SUPERSYMMETRY AND QUANTUM-MECHANICS

In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In parti cular, there is now a much deeper understanding of why certain potenti als are analytically solvable and an array of powerful new approximati on methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulation of supersymmetric q uantum mechanics and discuss many applications. Exactly solvable poten tials can be understood in terms of a few basic ideas which include su persymmetric partner potentials, shape invariance and operator transfo rmations. Familiar solvable potentials all have the property of shape invariance. We describe new exactly solvable shape invariant potential s which include the recently discovered self-similar potentials as a s pecial case. The connection between inverse scattering, isospectral po tentials and supersymmetric quantum mechanics is discussed and multiso liton solutions of the KdV equation are constructed. Approximation met hods are also discussed within the framework of supersymmetric quantum mechanics and in particular it is shown that a supersymmetry inspired WKB approximation is exact for a class of shape invariant potentials. Supersymmetry ideas give particularly nice results for the tunneling rate in a double well potential and for improving large N expansions. We also discuss the problem of a charged Dirac particle in an external magnetic field and other potentials in terms of supersymmetric quantu m mechanics. Finally, we discuss structures more general than supersym metric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a para-fermion of ord er p.