Supersymmetry of Generalized Linear Schrödinger Equations in (1+1) Dimensions

Abstract

We review recent results on how to extend the supersymmetry SUSY formalism in Quantum Mechanics to linear generalizations of the time-dependent Schrödinger equation in (1+1) dimensions. The class of equations we consider contains many known cases, such as the Schrödinger equation for position-dependent mass. By evaluating intertwining relations, we obtain explicit formulas for the interrelations between the supersymmetric partner potentials and their corresponding solutions. We review reality conditions for the partner potentials and show how our SUSY formalism can be extended to the Fokker-Planck and the nonhomogeneous Burgers equation.

Quantum surfing

“…We recommend time-dependent supersymmetry as an excellent starting place for British undergraduate final year projects. This procedure of deriving a new Hamiltonian and eigenfunctions for the Schrödinger equation requires detailed study of very mathematical research papers [3,5] which deepen and reinforce understanding of some of the formal aspects of quantum theory. This project involved the use of Matlab and Maple or Mathematica and so students gain experience in applying mathematical software and finally calculation and exploration of observables requires students to perform some original mathematics and to think creatively about its interpretation.…”

Bouncing in an oscillatory gravitational field

“…Non-stationary supersymmetric quantum theory has been derived [15][16][17] and extended [18] and provides a strategy for finding new solutions of the time-dependent Schrödinger equation if we know one solution. This work is a natural extension of the time independent supersymmetric methods discussed in very readable form by Cooper et al [19].…”

Optical Supersymmetry in the Time Domain

“…Nevertheless, to our knowledge, the SUSYQM formalism has never been applied in the time domain, whether in QM, optics, or any other field (SUSY quantum field theory is a multidimensional spacetime theory, but the formalism is considerably different and more complex than that of SUSYQM). This is probably due to the fact that the vast majority of 1D SUSY work has been developed within the realm of QM, and the time derivative in Schrödinger's equation is of first order, preventing a similar decomposition to that of equation (1) in the time domain (timedependent potentials have been considered in SUSYQM, but also using SUSY operators based on first-order spatial derivatives [14,15], making it impossible to exploit the potential of the standard spatial SUSY (S-SUSY) factorization in the time domain. In fact, none of the results we will derive here could be obtained with such operators).…”