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# Quantum Density Matrix

DOI link for Quantum Density Matrix

Quantum Density Matrix book

# Quantum Density Matrix

DOI link for Quantum Density Matrix

Quantum Density Matrix book

## 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.