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Proof of expectation value for a dynamic observable

  1. Apr 8, 2017 #1
    1. The problem statement, all variables and given/known data
    Show that:
    d<A(q,p)>/dt=<{A,H}>, where {A,H} is a Poisson Bracket

    2. Relevant equations
    Liouville theorem

    3. The attempt at a solution
    <A>=Tr(Aρ)⇒d<A>/dt=Tr(Adρ/dt)=Tr(A{H,ρ})
    So, in order to get the correct result, Tr(A{H,ρ}) must be equal to Tr({A,H}ρ), but I don't think I can do that substitution. Is it valid? How can I prove that?
     
  2. jcsd
  3. Apr 9, 2017 #2

    vela

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    Use the cyclic property of the trace: Tr(ABC) = Tr(BCA) = Tr(CAB).
     
  4. Apr 9, 2017 #3
    But there is a Poisson Bracket in it...
     
  5. Apr 9, 2017 #4

    vela

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    Good point.
     
  6. Apr 9, 2017 #5
    So? lol
     
  7. Apr 9, 2017 #6

    vela

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    Try expanding out the Poisson bracket and see if there's something you can do with the derivatives.
     
  8. Apr 9, 2017 #7
    No luck :(
     
  9. Apr 13, 2017 #8
    Bump...
     
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