How Does Time Evolution Affect Quantum Oscillator Wave Functions?

[arenaninja]

Homework Statement

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

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 $$\frac{1}{5}\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}$$, but apparently it wasn't needed.
For part a I got $$\frac{\hbar\omega}{2}\left(2n+1\right)$$, 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.

 

[fzero]

Homework Statement

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

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

There's no $$n$$ in $$\psi(x,0)$$, 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

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

You can use this formula to determine $$\psi(x,t)$$.

 

[arenaninja]

Well since I was using ladder operators I was eventually left with $$<\Psi_{n}|H\Psi_{n}>=const*<\Psi_{n}|(2n+1)\Psi_{n}>$$, 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 $$\psi(x,T)=B\left(1+2\sqrt(\frac{m\omega}{\hbar})x\right)^{2}e^{\frac{-m\omega}{2\hbar}x^{2}}$$ 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.

 

[fzero]

Well since I was using ladder operators I was eventually left with $$<\Psi_{n}|H\Psi_{n}>=const*<\Psi_{n}|(2n+1)\Psi_{n}>$$, 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 $$\psi(x,0)$$ an eigenstate? Again, what is $$n$$?

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

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