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Suppose $$\Psi (x, t=0)=Ae^{i\alpha _1}\psi _1(x)+Be^{i\alpha _2}\psi_2(x)+Ce^{i\alpha _3}\psi_3(x).$$ If $\psi _n$ are the energy eigenfunctions how would I derive $\Psi (x,t)$?

I am having trouble with the $\psi_n$ for which I don't know how to deal with when I use Fourier transform

$$\phi(k)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \left(Ae^{i\alpha _1}\psi _1(x)+Be^{i\alpha _2}\psi_2(x)+Ce^{i\alpha _3}\psi_3(x)\right)\, dx $$

Any hint would be appreciated.

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The time dependence is given by $$ \psi(x,t) = \psi_n(x) e^{-iE_nt/\hbar}$$ where $E_n$ is the energy of the system.

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I would put subscript $n$ on your $\psi$s to indicate it is only valid when $\psi$ is an energy eigenfunction. –  BMS Oct 3 '13 at 18:53
    
See Griffiths' QM. Chapter 2. –  Ome Mar 12 at 17:57
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