Negative Kinetic Energy Supersymmetric Quantum Mechanics

Abstract

Supersymmetric quantum mechanics(SUSYQM) has been proven to be an extremely effective tool for solving the Schrödinger equation. In recent years, the negative kinetic energy Schrödinger equation has been proposed based on the low-momentum approximation of the Klein-Gordon equation. This paper focuses on the negative kinetic energy SUSYQM. First, the authors briefly review the derivation of the negative kinetic energy Schrödinger equation based on the low-momentum approximation of the Klein-Gordon equation. Second, by solving the negative kinetic energy Schrödinger equation with the linear harmonic oscillator potential, the authors constructed the basic framework of SUSYQM for the negative kinetic energy system. Third, the authors derived the shape invariance, isospectral properties, and the eigen-energies and eigen-wave functions, thereby establishing the theory of the negative kinetic energy SUSYQM of the system. Fourth, the authors investigated the potential algebraic form of shape invariance of the negative kinetic energy SUSYQM. Fifth, the authors discussed the physical significance of the negative kinetic energy SUSYQM in depth. Sixth, as an example, the authors further demonstrated the application of the negative kinetic energy SUSYQM in solving the Schrödinger equation with the trigonometric Pöschl-Teller potential. Finally, the paper concludes with a summary.