Finite difference approximation of quantum mechanical wave packets

We consider the numerical approximation of quantum mechanical wave packets by a finite difference method for the Schrödinger equation. We discuss known results for the solutions of the equations for N coupled harmonic oscillators and separation of variables solutions of finite difference equations for the heat and wave equations. We find separation of variables solutions of the Schrödinger finite difference equations with the same spatial part as the above solutions. We compare approximating a stationary state of the Schrödinger equation via four approaches: full finite difference equations for the Schrödinger and wave equations, and individual separation of variables solutions for each set of difference equations. We approximate the Fourier integral representation of a Gaussian wave packet by a Riemann sum, thus writing the wave packet as a finite superposition of plane waves. We then approximate the wave packet using the Schrödinger and wave finite difference equations.