
the restriction that the sum over them yield the number of particlesin the system:
∑m∑
nfnm=N, (11.2.16)where
∑
n≡
∑∞
nx=−∞∑∞
ny=−∞∑∞
nz=−∞, (11.2.17)
and
∑
m≡
∑sm=−s(11.2.18)
runs over the (2s+1) possible values ofmfor a spin-sparticle. The occupation numbers
can be used to characterize the total energy eigenvalues of the system. The total energy
eigenvalue can be expressed as
(^1) The permanent of a 2×2 matrix
A =
(
a b
c d)
is perm(A) =ad+bc.