Solving the time dependent Schrödinger equation (here in 1D) with time independent potential: $$i\hbar\frac{\partial}{\partial t}\Psi(x,t)=\hat{H}\Psi(x,t),$$ using Ansatz: $$\Psi(x,t)=\psi(x)\phi(t),$$ and separation of variables, gives: $$\phi(t)=e^{-iE_nt/\hbar}$$ $\psi(x)$ are the eigenfunctions and $E_n$ the energy eigenvalues. With the superposition principle: $$\Psi(x,t)=\displaystyle\sum_n^{+\infty}c_n\psi_n(x)e^{-iE_nt/\hbar}$$ Assume that at $t=0$ the wave function was $\Psi_0$, then: $$\Psi(x,0)=\Psi_0=\displaystyle\sum_n^{+\infty}c_n\psi_n(x)$$ Assuming the system has domain $\Delta x$, we should then be able to determine the coefficients with Fourier: $$c_n=\frac{2}{\Delta x}\int_{\Delta x}\Psi_0\psi_n(x)dx$$
Is this true? If so, is it a 'practical' way of determining $c_n$?