“The Kronig-Penney model extended to arbitrary potentials via numerical matrix mechanics”

The American Journal of Physics just published a paper I wrote with my supervisor, Frank Marsiglio. The link to the gated paper is here, and an older version is available on the ArXiv.

Kronig-Penney model (Source: Wikimedia)

The Kronig-Penney model is a first attempt at extending quantum mechanics to solid systems with many particles. It is notable since it can be solved analytically, and the solution demonstrates some important salient features of electronic bandstructure, which encodes the electronic properties of various materials.

However, the model is fairly limited. It consists of a series of rectangular wells and barriers (or in a certain limit called the Dirac comb, the wells or barriers are delta functions), whereas more “realistic” potentials would have more interesting shapes.

What I did is applied a method that Frank invented for solving certain problems in quantum mechanics in a 2009 paper (ArXiv version) using what’s called matrix mechanics. I found a way to solve for a single “unit cell” of the potential using matrix mechanics and thereafter applied something called Bloch’s theorem to generate bandstructure for any one-dimensional repeating potential.

In summary, it’s a new tool for investigating a variety of problems in quantum mechanics that were previously extremely difficult (if not outright impossible) to solve given a standard undergraduate education in quantum mechanics and solid state physics.

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