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QM, Heisenberg's motion equations, harmonic oscillator

  1. Apr 20, 2013 #1

    fluidistic

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    Gold Member

    1. The problem statement, all variables and given/known data
    Hi guys, I don't really know how to solve the first part of a problem which goes like this:
    Consider a 1 dimensional harmonic oscillator of mass m, Hooke's constant k and angular frequency ##\omega = \sqrt{\frac{k}{m} }##.
    Remembering the classical solutions, solve the Heisenberg's motion equations for the operators ##\hat x## and ##\hat p##. Why does the quantum evolution match the classical one?


    2. Relevant equations
    Heisenberg's motion equation: ##\frac{dA(t)}{dt}=\frac{i}{h}[\hat H, \hat A (t)]=\frac{i}{h}[\hat H, \hat A](t)##. (eq.1)
    Where ##\hat A(t)=U_t^*AU_t##.
    Where ##U_t=\exp \{ -i\hat H t/\hbar \}##. Since ##\hat H## is self-adjoint, it follows that ##U_t## is unitary. Thus ##U_t^*=U_t^{-1}##. Also ##U_t^*=U_{-t}##.
    3. The attempt at a solution
    From Classical mechanics, ##H=\frac{p^2}{2m}+\frac{m\omega^2 q^2}{2}##.
    This leads to ##\dot p =-m\omega ^2 q## and ##\dot q=\frac{p}{m}##. I know I must solve them but before proceeding, let's see if I can find some similar equations to solve using Heisenberg's motion equations.
    In QM, ##\hat H=\frac{\hat p ^2}{2m}+\frac{m\omega ^2}{2} \hat x##.
    Using eq.1, skipping arithmetical steps I reached that ##\frac{d}{dt} \hat A (t) = \frac{i}{\hbar} \{ \frac{1}{2m} [\hat p ^2 , \hat A](t) +\frac{m \omega ^2}{2} [\hat x ^2 , A](t) \}## where ##\hat p=-i\hbar \frac{d}{dx}## and ##\hat x =x##.
    Now, replacing ##\hat A## by ##\hat x##, I get that ##\frac{d\hat x}{dt}=\frac{i}{\hbar} \{ \frac{1}{2m}[\hat p^2,\hat x](t) \}=\frac{i}{2m\hbar}(-\hat x \hat p^2)(t)=-\frac{i}{2m\hbar}U_t^* (\hat x \hat p ^2)U_t##.
    Here I am unsure of my step. I think that it's worth ##-\frac{i}{2m\hbar}U_t ^* U_t \hat x \hat p^2=-\frac{i}{2m\hbar} \hat x \hat p ^2##.
    So I get the differential equation ##\frac{d\hat x}{dt}=-\frac{i}{2m\hbar}\hat x \hat p ^2## which differs from the one of classical mechanics for ##\dot q##. I don't know what I did wrong.

    Similarly for ##\frac{d \hat p}{dt}##, I found that it's worth ##\frac{m \omega ^2 }{2}(\hat x \hat p -1)## which again does not match the DE of CM.
    I guess my approach is wrong?
    I would like any pointer. Thank you.
     
  2. jcsd
  3. Apr 21, 2013 #2

    BruceW

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    Homework Helper

    I don't think this step is correct. try checking this again.
     
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