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Quantum physics problem: SE and operators

  1. Jun 29, 2012 #1
    Hello.
    I have this problem at hand:

    1. The problem statement, all variables and given/known data
    A quantum mechanical system has a hamilton operator [itex]\hat{H}[/itex] and another, time independent operator [itex]\hat{A}_{0}[/itex].
    Construct a time dependent operator [itex]\hat{A}(t)[/itex] so that:
    <ψ(t)|[itex]\hat{A}_{0}[/itex]|ψ(t)> = <ψ(0)|[itex]\hat{A}(t)[/itex]|ψ(0)>
    for all states ψ(t) that develop in time according to the SE.


    3. The attempt at a solution

    In the derivation of the Schrödinger equation, we use the unitary operator [itex]\hat{U}(t)[/itex] to calculate the effect of time on the state ψ(0)...
    So
    ψ(t) = [itex]\hat{U}(t)[/itex] ψ(0) = exp(-i/[itex]\hbar \hat{H}[/itex] t) ψ(0).

    In other words:
    <ψ(t)|[itex]\hat{A}_{0}[/itex]|ψ(t)> = <[itex]\hat{U}(t)[/itex] ψ(0)|[itex]\hat{A}_{0}[/itex]|[itex]\hat{U}(t)[/itex] ψ(0)>
    =<ψ(0) |[itex]\hat{U}(t)^{+} \hat{A}_{0} \hat{U}(t)[/itex] | ψ(0)>.

    so my "solution" is that
    [itex]\hat{A}(t)[/itex] = [itex]\hat{U}(t)^{+} \hat{A}_{0} \hat{U}(t)[/itex]...

    But this is way too simple to be correct...

    So what am I missing?

    Thanks

    /D
     
  2. jcsd
  3. Jun 29, 2012 #2

    vela

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    Looks fine to me. I suppose you could write explicitly what ##U(t)^\dagger## is equal to.
     
  4. Jun 29, 2012 #3
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