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.
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 process that begins by inserting the Coulomb potential into the Schrödinger equation.
Following use of multiple differential equations, the analysis produces a recursion relation for the Laguerre polynomials.
The 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 greater ease, in much the same way that operator methods are used to solve the harmonic oscillator.
[1] A similar supersymmetric approach can also be used to more accurately find the hydrogen spectrum using the Dirac equation.
[2] Oddly enough, this approach is analogous to the way Erwin Schrödinger first solved the hydrogen atom.
[3][4] He did not call his solution supersymmetric, as SUSY was thirty years in the future.
The SUSY solution of the hydrogen atom is only one example of the very general class of solutions which SUSY provides to shape-invariant potentials, a category which includes most potentials taught in introductory quantum mechanics courses.
This fact can be exploited to deduce many properties of the eigenstate spectrum.
It is analogous to the original description of SUSY, which referred to bosons and fermions.
Each boson would have a fermionic partner of equal energy—but, in the relativistic world, energy and mass are interchangeable, so we can just as easily say that the partner particles have equal mass.
SUSY concepts have provided useful extensions to the WKB approximation in the form of a modified version of the Bohr-Sommerfeld quantization condition.
In addition, SUSY has been applied to non-quantum statistical mechanics through the Fokker–Planck equation, showing that even if the original inspiration in high-energy particle physics turns out to be a blind alley, its investigation has brought about many useful benefits.
The Schrödinger equation for the harmonic oscillator takes the form where
would satisfy the equation Assuming that we know the ground state of the harmonic oscillator
as We then find that We can now see that This is a special case of shape invariance, discussed below.
Taking without proof the introductory theorem mentioned above, it is apparent that the spectrum of
(Here, we use "natural units" where the Planck constant is set equal to 1.)
A more intricate case is the algebra of angular momentum operators; these quantities are closely connected to the rotational symmetries of three-dimensional space.
To generalize this concept, we define an anticommutator, which relates operators the same way as an ordinary commutator, but with the opposite sign: If operators are related by anticommutators as well as commutators, we say they are part of a Lie superalgebra.
We shall call this system supersymmetric if the following anticommutation relation is valid for all
This is only in analogy to quantum field theory and should not be taken literally.
To avoid this problem, define the self-adjoint operator
with the latter being the adjoint of the former such that and both of them commute with bosonic operators but anticommute with fermionic ones.
Next, we define a construct called a superfield: f is self-adjoint.
For example, the hydrogen atom potential with angular momentum
We can continue this process of finding partner potentials with the shape invariance condition, giving the following formula for the energy levels in terms of the parameters of the potential where