582
10 Quantum Statistics
Substitution of Eq. (10.7.2b) into Eq. (10.7.3) gives the ensemble average A of
the operator A as
A =
N
α=1
i,j
i|ϕ α p α ϕ α |j j |A|i .
Upon defining a density matrix operator ρ as
ρ ≡
α
|ϕ α p α ϕ α |,
(10.7.4a)
with matrix elements ρ ij given by
ρ ij ≡ ≡i|ρ|j =
N
α=1
i|ϕ α p α ϕ α |j ,
(10.7.4b)
then gives the ensemble average of A as
A =
i,j
i|ρ|j j |A|i ≡
i
i|ρA|i ,
(10.7.5)
upon utilizing the completeness of the (orthonormal) basis set. Thus, the ensemble
average A of the operator A is obtained as the sum of the diagonal elements, or
trace T r the product of operator ρA, or
A = T r{ ρA} .
(10.7.6)
As the density matrix ρ is by definition an Hermitian operator, it follows that
the density matrix corresponding to a system eigenfunction φ that is normalized to
unity must satisfy the condition
T r ρ =
i
i|ρ|i = 1 .
Diagonal elements of ρ may thus be interpreted as probabilities of occurrence of
the energy eigenstates of H in the ensemble, thereby providing a basis for utilizing
density matrix calculations of ensemble averages.
Although a time-independent density matrix suffices for the description of
equilibrium systems because their observable properties are typically independent
of time, a time dependent density matrix would be needed for the description of
observable properties in a nonequilibrium system. For additional material on the
density matrix and its applications, see Bransden and Joachain [10].