Superpotential

In theoretical physics, the superpotential is a parameter in supersymmetric quantum mechanics.

Consider a one-dimensional, non-relativistic particle with a two state internal degree of freedom called "spin". (This is not quite the usual notion of spin encountered in nonrelativistic quantum mechanics, because "real" spin applies only to particles in three-dimensional space.) Let b and its Hermitian adjoint b^† signify operators which transform a "spin up" particle into a "spin down" particle and vice versa, respectively. Furthermore, take b and b^† to be normalized such that the anticommutator {b,b^†} equals 1, and take that b² equals 0. Let p represent the momentum of the particle and x represent its position with [x,p]=i, where we use natural units so that ${\displaystyle \hbar =1}$. Let W (the superpotential) represent an arbitrary differentiable function of x and define the supersymmetric operators Q₁ and Q₂ as

Note that Q₁ and Q₂ seem self-adjoint. Let the Hamiltonian be

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