Uday P. Sukhatme

Supersymmetry and quantum mechanics

In the past ten years, the ideas of supersymmetry have been profitably applied to many nonrelativistic quantum mechanical problems. In particular, there is now a much deeper understanding of why certain potentials are analytically solvable and an array of powerful new approximation methods for handling potentials which are not exactly solvable. In this report, we review the theoretical formulation of supersymmetric quantum mechanics and discuss many applications. Exactly solvable potentials can be understood in terms of a few basic ideas which include supersymmetric partner potentials, shape invariance and operator transformations. Familiar solvable potentials all have the property of shape invariance. We describe new exactly solvable shape invariant potentials which include the recently discovered self-similar potentials as a special case. The connection between inverse scattering, isospectral potentials and supersymmetric quantum mechanics is discussed and multisoliton solutions of the KdV equation are constructed. Approximation methods 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 quantum mechanics. Finally, we discuss structures more general than supersymmetric quantum mechanics such as parasupersymmetric quantum mechanics in which there is a symmetry between a boson and a para-fermion of order p.

Dual parton model

Soft multiparticle production is a dominant feature of most events in high energy hadronic collisions. Since soft processes have no large momentum transfer, perturbative QCD expansions in the strong coupling constant are not applicable. However, suitable large N expansions of QCD provide a topological classification of diagrams and a potentially useful non-perturbative approach. This topological expansion, when supplemented with generally accepted theoretical principles like duality, unitarity, Regge behavior and the parton structure of hadrons, provides the basis underlying the dual parton model (DPM). This model has been extensively studied and gradually extended over the past twelve years. It has been shown that DPM provides a complete, phenomenological description of all facets of soft processes. This is a non-trivial achievement in view of the large amount of soft multiparticle data available from both hadronic as well as nuclear beams and targets. Here, we describe the basic ideas of the model and review the main results coming from DPM.

Supersymmetry, shape invariance, and exactly solvable potentials

It is well known that the harmonic oscillator potential can be solved by using raising and lowering operators. This operator method can be generalized with the help of supersymmetry and the concept of ‘‘shape‐invariant’’ potentials. This generalization allows one to calculate the energy eigenvalues and eigenfunctions of essentially all known exactly solvable potentials in a simple and elegant manner.

Explicit wavefunctions for shape-invariant potentials by operator techniques

The authors obtain explicit expressions for the wavefunctions of all the known shape-invariant potentials by using the operator method.

Exactness of Supersymmetric {WKB} Spectra for Shape Invariant Potentials

Abstract Potentials which have the property of “shape-invariance” are known to be exactly solvable. For all these potentials, it is proved that the supersymmetric WKB (SWKB) quantization condition (to leading order in ħ) also has the nice property of reproducing the exact bound-state spectra. These results are not modified even if the next higher-order correction in ħ is taken into account.

Scattering Amplitudes for Supersymmetric Shape Invariant Potentials by Operator Methods

The scattering amplitudes for all currently known supersymmetric shape-invariant potentials are calculated by analytically continuing the explicit wavefunctions obtained via supersymmetric operator techniques. The procedure is illustrated in detail for the superpotential W(x)=A tanh alpha x+B sech alpha x, for which the S matrix has not been previously calculated.

New solvable and quasiexactly solvable periodic potentials

Using the formalism of supersymmetric quantum mechanics, we obtain a large number of new analytically solvable one-dimensional periodic potentials and study their properties. More specifically, the supersymmetric partners of the Lame potentials ma(a+1)sn2(x,m) are computed for integer values a=1,2,3,… . For all cases (except a=1), we show that the partner potential is distinctly different from the original Lame potential, even though they both have the same energy band structure. We also derive and discuss the energy band edges of the associated Lame potentials pm sn2(x,m)+qm cn2(x,m)/dn2(x,m), which constitute a much richer class of periodic problems. Computation of their supersymmetric partners yields many additional new solvable and quasiexactly solvable periodic potentials.

ACCURACY OF SEMICLASSICAL METHODS FOR SHAPE-INVARIANT POTENTIALS

We study the accuracy of several alternative semiclassical methods by computing analytically the energy levels for many large classes of exactly solvable shape-invariant potentials. For these potentials, the ground-state energies computed via the WKB method typically deviate from the exact results by about 10{percent}, a recently suggested modification, using nonintegral Maslov indices, is substantially better, and the supersymmetric WKB quantization method gives exact answers for all energy levels. {copyright} {ital 1997} {ital The American Physical Society}

New shape-invariant potentials in supersymmetric quantum mechanics

Quantum mechanical potentials satisfying the property of shape invariance are well known to be algebraically solvable. Using a scaling ansatz for the change of parameters, the authors obtain a large class of new shape-invariant potentials which are reflectionless and possess an infinite number of bound states. They can be viewed as q-deformations of the single soliton solution corresponding to the Rosen-Morse potential. Explicit expressions for energy eigenvalues, eigenfunctions and transmission coefficients are given. Included in the potentials as a special case is the self-similar potential recently discussed by Shabat (1992) and Spiridonov (1992).

Bound states in the continuum from supersymmetric quantum mechanics

Starting from a potential with a continuum of energy eigenstates, we show how the methods of supersymmetric quantum mechanics can be used to generate families of potentials with bound states in the continuum (BICs). We also find the corresponding wave functions. Our method preserves the spectrum of the original potential except it adds these discrete BICs at selected energies. Specifically, we compute and graph potentials which have bound states in the continuum starting from a null potential representing a free particle and from both the attractive and the repulsive Coulomb potentials