# Best way to solve Schrodinger's wave equation numerically.

 Sci Advisor PF Gold P: 1,354 Best way to solve Schrodinger's wave equation numerically. Since your potential is time independent, the fastest way to solve your problem is to first find the eigenfunctions of the Hamiltonian $$H \phi_i = E_i \phi_i$$ To do this, discretize space and write the Hamiltonian as a matrix. As you said, you can use a finite difference approximation for the momentum operator. Once you have the $\phi_i$, find the initial coefficients of your wave function in this basis, $$\psi(x,t=0) = \sum_i c_i \phi_i(x)$$ by calculating $$c_i = \int \phi_i^*(x) \psi(x,t=0) dx$$ Then, the wave function at any time time is simply given by $$\psi(x,t) = \sum_i c_i \phi_i(x) \exp(-i E_i t / \hbar)$$ By advancing $t$ and refreshing the plot, you will get your animation. I have no idea how to do this in Maple