1549380323-Statistical Mechanics Theory and Molecular Simulation

398 Quantum ensembles

〈Pˆk〉= Tr( ̃ρ|wk〉〈wk|)

=

∑

l

〈wl|ρ ̃|wk〉〈wk|wl〉

=

∑

l

wkδkl

=wk, (10.2.9)

where we have used eqn. (10.2.7) and the orthogonality of the eigenvectors of ̃ρ. Note,
however, that

〈Pˆk〉=

1

Z

∑Z

λ=1

〈Ψ(λ)|wk〉〈wk|Ψ(λ)〉

=

1

Z

∑Z

λ=1

|〈Ψ(λ)|wk〉|^2 ≥ 0. (10.2.10)

Eqns. (10.2.9) and (10.2.10) imply thatwk ≥0. Combining the facts thatwk≥ 0
and

∑

kwk = 1, we see that 0≤wk ≤1. Thus, thewksatisfy the properties of
probabilities.
With this key property ofwkin mind, we can now assign a physical meaning to the
density matrix. Let us now consider the expectation value of a projector|ak〉〈ak|≡ˆPak
onto one of the eigenstates of the operatorAˆ. The expectation value of this operator
is given by

〈Pˆak〉=

1

Z

∑Z

λ=1

〈Ψ(λ)|Pˆak|Ψ(λ)〉=

1

Z

∑Z

λ=1

〈Ψ(λ)|ak〉〈ak|Ψ(λ)〉=

1

Z

∑Z

λ=1

|〈ak|Ψ(λ)〉|^2.

(10.2.11)

However,|〈ak|Ψ(λ)〉|^2 ≡Pa(λk)is just the probability that a measurement of the operator
Aˆin theλth member of the ensemble will yield the eigenvalueak. Similarly.

〈Pˆak〉=

1

Z

∑Z

λ=1

Pa(λk) (10.2.12)

is just the ensemble average of the probability of obtaining the valuealin each member
of the ensemble. However, note that the expectation value ofˆPakcan also be written
as

〈Pˆak〉= Tr( ̃ρˆPak)

=

∑

l

〈wl|ρ ̃ˆPak|wl〉

=

∑

l

wl〈wl|ak〉〈ak|wl〉