Shape invariance of solvable Schrödinger equations with a generalized hyperbolic tangent superpotential

Abstract

Supersymmetric quantum mechanics provides a powerful method for solving the Schrödinger equation in quantum mechanics problems. In this paper, a hyperbolic tangent superpotential is generalized according to a new hyperbolic tangent superpotential. Shape invariances of partner potentials of the generalized superpotential with additive partner potential are derived based on supersymmetric quantum mechanics. Their coefficient-dependent eigenfunctions and eigenvalues have been calculated according to those potentials. It is found that only the ground state exists when one of the two parameters satisfies additivity. But if the two parameters meet additivity, the number of excited states has an upper limit, then the energy levels are finite. These potentials increase the number of exactly solvable potentials and could be used as good models in quantum physics, atomic physics, mathematical physics, etc. Moreover, the consistent eigenvalues are obtained through the potential algebra method.