Supersymmetric quantum mechanics and its applications

The Hamiltonian in Supersymmetric Quantum Mechanics is defined in terms of charges that obey the same algebra as that of the generators of supersymmetry in field theory. The consequences of this symmetry for the spectra of the component parts that constitute the supersymmetric system are explored. The implications of supersymmetry for the solutions of the Schrödinger equation, the Dirac equation, the inverse scattering theory and the multi-soliton solutions of the KdV equation are examined. Applications to scattering problems in Nuclear Physics with specific reference to singular potentials which arise from considerations of supersymmetry will be discussed. 1. SUPERSYMMETRIC QUANTUM MECHANICS OF ONE-DIMENSIONAL SYSTEMS It is shown that every one-dimensional quantum mechanical Hamiltonian H can have a partner H̃ such that H and H̃ taken together may be viewed as the components of a supersymmetric Hamiltonian H. The term ‘supersymmetric Hamiltonian’ is taken to mean a Hamiltonian defined in terms of charges that obey the same algebra as that of the generators of supersymmetry in field theory. The consequences of this symmetry for the spectra of H and H̃ are explored. It is shown how the supersymmetric pairing may be used to eliminate the ground state of H, or add a state below the ground state of H or maintain the spectrum of H. It is also explicitly demonstrated that the supersymmetric pairing may be used to generate a class of anharmonic potentials with exactly specified spectra.