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Quantum Mechanics Question

  • Thread starter Shomy
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18
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1. Homework Statement

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. Homework 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
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Answers and Replies

nrqed
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1. Homework Statement

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. Homework 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
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.
 
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There was no other information given. What do you mean by shifting the time??
 
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?
 
nrqed
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There was no other information given. What do you mean by shifting the time??
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".
 
nrqed
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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?
Well, it could be done this way, yes. But I had something simpler in mind...see my previous post.
 
Redbelly98
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Well, it could be done this way, yes. But I had something simpler in mind...see my previous post.
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.
 
nrqed
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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.
That's what I wanted him/her to realize: that there are factors [tex] e^{-iE_n t/\hbar} [/tex]
 

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