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I want to calculate the function $$\varphi(x,0) = \frac15(3 \Psi_0 + 4 \Psi_1)$$ for a later point in time. I know the formula for the time evolution is $$\Psi(x,t)=\sum_{n=0}^N c_n \Psi_n e^{-i \frac{E_n}{\hbar} t}.$$

How do I calculate the evolution coefficients? In my case $c_n$ seems to be one but why?

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Set $t=0$ to get on the one hand $$ \Psi(x,0)=\sum_n c_n \Psi_n $$ and compare with $\varphi(x,0)= \frac{3}{5}\Psi_0 +\frac{4}{5}\Psi_1$. Alternatively, set $t=0$ and use orthogonality of the wavefunctions to find $$ c_n=\int dx\, \varphi(x,0)\,\Psi^*_n(x)\, . $$

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