Editing Superpotential (section)

==One-dimensional example==

Consider a [[one-dimensional]], non-relativistic particle with a two state internal degree of freedom called "[[Spin (physics)|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''<sup>†</sup> signify [[operator (physics)|operators]] which transform a "spin up" particle into a "spin down" particle and vice versa, respectively. Furthermore, take ''b'' and ''b''<sup>†</sup> to be normalized such that the [[anticommutator]] {''b'',''b''<sup>†</sup>} equals 1, and take that ''b''<sup>2</sup> equals 0. Let ''p'' represent the [[momentum]] of the particle and ''x'' represent its [[position vector|position]] with [''x'',''p'']=i, where we use [[natural units]] so that <math>\hbar=1</math>. Let ''W'' (the superpotential) represent an arbitrary [[differentiable function]] of ''x'' and define the supersymmetric operators ''Q''<sub>1</sub> and ''Q''<sub>2</sub> as

:<math>Q_1=\frac{1}{2}\left[(p-iW)b+(p+iW)b^\dagger\right]</math>
:<math>Q_2=\frac{i}{2}\left[(p-iW)b-(p+iW)b^\dagger\right]</math>

The operators ''Q''<sub>1</sub> and ''Q''<sub>2</sub> are self-adjoint. Let the [[Hamiltonian (quantum mechanics)|Hamiltonian]] be

:<math>H=\{Q_1,Q_1\}=\{Q_2,Q_2\}=\frac{p^2}{2}+\frac{W^2}{2}+\frac{W'}{2}(bb^\dagger-b^\dagger b)</math>

where ''W''' signifies the derivative of ''W''. Also note that {''Q''<sub>1</sub>,''Q''<sub>2</sub>}=0. Under these circumstances, the above system is a [[toy model]] of ''N''=2 supersymmetry. The spin down and spin up states are often referred to as the "[[boson]]ic" and "[[fermion]]ic" states, respectively, in an analogy to [[quantum field theory]]. With these definitions, ''Q''<sub>1</sub> and ''Q''<sub>2</sub> map "bosonic" states into "fermionic" states and vice versa. Restricting to the bosonic or fermionic sectors gives two [[Supersymmetric Quantum Mechanics|partner potentials]] determined by

:<math> H = \frac{p^2}{2}+\frac{W^2}{2} \pm \frac{W'}{2}</math>