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Time evolution of density matrix

  1. Apr 22, 2013 #1
    1. The problem statement, all variables and given/known data

    Hi there. just working on a problem from sakurai's modern quantum mechanics. it is:

    A) Prove that the time evolution of the density operator ρ (in the Schrodinger picture) is given by
    [itex]ρ(t)=U(t,t_{0})ρ(t_{0})U^\dagger(t,t_{0})[/itex]

    B) Suppose that we have a pure ensemble at t=0. Prove that it cannot evolve into a mixed ensemble as long as the time evolution is governed by the Schrodinger equation.


    2. Relevant equations



    3. The attempt at a solution

    Working out:

    part a) ok, so what i've done is simply say the state |α[itex]^{i}>[/itex] at some time t can be described as:

    [itex]|α^{i};t>=U(t)|α^{i};t_{0}>[/itex]

    Knowing that:
    [itex]ρ(t)=\sum w_{i}|α^{i}><α^{i}|[/itex]

    then
    [itex]ρ(t)=\sum w_{i}U(t)|α^{i};t_{0}><α^{i};t_{0}|U^\dagger(t)[/itex]
    [itex]ρ(t)=U(t,t_{0})ρ(t_{0})U^\dagger(t,t_{0})[/itex]

    part b)
    for this i looked at the trace of ρ[itex]^{2}[/itex]

    [itex]tr(ρ^{2}))=tr(U(t)ρ(t_{0})U^\dagger(t)Uρ(t_{0})U^\dagger)[/itex]
    [itex]tr(ρ^{2}))=tr(ρ(t_{0})ρ(t_{0})U^\dagger(t)U(t)[/itex]
    [itex]tr(ρ^{2}))=tr(ρ^{2}(t_{0}))[/itex]


    all the other questions i have been given in this class have taken a couple of pages worth of working out, and that has made me paranoid that i'm over-simplifying this problem and possibly missing something. any feedback would be much appreciated.

    cheers guys!
     
    Last edited by a moderator: Apr 22, 2013
  2. jcsd
  3. Apr 22, 2013 #2

    TSny

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    That all looks good to me. Alternatively, for part (b) you could try to show that the pure state condition [itex]ρ^{2}=ρ[/itex] holds at all times if it holds at ##t = t_0##.
     
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