How Does Time Evolution Affect Quantum Oscillator Wave Functions?

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The discussion centers on the time evolution of a quantum harmonic oscillator's wave functions. The initial wave function is given, and the expectation value of energy is calculated using ladder operators, yielding the result \(\frac{\hbar\omega}{2}(2n+1)\). However, confusion arises regarding the time-dependent wave function and the variable \(T\), as neither wave function shows explicit time dependence. Participants suggest using the time-evolution operator to understand how the wave function changes over time, emphasizing the need to clarify whether the initial state is an energy eigenstate. The conversation highlights the complexities of quantum mechanics and the importance of understanding time evolution in wave functions.
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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.
 
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arenaninja said:

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).
 
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.
 
arenaninja said:
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.
 
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