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Basic calculation problem with commutators

  1. Jul 26, 2009 #1
    1. The problem statement, all variables and given/known data
    A is a Hermitian operator which commutes with the Hamiltonian: [itex]\left[A,H\right]=AH-HA=0[/itex]

    To be shown: [itex]\frac{d}{dt}A=0[/itex]

    2. Relevant equations

    Schrödinger equation: [itex]i\hbar\frac{\partial}{\partial t}\psi=H\psi[/itex] with the Hamilton operator H.

    3. The attempt at a solution
    I have seen this solution on many sites:

    [itex]\frac{d}{dt}<A>=\frac{d}{dt}<\psi|A|\psi>=<\psi|\frac{\partial A}{\partial t}|\psi>+<\frac{d\psi}{dt}|A|\psi>+<\psi|A|\frac{d\psi}{dt}>=<\frac{\partial A}{\partial t}>+\frac{1}{i\hbar}<\left[ A, H\right] >=0[/itex]

    I have a problem with this: [itex]<\frac{d\psi}{dt}|A|\psi>+<\psi|A|\frac{d\psi}{dt}>=\frac{1}{i\hbar}<\left[ A, H\right] >[/itex]

    Okay, obviously we have from the Schrödinger equation:
    [itex]H=i\hbar\frac{\partial}{\partial t}[/itex]
    and thus
    [itex]\frac{\partial}{\partial t}=\frac{1}{i\hbar}H[/itex]
    and thus
    [itex]<\frac{d\psi}{dt}|A|\psi>+<\psi|A|\frac{d\psi}{dt}>=\frac{1}{i\hbar}(<H\psi|A|\psi>+<\psi|A|H\psi>)=\frac{1}{i\hbar}<\psi|HA+AH|\psi>[/itex]

    But this is not the commutator but the anti-commutator. It is plus and not minus! What did I do wrong here?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jul 26, 2009 #2

    Dick

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    Remember <a|ib>=i*<a|b> but <ia|b>=(-i)*<a|b>.
     
  4. Jul 26, 2009 #3
    Oh, of course! :yuck:
    Thank you.
     
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