How Does Time Periodicity Affect Wavepacket Position in Quantum Mechanics?

[Shomy]

Homework Statement

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

Homework Equations

The Attempt at a Solution

I was thinking of using the position operator on the function and subbing t = 2n \pi/\omega as the time but i don't really know where to start

 

[nrqed]

Homework Statement

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

Homework Equations

The Attempt at a Solution

I was thinking of using the position operator on the function and subbing t = 2n \pi/\omega as the time but i don't really know where to start

WHat is the time dependence of each term in your sum? Consider shifting the time t by 2 \pi / \omega and see what happens.

 

[Shomy]

There was no other information given. What do you mean by shifting the time??

 

[Redbelly98]

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]

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]

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]

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]

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 e^{-iE_n t/\hbar}