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Homework Help: Time evolution of Expectation

  1. Sep 28, 2010 #1
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 28, 2010 #2
    Are you asking about how quantum mechanical expectation values evolve with time? If so, then it evolves according to the differential equation

    [tex]\frac{d}{dt}<{\psi}|O|{\psi}> = \frac{i}{\hbar}<{\psi}|[H,O]|{\psi}> + <{\psi}|\frac{{\partial}O}{{\partial}t}|{\psi}>[/tex]

    With O being a Hermitian operator.
     
    Last edited: Sep 28, 2010
  4. Sep 28, 2010 #3
    i don't know what happened to my original post, but I am having an issue with the following problem:

    Show that:

    \frac{d}{dt}<x^2> =\frac{1}{m}(< xp_x> +<p_xx>)

    for a three dimensional wave packet.

    relevant equations:

    Ehrenfest Theorem:(1)
    [tex]i\hbar\frac{d}{dt}<O>=<[O,H]>+i\hbar<\frac{\partial }{\partial t}O>[/tex]

    where O is an operator
    (2)
    [tex]\frac{d}{dt}\int_{V}d^3r\psi ^*O\psi[/tex]

    I tried using both ways illustrated above and I arrived at the same answer:

    [tex]\frac{-i\hbar}{2m}\int_{V}d^3r\psi ^*[\bigtriangledown ^2(x^2\psi)-x^2\bigtriangledown ^2\psi][/tex]

    [tex]=\frac{-i\hbar}{2m}\int_{V}d^3r\psi ^*[\frac{\partial }{\partial x}
    (x^2\psi'+2x\psi)-x^2\frac{\partial^2 }{\partial x^2}\psi][/tex]

    [tex]=\frac{-i\hbar}{2m}\int_{V}d^3r\psi ^*[
    2\psi+2x\frac{\partial }{\partial x}\psi+2x\frac{\partial }{\partial x}\psi+(x^2-x^2)\frac{\partial^2 }{\partial x^2}\psi][/tex]

    [tex]=\frac{1}{m}\int_{V}d^3r[\psi ^*(-i\hbar)\psi+\psi^*x(-i\hbar\frac{\partial }{\partial x})\psi+\psi^*(-i\hbar\frac{\partial }{\partial x})x\psi][/tex]

    [tex]=\frac{1}{m}(<xp_x>+<p_xx>)-\frac{i\hbar}{m}[/tex]

    Am I doing anything wrong? Where does the extra term come from, and does it mean anything?
     
  5. Sep 28, 2010 #4
    nvm, I got it:

    [tex]
    =\frac{-i\hbar}{2m}\int_{V}d^3r\psi ^*[\frac{\partial }{\partial x}
    (x^2\psi'+2x\psi)-x^2\frac{\partial^2 }{\partial x^2}\psi]
    [/tex]

    [tex]=\frac{-i\hbar}{2m}\int_{V}d^3r\psi ^*[x^2\frac{\partial^2 }{\partial x^2}\psi+2x\frac{\partial }{\partial x}\psi+2\frac{\partial }{\partial x}(x\psi)-x^2\frac{\partial^2 }{\partial x^2}\psi][/tex]

    [tex]=\frac{1}{m}\int_{V}d^3r[\psi^*x(-i\hbar\frac{\partial }{\partial x})\psi+\psi^*(-i\hbar\frac{\partial }{\partial x})x\psi][/tex]

    [tex]=\frac{1}{m}(<xp_x>+<p_xx>)[/tex]

    I went wrong thinking I could just rearrange the derivatives.
     
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