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Quantum Mechanics Question

  1. Apr 18, 2008 #1
    1. The problem statement, all variables and given/known data

    I'm supposed to show that whatever superposition of harmonic oscillator states is used to construct wavepacket of the form [tex]\sum[/tex] cv[tex]\Psi[/tex] (x,t) (cv are arbitary complex coefficients), it is at the same place at the times 0, T, 2T,.. where T = 2 [tex]\pi[/tex]/[tex]\omega[/tex]


    2. Relevant equations


    3. The attempt at a solution

    I was thinking of using the position operator on the function and subbing t = 2n [tex]\pi[/tex]/[tex]\omega[/tex] as the time but i dont really know where to start
    PHP:
     
     
  2. jcsd
  3. Apr 18, 2008 #2

    nrqed

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    WHat is the time dependence of each term in your sum? Consider shifting the time t by [tex] 2 \pi / \omega [/tex] and see what happens.
     
  4. Apr 19, 2008 #3
    There was no other information given. What do you mean by shifting the time??
     
  5. Apr 19, 2008 #4

    Redbelly98

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    It has been a while for me, but I believe nrged is saying to apply the time-evolution operator,

    e^(iHt)
    or maybe it was
    e^(-iHt)

    Have the covered this concept in your class?
     
  6. Apr 19, 2008 #5

    nrqed

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    First things first. What is the time dependence of the total wavefunction? Psi is a linear combination of the eigenstates of the Hamiltonian, right? What is the time dependence of each eigenstate? What i sthe time dependence of the total wavefunction? Can you writ edown the total wavefunction, showing explicitly its time dependence?

    Then you should simply replace t by t+2 pi/omega in you expression and you should see that the total wavefunction remains unchanged. That's what I meant by "shifting the time".
     
  7. Apr 19, 2008 #6

    nrqed

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    Well, it could be done this way, yes. But I had something simpler in mind...see my previous post.
     
  8. Apr 19, 2008 #7

    Redbelly98

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    Okay. I didn't see the necessary e^iwt factors explicitly in the original description, but perhaps they are in Shomy's textbook or class notes description of the H.O. wavefunctions.
     
  9. Apr 19, 2008 #8

    nrqed

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    That's what I wanted him/her to realize: that there are factors [tex] e^{-iE_n t/\hbar} [/tex]
     
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