584
10 Quantum Statistics
in terms of the partition function
Z(T ) ≡
i
e
−βE i ,
(10.7.9b)
with the sum being over the energy states of the system. Thus, for the canonical
ensemble the density matrix takes the form
ρ =
e −βH
T r{e −βH }
=
1
Z
e
−βH .
(10.7.9c)
The ensemble average of the expectation value of an operator A is thus given in the
canonical ensemble by
A =
1
Z(T )
T r
e
−βH
A
.
(10.7.10)
Most applications of the density matrix approach involve the canonical ensemble
version. For completeness, however, the density matrix operator for the grand
ensemble, has been shown by Greiner et al. [11] to have the form
ρ =
1
e
−β(H−μN ) ,
(10.7.11)
in which is the grand partition function, μ is the chemical potential, and N is the
particle number operator.
10.8 Problems for This Chapter
1. In how many ways may N identical, yet distinguishable, objects be arranged in
M different containers, with no limitation on the number of objects in any given
container? Such arrangements apply to a classical, or Maxwell–Boltzmann,
system consisting of distinguishable particles.
2. In how many ways may N identical, indistinguishable objects be arranged in M
different containers (N < M) such that no more than a single object occupies
any given container? This process corresponds to the conditions that apply to a
system of Fermi–Dirac particles.
3. In how many ways may N identical, indistinguishable objects be arranged in
M different containers, with no limitation placed upon the number of objects
in a given container? This process corresponds to the conditions that apply to a
system of Bose–Einstein particles.
4. Employ the combinatoric formulae (10.1.1a) and (10.1.1b) for the numbers of
microstates accessible to multi-particle Bose–Einstein and Fermi–Dirac sys-