Supersymmetric quantum mechanics

In theoretical physics, supersymmetric quantum mechanics is an area of research where supersymmetry are applied to the simpler setting of plain quantum mechanics, rather than quantum field theory. Supersymmetric quantum mechanics has found applications outside of high-energy physics, such as providing new methods to solve quantum mechanical problems, providing useful extensions to the WKB approximation, and statistical mechanics. Understanding the consequences of supersymmetry (SUSY) has proven mathematically daunting, and it has likewise been difficult to develop theories that could account for symmetry breaking, i.e., the lack of observed partner particles of equal mass. To make progress on these problems, physicists developed supersymmetric quantum mechanics, an application of the supersymmetry superalgebra to quantum mechanics as opposed to quantum field theory. It was hoped that studying SUSY's consequences in this simpler setting would lead to new understanding; remarkably, the effort created new areas of research in quantum mechanics itself. For example, students are typically taught to "solve" the hydrogen atom by a laborious process which begins by inserting the Coulomb potential into the Schrödinger equation. After a considerable amount of work using many differential equations, the analysis produces a recursion relation for the Laguerre polynomials. The final outcome is the spectrum of hydrogen-atom energy states (labeled by quantum numbers n and l). Using ideas drawn from SUSY, the final result can be derived with significantly greater ease, in much the same way that operator methods are used to solve the harmonic oscillator. A similar supersymmetric approach can also be used to more accurately find the hydrogen spectrum using the Dirac equation. Oddly enough, this approach is analogous to the way Erwin Schrödinger first solved the hydrogen atom. Of course, he did not call his solution supersymmetric, as SUSY was thirty years in the future.