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In a question in the situation of a harmonic oscillator with the ladder-operators $a$ and $a^\dagger$, I'm asked to find the full wave function at any $t$, with this inital wave function:

$\Psi(x,0) = \frac{1}{\sqrt2}(\psi_0 (x) + \psi_2 (x))$

A given hint is to show that $c_n (t) = 0$ when $n \neq 0,2$

I know $c_n (t) = c_n(0) \,e^{-iE_nt/\hbar}$, so I guess $c_0(0) = \frac{1}{\sqrt2}$ and the same for $c_2$. I don't know where to from there.

Oh and the energy is given by $E_n = (n+1)\hbar\omega$

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closed as off-topic by ACuriousMind, Wolpertinger, user36790, Jon Custer, John Rennie Oct 1 '16 at 18:38

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Thank you for the answer. I am not yet familiar with the notation used in the first part of your answer. I figured out the reply for this exercise on my own though, inspired by the second part of your answer. As $c_0=c_2=1/\sqrt2$,

$|c_0|^2 + |c_2|^2 = 1$

Meaning $E_0$ and $E_2$ are the only possible energy states. Using these values for $n$ and $c_0(0) = 1/\sqrt2$, you can figure out the full wave function using $\psi_n (x) = A_n(a_+)^n\psi_0(x)$, which is valid for the harmonic oscillator.

Thanks for the help [edit: refers to previous answer that was deleted], I hope my answer is clear enough for others to follow.

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