Ab Initio Method for Obtaining Exactly Solvable Quantum Mechanical Potentials

The shape invariance condition is the integrability condition in supersymmetric quantum mechanics (SUSYQM). It is a difference-differential equation connecting the superpotential W and its derivative at two different values of parameters. We show that this difference equation is equivalent to a non-linear partial differential equation whose solutions are translational shape invariant superpotentials. In lieu of trial and error, this method provides the first ab initio technique for generating shape invariant superpotentials. Supersymmetric quantum mechanics (SUSYQM)[1] extends Dirac’s factorization method for the harmonic oscillator to a large number of other potentials, whose energy eigenvalues can then be obtained algebraically. The SUSYQM extension consists of introducing a superpotential W (x, a) that generates two partner Hamiltonians, both with the same energy eigenvalues. These partner Hamiltonians are given by H∓ = A ± A = ( ∓ h̄ d dx +W (x, a) )( ± h̄ d dx +W (x, a) )