# Coefficients $c_n$ in superposition of wave functions?

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 $c_n$?

• Except for the issue with complex conjugation, this is really the only way for an arbitrary wavefunction. Nov 16, 2016 at 21:42

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, $c_m=\int_{-\infty}^{+\infty}\psi_m^*\Psi_0 dx$.