 Supersymmetric quantum mechanics

Supersymmetric quantum mechanics organizes two sectors through an operator $A$ and its adjoint operator, with partner Hamiltonians $H_{-}=A^{†}A/2$ and $H_{+}=A{A}^{†}/2$. An off-diagonal supercharge maps between the sectors, and its anticommutator with its adjoint gives the block Hamiltonian. This makes both partner energies nonnegative.

For a one-dimensional example, $A={\partial }_x+w(x)$ in suitable dimensionless units gives partner potentials proportional to $w^2-w^{\prime }$ and $w^2+w^{\prime }$. A positive-energy state of one partner maps to a state of the other with the same energy. Zero modes need not pair, because $A$ or $A^{†}$ can annihilate them; supersymmetric factorization and zero-mode normalizability determines which candidates are actual states.

The method relates apparently different potential problems and isolates the role of boundary conditions. In a geometric realization, differential forms provide the graded Hilbert space and the exterior derivative and its adjoint play the role of supercharges.