ABSTRACT
Up until this point, we have concerned ourselves with a description of quantum mechanics centered upon how a state |ψ〉 evolves in time. Using this we can show that the expectation value of an operator evolves as
〈A(t)〉 = 〈ψ(t)| ˆA|ψ(t)〉 (6.1) However, for many instances, especially if we are interested in describing relaxation processes, it is useful to introduce the density operator
ρˆ(t) = |ψ(t)〉〈ψ(t)| (6.2) taken as the outer product of the state vector with itself. From this definition we can write
ρˆ = ∑ mn
c∗ncm |m〉〈n| (6.3)
= ∑ mn
ρmn|m〉〈n| (6.4)
where the ρmn are the density matrix elements. Expectation values of operator are then given by the trace
〈A(t)〉 = ∑ mn
Amnρmn = Tr [ ˆAρ(t)] (6.5)
where Tr [ ˆAρ(t)] denotes the trace operation:
Tr [ ˆAρ(t)] = ∑ nn′
If ρ is diagonal, then
Tr [ ˆAρ(t)] = ∑
Annρnn = ∑
〈n|A|n〉Pn
where Pn = ρnn is the statistical probability of finding the system in state n. These statistical weights must be such that Pn ≤ 1 and∑
Pn = 1
Hence, we conclude that knowing ρ we can compute the statistical average of an operator A.
