Generalized numerical solution of the time-independent Schrödinger equation

We will try to numerically solve the unidimensional time-independent Schrödinger equation for constant potentials using Numerov’s method. The standard approach is to find the energy eigenvalues and infer the wave functions thereon. In contrast to that, we will try to find the generalized wave functions and then infer the energy eigenvalues. We start with a constant potential in infinite space, find the general solution then particularise the wave function to a potential well.