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I am given that a wave function at time 0 is given as $$\psi(x,t=0)=\sin^6(\frac{\pi x}{2L})\cos(\frac{\pi x}{2L})$$ I am asked to find this wave function as a function of time. In order to do this, I feel like I need to find the combinations of eigenfunctions that make this function and then I can find how they each evolvewith time, so that $$\psi(x,t)=\sum_{n=1}^{\infty}c_n\psi_n(x)e^{-iE_nt/\hbar}$$

Am I correct in saying this? I feel that this makes sense, and then I can find $c_n$ as $$c_n=\int_0^{2L}\psi(x,t=0)\sin(\frac{n\pi x}{2L})dx$$

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That's all fine except, that you need properly normalized eigenfunctions to calculated the $c_n$.

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Yes,you are correct in the approach.However,the $c_n$ must be appropriately normalized.

$$c_n=\frac{1}{L}\int_0^{2L}\psi(x,t=0)\sin(\frac{n\pi x}{2L})dx$$

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