Wavefunctions describe the motion of particles such as electrons. However, electrons are usually confined by some manner of potential distributions. This is the case for atoms as well as arrays of atoms that form solids. In such cases, the wavefunction extends throughout the solid. Solving the Schroedinger’s equation analytically for such cases is incredibly complicated, even for periodically repeating lattices. One pattern that occurs however is a potential well, or a depression in potential which confines an electron. By coupling multiple wells, a periodic potential can be described that more closely resembles a periodic wavefunction in real crystals.
In an approximation, electrons in solids behave similarly to free electrons, but with an altered mass or effective mass. These electrons in solids have solutions which are described using the E-k relationship.
For a one-dimensional potential well, we can recognize three regions. Lets assume the well is 6nm in width and the height of the well is 1eV. We can find the general solutions of the potential well for each region.
General solutions to the potential well:
The potential well can have two solutions: an odd and even solution.
The energy levels associated with the wavefunctions of the potential well are then calculated.
The following matlab code further demonstrates an important concept, which is the number of allowed states in a potential well.