operator . Then, and we see that we have a linear SUSY representation. Now let's introduce two "formal" quantities, ; and 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, of course. Then, Incidentally, there's also a U(1)R symmetry, with p and x and W having zero R-charges and having an R-charge of 1 and b having an R-charge of -1. Shape invariance Suppose is real for all real . Then we can simplify the expression for the Hamiltonian to There are certain classes of superpotentials such that both the bosonic and fermionic Hamiltonians have similar forms. Specifically where the 's are parameters. For example, the hydrogen atom potential with angular momentum can be written this way. This corresponds to for the superpotential This is the potential for angular momentum shifted by a constant. After solving the ground state, the supersymmetric operators can be used to construct the rest of the bound state spectrum. In general, since and are partner potentials, they share the same energy spectrum except the one extra ground energy. 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 are the parameters for the multiple partnered potentials. Applications In 2021, supersymmetric quantum mechanics was applied to option pricing and the analysis of markets in quantum finance, and to financial networks. See also Supersymmetry algebra Superalgebra Supersymmetric gauge theory References Sources F. Cooper, A. Khare and U. Sukhatme, "Supersymmetry and Quantum Mechanics", Phys.Rept.251:267-385, 1995. D.S. Kulshreshtha, J.Q. Liang and H.J.W. Muller-Kirsten, "Fluctuation equations about classical field configurations and supersymmetric quantum mechanics", Annals Phys. 225:191-211, 1993. G. Junker, "Supersymmetric Methods in Quantum and Statistical Physics", Springer-Verlag, Berlin, 1996 B. Mielnik and O. Rosas-Ortiz, "Factorization: Little or great algorithm?", J. Phys. A: Math. Gen. 37: 10007-10035, 2004 External links References from INSPIRE-HEP Quantum mechanics Supersymmetry