Ranabir Dutt

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.

Mapping of shape invariant potentials under point canonical transformations

The authors give explicit point canonical transformations which map twelve types of shape invariant potentials (which are known to be exactly solvable) into two potential classes. The eigenfunctions in these two classes are given by hypergeometric and confluent hypergeometric functions respectively.

New class of conditionally exactly solvable potentials in quantum mechanics

Motivated by an idea of Dutra (1993), we obtain a new class of one-dimensional conditionally exactly solvable potentials for which the entire spectra can be obtained in an algebraic manner provided one of the potential parameters is assigned a fixed negative value. It is shown that using shape-invariant potentials as input, one may generate different classes of such potentials even in more than one dimension. We also illustrate that WKB and supersymmetry inspired WKB methods provide very good approximations for these potentials with the latter doing comparatively better.

Noncentral potentials and spherical harmonics using supersymmetry and shape invariance

It is shown that the operator methods of supersymmetric quantum mechanics and the concept of shape invariance can profitably be used to derive properties of spherical harmonics in a simple way. The same operator techniques can also be applied to several problems with noncentral vector and scalar potentials. As examples, we analyze the bound state spectra of an electron in a Coulomb plus an Aharonov–Bohm field and/or in the magnetic field of a Dirac monopole.

An improved calculation for screened Coulomb potentials in Rayleigh-Schrodinger perturbation theory

Earlier works on screened Coulomb potentials using Rayleigh-Schrodinger perturbation theory have been re-examined. Instead of working with the usual Hulthen potential as the unperturbed Hamiltonian, the authors propose that a scaled Hulthen potential with modified strength and screening coefficient represents the lowest-order approximation for the static-screened Coulomb and exponential cosine-screened Coulomb potentials. The scale parameter appearing in the new Hulthen potential is then determined from the notion of the virial theorem and intuitive physical arguments. It is found that the accuracy of the predicted energy eigenvalues for the bound s states improves significantly even when the screening parameter is large and quite close to its critical values for which the quantum state becomes just bound. In spite of the simplicity of the approach, the numerical results compare fairly well with those obtained from rigorous analytic approximation methods.

Supersymmetry‐inspired WKB approximation in quantum mechanics

The supersymmetry‐inspired WKB approximation (SWKB) in quantum mechanics is discussed in detail. The SWKB method can be easily applied to any potential whose ground‐state wave function is known. It yields eigenvalues that are exact for large quantum numbers n (as any WKB approximation should in the classical limit). Furthermore, for the important special case of ‘‘shape‐invariant’’ potentials, the SWKB approach gives the exact analytic expressions for the entire bound‐state spectra. A study of some nonshape‐invariant, but solvable, potentials suggests that shape invariance is not only sufficient but perhaps even necessary for the SWKB approximation to be exact. A comparison of the WKB and SWKB predictions for the bound‐state spectra of a number of potentials reveals that in many cases the SWKB approach does better than the usual WKB approximation.

Dipole polarizability of hydrogen atom at high pressures

Dipole polarizability of hydrogen atom at high pressures is investigated using the model of a trapped atom inside a spherical box with impenetrable surface. Both upper and lower bounds for the polarizability are obtained using accurate variational wavefunctions proposed recently. Buckingham polarizabilities calculated from the 1s state wavefunction are shown to be in good agreement with those calculated from the exact values of 〈rn〉 for cage radii less than 2.5.

Exact solutions for polynomial potentials using supersymmetry inspired factorization method

Abstract Supersymmetry inspired factorization method is used to obtain exact analytic solutions for one or a few quantum states of general polynomial potentials with or without Coulomb term under suitable constraints on the potential parameters. Illustrative examples and numerical implications of our approach are discussed.

Bound states of screened coulomb potentials

Abstract We propose an extension of the Ecker-Weizel approximation to treat the non-zero angular momentum bound states of a class of screened Coulomb potentials. As an illustration of our prescription, we have calculated the discrete energies E nl of the Yukawa potential, which are in excellent agreement with those of Rogers et al.

Algebraic shape invariant models

Motivated by the shape invariance condition in supersymmetric quantum mechanics, we develop an algebraic framework for shape invariant Hamiltonians with a general change of parameters. This approach involves nonlinear generalizations of Lie algebras. Our work extends prsvious results showing the equivalence of shape invariant potentials involving a translational change of parameters with a standard SO(2, 1) potential algebra for Natanzon type potentials.