Another quantum oscillator

  • Thread starter arenaninja
  • Start date
  • #1
arenaninja
26
0

Homework Statement


A particle mass m in the harmonic oscillator potential starts out in the state [tex]\psi(x,0)=A\left(1-2\sqrt(\frac{m\omega}{\hbar})x\right)^{2}e^{\frac{-m\omega}{2\hbar}x^{2}}[/tex] for some constant A.
a) What is the expectation value of the energy?
b) At some time later T the wave function is [tex]\psi(x,T)=B\left(1+2\sqrt(\frac{m\omega}{\hbar})x\right)^{2}e^{\frac{-m\omega}{2\hbar}x^{2}}[/tex]

Homework Equations


I used ladder operators (wayyy better than doing integrals, though I figured that out only minutes ago).

The Attempt at a Solution


I solved that the constant A should be [tex]\frac{1}{5}\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}[/tex], but apparently it wasn't needed.
For part a I got [tex]\frac{\hbar\omega}{2}\left(2n+1\right)[/tex], which has units of energy so it could be ok. But part b has me baffled. Neither equation has a time dependence, so I have no clue what T should be. I do notice that the constant has changed, and that there seems to be a sign reversal. Unfortunately I'm stuck as to how this would help me resolve what T should be.
 

Answers and Replies

  • #2
fzero
Science Advisor
Homework Helper
Gold Member
3,119
290

Homework Statement


A particle mass m in the harmonic oscillator potential starts out in the state [tex]\psi(x,0)=A\left(1-2\sqrt(\frac{m\omega}{\hbar})x\right)^{2}e^{\frac{-m\omega}{2\hbar}x^{2}}[/tex] for some constant A.
a) What is the expectation value of the energy?
b) At some time later T the wave function is [tex]\psi(x,T)=B\left(1+2\sqrt(\frac{m\omega}{\hbar})x\right)^{2}e^{\frac{-m\omega}{2\hbar}x^{2}}[/tex]

Homework Equations


I used ladder operators (wayyy better than doing integrals, though I figured that out only minutes ago).

The Attempt at a Solution


I solved that the constant A should be [tex]\frac{1}{5}\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}[/tex], but apparently it wasn't needed.
For part a I got [tex]\frac{\hbar\omega}{2}\left(2n+1\right)[/tex], which has units of energy so it could be ok.

There's no [tex]n[/tex] in [tex]\psi(x,0)[/tex], so that's not the complete answer. You might want to show some of your work if you can't figure it out.

But part b has me baffled. Neither equation has a time dependence, so I have no clue what T should be. I do notice that the constant has changed, and that there seems to be a sign reversal. Unfortunately I'm stuck as to how this would help me resolve what T should be.

You don't actually state a question for part b, so I can't make a specific suggestion. But in general, the energy eigenstates evolve in time according to

[tex] | n, t\rangle = e^{-iE_nt/\hbar} |n\rangle .[/tex]

You can use this formula to determine [tex]\psi(x,t)[/tex].
 
  • #3
arenaninja
26
0
Well since I was using ladder operators I was eventually left with [tex]<\Psi_{n}|H\Psi_{n}>=const*<\Psi_{n}|(2n+1)\Psi_{n}>[/tex], and here I used the fact that the time dependence wouldn't matter in the inner product (since, once I take out 2n+1, it must be 1)

Part b) is actually "b) At some time later T the wave function is [tex]\psi(x,T)=B\left(1+2\sqrt(\frac{m\omega}{\hbar})x\right)^{2}e^{\frac{-m\omega}{2\hbar}x^{2}}[/tex] for some constant B. What is the smallest possible value of T?"

Sorry about the confusion. I'm pretty confident on my answer to a) (though if there's something blatantly wrong please let me know), it's only part b) that I'm having issues with.
 
  • #4
fzero
Science Advisor
Homework Helper
Gold Member
3,119
290
Well since I was using ladder operators I was eventually left with [tex]<\Psi_{n}|H\Psi_{n}>=const*<\Psi_{n}|(2n+1)\Psi_{n}>[/tex], and here I used the fact that the time dependence wouldn't matter in the inner product (since, once I take out 2n+1, it must be 1)

That's a formula for the energy of the eigenstates. Is [tex]\psi(x,0)[/tex] an eigenstate? Again, what is [tex]n[/tex]?

Part b) is actually "b) At some time later T the wave function is [tex]\psi(x,T)=B\left(1+2\sqrt(\frac{m\omega}{\hbar})x\right)^{2}e^{\frac{-m\omega}{2\hbar}x^{2}}[/tex] for some constant B. What is the smallest possible value of T?"

Well time-dependence can be worked out using the time-evolution operator [tex]\hat{U}(t) = e^{-i\hat{H}t/\hbar}[/tex]. The action is simplest on energy eigenstates as I wrote before.
 

Suggested for: Another quantum oscillator

Replies
1
Views
514
Replies
5
Views
614
Replies
21
Views
991
Replies
2
Views
494
Replies
9
Views
208
Replies
6
Views
91
  • Last Post
Replies
4
Views
515
  • Last Post
Replies
9
Views
740
Replies
3
Views
177
Top