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 $$e^{-iE_n t/\hbar}$$