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## Homework Statement

A harmonic oscillator is initially in the state [itex]\psi (x,0)=Ae^{-\frac{\alpha ^2 x^2}{2}} \alpha x (2\alpha x +i)[/itex]. Where [itex]\alpha ^2 =\frac{m \omega}{\hbar}[/itex].

1)Find the wavefunction for all t>0.

2)Calculate the probability to measure the values [itex]\frac{5\hbar \omega }{2}[/itex] and [itex]\frac{7\omega \hbar }{2}[/itex] for the energy.

3)Calculate the mean value of the energy of this oscillator.

## Homework Equations

[itex]\Psi (x,t)= \psi (x)e^{-\frac{iEt}{\hbar}}[/itex].

[itex]E=(n+1)\frac{\hbar }{2}[/itex].

Probability to measure [itex]E_n[/itex] associated to the wavefunction [itex]\psi _n[/itex] is [itex]|c_n|^2[/itex] in the expression [itex]\Psi (x,t)= \sum _i ^{\infty } c_i \psi _i (x,t)[/itex]. I'm only using memory for all of this, so I might be wrong for some things.

## The Attempt at a Solution

Now that I think about it... can I consider that the particle is in a stationary state? I.e. that [itex]\psi (x,0)=\psi (x,t)[/itex]? Hmm I think not, that would be too easy to answer to 1).

Now even more confused. Stationary state would imply that [itex]\Psi (x,t)=\Psi(x)[/itex] and so the equation [itex]\Psi (x,t)= \psi (x)e^{-\frac{iEt}{\hbar}}[/itex] looks wrong, which is I think impossible.

I hope someone can shred some light on my understanding of intro to QM.

Thank you very much.