Potential Wells and Bound Electrons

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.

Wavefunctions and Uncertainty Principle

The wave or state functions describes the motion and properties of a particle, such as an electron or photon. The magnitude squared of the wave-function is the probability density of finding a particle in a volume.

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The Uncertainty Principle defines the limits of accuracy as a relationship between position and momentum.

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The expected value of an observation, such as the presence of a particle in a volume is calculated via an operator.

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The Schroedinger equation describes the motion of particles according to quantum mechanics.

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The Schroedinger equation can be separated into the time dependent and and spatially dependent forms. The time-independent, spatially dependent form solution is as follows.

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