A. Lahiri

SUPERSYMMETRY IN QUANTUM MECHANICS

A pedagogical review on supersymmetry in quantum mechanis is presented which provides a comprehensive coverage of the subject. First, the key ingredients on the quantization of the systems with anticommuting variables are discussed. The supersymmetric Hamiltotian in quantum mechanics is then constructed by emphasizing the role of partner potentials and the superpotentials. We also make explicit the mathematical formulation of the Hamiltonian by considering in detail the N=1 and N=2 supersymmetric (quantum) mechanics. Supersymmetry is then discussed in the context of one-dimensional problems and the importance of the factorization method is highlighted. We treat in detail the technique of constructing a hierarchy of Hamiltonians employing the so-called ‘shape-invariance’ of potentials. To make transparent the relationship between supersymmetry and solvable potentials, we also solve several examples. We then go over to the formulation of supersymmetry in radial problems, paying a special attention to the Co...

Supersymmetry in atomic physics and the radial problem

A correct procedure for constructing supersymmetry in three dimensions is presented. The degeneracies are found between states of the same l but different n and Z and the previous results on the Coulomb and the three-dimensional isotropic oscillator problems are reestablished. The authors also consider the hydrogen-helium problem and find supersymmetry to hold to a good approximation.

Supersymmetry and the three-dimensional isotropic oscillator problem

By transforming the radial equation of the isotropic oscillator to the Coulomb form, the authors show that the energy degeneracy of the three-dimensional oscillator may be understood on account of supersymmetry.

Nonuniqueness of the factorization scheme in quantum mechanics

We enquire into the consequences of the nonuniqueness of the factorizability of a quantum mechanical Hamiltonian in one dimension. This leads to a hierarchy of potentials, a particular class of which is endowed with the energy spectrum of the harmonic oscillator.

A scenario of supersymmetry in quantum mechanics

SummaryWe construct a scenario of supersymmetry in quantum mechanics by imposing a structure on the raising and lowering operators. Possible consequences are worked out in one dimension which includes relating supersymmetry to the infinite deep well problem

Supersymmetry and the ladder operator technique in quantum mechanics: the radial Schrödinger equation

Using the ladder operator technique, a construction of the supersymmetric Hamiltonian is proposed. We show that the accidental degeneracies associated with the Coulomb and isotropic oscillator problems may be attributed to the existence of a supersymmetry of the Hamiltonians.