We are interested in the motion of a particle in a potential landscape. Thus the Hamilton operator is replaced with the Hamilton operator for the free electron plus the potential function describing the potential landscape.

It should be noted that the time-dependent Schrödinger wave equation is, in fact, a `diffusion' not a `wave' equation, the time derivative, being of the first and not of the second order. A wave equation and a diffusion equation have the form

Thus indeed the Schrödinger equation (C.2) has the form of a diffusion equation. With the separation Ansatz

one derives the time-invariant one-dimensional Schrödinger equation

For an easier notation in the following examples, stands for .