QuantumPhysics.dvi

16.2.1 Ensemble averages of expectation values in mixtures

In a pure normalized state|ψ〉, we defined theexpectation value of an observableAby the matrix
element〈ψ|A|ψ〉= Tr

(

A|ψ〉〈ψ|

)
, a quantity that gives the quantum mechanical weighed probability
average of the eigenvalues ofA.

In a mixed state, these quantum mechanical expectation values must be further weighed by
the population fraction of each pure state in the mixture. One defines theensemble averageof the
observableAby

Tr(ρA) =

∑

i

Tr

(

A|ψi〉wi〈ψi|

)

=

∑

i

wi〈ψi|A|ψi〉 (16.10)

It follows from the properties of the density operator that the ensemble average of any self-adjoint
operator is real.

16.2.2 Time evolution of the density operator

Any quantum state|ψ(t)〉in the Schr ̈odinger picture evolves in time under the Schr ̈odinger equation,

i ̄h

d

dt

|ψ(t)〉=H|ψ(t)〉 (16.11)

Assuming that the population fractionswiandpido not change in time during this evolution, the
density operatorρ(t) then obeys the following time evolution equation,

i ̄h

d

dt

ρ(t) = [H,ρ(t)] (16.12)

To prove this formula, one first derives the time evolution equation for|ψ(t)〉〈ψ(t)|and then takes
the weighed average with the time-indepenent population fractions. This time evolution may be
solved for in terms of the unitary evolution operatorU(t) =e−itH/ ̄h, by

ρ(t) =U(t)ρ(0)U(t)† (16.13)

Notice that the normalization condition Tr(ρ) = 1 is automatically preserved under time evolution.

16.3 Example of the two-state system

Consider the two-state system of a spin 1/2, expressed in theeigenbasis ofSz, given by the states
|z+〉and|z−〉. Below are three examples of the density matrix evaluated for pure states.

• Polarized along theSzdirection with eigenvalue + ̄h/2, the state is|z+〉, and the correspond-
  ing density operator isρ=|z+〉〈z+|. The ensemble averages are tr(ρSx) = tr(ρSy) = 0,
  and tr(ρSz) = + ̄h/2.