C Solutions to Schrödinger's Equation

  The time-dependent Schrödinger wave equation which describes the motion of a particle under the influence of a potential, reads
$$\begin{gather}\text{i}\hbar\frac{\partial\psi(t,x,y,z)}{\partial t}=H\psi(t,x,y,z) \end{gather}$$
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.
 $$\begin{gather} \text{i}\hbar\frac{\partial\psi(t,x,y,z)}{\partial t}= -\frac{\hbar^2}{2m^*}\nabla^2\psi(t,x,y,z) + V(x,y,z)\psi(t,x,y,z) \end{gather}$$
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
$$\begin{gather}% latex2html id marker 2864 \text{wave equation:\hspace{1em}} \nab... ...ation:\hspace{1em}} \nabla^2\psi= \frac{\partial\psi}{\partial t}. \end{gather}$$
Thus indeed the Schrödinger equation (C.2) has the form of a diffusion equation. With the separation Ansatz
$$\begin{gather}\psi(t,x,y,z)=e^{\frac{\text{i}}{\hbar}Et}\psi_E(x,y,z) \end{gather}$$
one derives the time-invariant one-dimensional Schrödinger equation
$$\begin{gather}\frac{\hbar^2}{2m^*}\frac{d^2\psi_E(x)}{dx^2}-V(x)\psi_E(x) = E\psi_E(x). \end{gather}$$
For an easier notation in the following examples, $\psi$ stands for $\psi_E(x)$.