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We consider the following wave function: $$\psi(x) = \frac{1}{\sqrt{2}}(\psi_2(x) + \psi_3(x))$$

where: $\psi_n(x) = \sqrt{\frac{2}{a}}sin\left(\frac{n\pi x}{a}\right)$ are the eigenfunctions.

How do we make $\psi(x)$ into a Time dependent wave function, so how do we find $\Psi(x,t) ? $

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Assuming what you currently have as $\psi(x)$ is the initial condition, i.e. $\Psi(x,0)$, then what you need to do is to just find the eigenvalues of $\psi_n(x)$, which for this case is just the eigenvalues of $\psi_2(x)$ and $\psi_3(x)$. The eigenvalues are the energies of the system. So once you got the energies, the final solution would be of the form: $$\Psi(x,t) = \frac{1}{\sqrt{2}}(\psi_2(x)e^{-iE_2t/\hbar} + \psi_3(x)e^{-iE_3t/\hbar})$$

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  • $\begingroup$ Yes $t = 0$ is my initial condition, but what should I do if it wasn't? $\endgroup$ – Leroy Jan 17 '18 at 2:16
  • $\begingroup$ I'm sorry. I'm not sure if I understand what you are asking. Since you are looking for how $\psi(x)$ progresses in time, you can treat it as your initial condition for this case or any other combinations of $\psi_n$. Can you explain a little bit what you meant by if it wasn't? $\endgroup$ – Kane Billiot Jan 17 '18 at 2:22

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