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SuperpotentialSuperpotential is a concept from particle physics' supersymmetry. Example of superpotentiality Let's look at the example of a one dimensional nonrelativistic particle with a 2D (i.e. two state) internal degree of freedom called "spin" (it's not really spin because "real" spin is for 3D particles). Let b be an operator which transforms a "spin up" particle into a "spin down" particle and its adjoint b† transforming a spin down particle into a spin up particle normalized such that the anticommutator {b,b†}=1. And of course, b2=0. Let p be the momentum of the particle and x be its position with x,p=i (let's use natural units where hbar=1). Let W (the superpotential) be an arbitrary differentiable function of x and let the supersymmetric operators -
-
Note that Q1 and Q2 are self-adjoint. Let the Hamiltonian -
where W' is the derivative of W. Also note that {Q1,Q2}=0. This is nothing other than N=2 supersymmetry. Let's also call the spin down state "bosonic" and the spin up state "fermionic". This is only in analogy to quantum field theory and should not be taken literally. Then, Q1 and Q2 maps "bosonic" states into "fermionic" states and vice versa.
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