Superpotential Guide, Meaning , Facts, Information and Description

In SUSY:

Example

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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