10.7 Density Matrix and Ensemble Averages
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is referred to as a ‘bra’, and is designated i|. In terms of these quantities, the inner
product between the two quantum states represented by |i and |j is given by the
‘bracket’ i|j . If a quantum mechanical system can be described in terms of a single
quantum state, it is said to be in a pure state; otherwise, it is said to be in a mixed
state, that is, a state that is represented by a linear combination of the |ϕ α , namely
|ψ =
α
p α |ϕ α ,
in which the p α represent (relative) probabilities satisfying
α
p α = 1.
An ensemble of pure quantum states is thus necessarily represented by a mixed
quantum state.
We shall employ Roman letters to identify eigenstates of the Hamiltonian and
Greek letters to label the members of an ensemble containing N subsystems, each
of which is in a pure quantum state |ϕ α that has been normalized to 1, i.e., ϕ α |ϕ α
= 1; α = 1, . . . , N. If we further choose a complete orthonormal basis set {|i; i =
1, . . . , } (such as the eigenfunctions of the system Hamiltonian), then each system
eigenfunction |ϕ α may be expressed in terms of this basis set as
|ϕ α =
i
c
(α)
i |i ≡
i
ϕ α |i |i .
(10.7.1)
Let us identify the expectation value of an operator A in the pure state |ϕ α
associated with an individual member of the ensemble as
A α ≡ ≡ϕ α |A|ϕ α =
i,j
c
(α)∗
i
c
(α)
i j |A|i
(10.7.2a)
or, in terms of Dirac notation,
A α =
i,j
i|ϕ α ϕ α |j j |A|i .
(10.7.2b)
We may now express the ensemble average A as
A ≡
N
α=1
p α A α ,
(10.7.3)
in which p α (0 ≤ p α ≤ 1, and
N
α=1
p α = 1) is the probability that the system is
found in quantum state |ϕ α .