Coefficients $c_n$ in superposition of wave functions?

Question

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 $c_n$?

Answer 1

I would say this is not always true (the eigenfunctions are not necessarily real). You should use the condition of orthonormality for the eigenfunctions (if it holds, otherwise use the relevant coefficients). The orthonormality conditions contain complex conjugates of the eigenfunctions.

EDIT: (11/16/2016) You have an orthonormality condition for the eigenfunctions: $<\psi_m|\psi_n>=\delta^m_n=\int^{+\infty}_{-\infty}\psi_m^*\psi_n dx$. Therefore, if $\Psi_0=\sum_n c_n\psi_n$, then $<\psi_m|\Psi_0>=\int_{-\infty}^{+\infty}\psi_m^*\Psi_0 dx=\int^{+\infty}_{-\infty}\psi_m^*\sum_n c_n\psi_n dx=\sum_n c_n\delta^m_n=c_m$. Therefore, $c_m=\int_{-\infty}^{+\infty}\psi_m^*\Psi_0 dx$.