Supersymmetric WKB approximation

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Supersymmetric WKB (SWKB) approximation is an extension of the standard WKB method that uses ideas from supersymmetric quantum mechanics to estimate energy levels. In SUSY quantum mechanics, many potentials V(x) can be written in terms of a superpotential W(x), so that the potential is built from W and its derivative. The SWKB approach rewrites the usual quantization condition of WKB using this superpotential.

For unbroken supersymmetry, to leading order in Planck’s constant ħ, the energy levels E_n are found from a one-dimensional integral between the classical turning points a and b. These turning points are the points where the motion would stop classically, roughly corresponding to where E_n equals W^2(x) (ignoring small ħ-dependent terms to locate them).

The supersymmetric method has several attractive features. First, the ground state energy is predicted exactly, which is an improvement over standard WKB at low energies. Second, for a class of potentials known as shape-invariant potentials, the entire energy spectrum is predicted exactly by this first-order condition.

In short, SWKB provides a simpler and often more accurate way to estimate energy levels in systems with supersymmetry.