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Full-Text Articles in Physics
Exactness Of Swkb For Shape Invariant Potentials, Asim Gangopadhyaya, Jeffry V. Mallow, Emeritus, Constantin Rasinariu, Jonathan Bougie
Exactness Of Swkb For Shape Invariant Potentials, Asim Gangopadhyaya, Jeffry V. Mallow, Emeritus, Constantin Rasinariu, Jonathan Bougie
Physics: Faculty Publications and Other Works
The supersymmetry-based semiclassical method (SWKB) is known to produce exact spectra for conventional shape invariant potentials. In this paper we prove that this exactness follows from their additive shape invariance.
Inter-Relations Between Additive Shape Invariant Superpotentials, Jeffry V. Mallow, Emeritus, Asim Gangopadhyaya, Jonathan Bougie, Constantin Rasinariu
Inter-Relations Between Additive Shape Invariant Superpotentials, Jeffry V. Mallow, Emeritus, Asim Gangopadhyaya, Jonathan Bougie, Constantin Rasinariu
Physics: Faculty Publications and Other Works
All known additive shape invariant superpotentials in nonrelativistic quantum mechanics belong to one of two categories: superpotentials that do not explicitly depend on ħ, and their ħ-dependent extensions. The former group themselves into two disjoint classes, depending on whether the corresponding Schrödinger equation can be reduced to a hypergeometric equation (type-I) or a confluent hypergeometric equation (type-II). All the superpotentials within each class are connected via point canonical transformations. Previous work [19] showed that type-I superpotentials produce type-II via limiting procedures. In this paper we develop a method to generate a type I superpotential from type II, thus …
Generation Of A Novel Exactly Solvable Potential, Jonathan Bougie, Asim Gangopadhyaya, Jeffry V. Mallow, Emeritus, Constantin Rasinariu
Generation Of A Novel Exactly Solvable Potential, Jonathan Bougie, Asim Gangopadhyaya, Jeffry V. Mallow, Emeritus, Constantin Rasinariu
Physics: Faculty Publications and Other Works
We report a new shape-invariant (SI) isospectral extension of the Morse potential. Previous investigations have shown that the list of “conventional” SI superpotentials that do not depend explicitly on Planck's constant ħ is complete. Additionally, a set of “extended” superpotentials has been identified, each containing a conventional superpotential as a kernel and additional ħ-dependent terms. We use the partial differential equations satisfied by all SI superpotentials to find a SI extension of Morse with novel properties. It has the same eigenenergies as Morse but different asymptotic limits, and does not conform to the standard generating structure for isospectral deformations.
Method For Generating Additive Shape Invariant Potentials From An Euler Equation, Jonathan Bougie, Asim Gangopadhyaya, Jeffrey Mallow
Method For Generating Additive Shape Invariant Potentials From An Euler Equation, Jonathan Bougie, Asim Gangopadhyaya, Jeffrey Mallow
Physics: Faculty Publications and Other Works
In the supersymmetric quantum mechanics formalism, the shape invariance condition provides a sufficient constraint to make a quantum mechanical problem solvable; i.e., we can determine its eigenvalues and eigenfunctions algebraically. Since shape invariance relates superpotentials and their derivatives at two different values of the parameter a, it is a non-local condition in the coordinate-parameter (x,a) space. We transform the shape invariance condition for additive shape invariant superpotentials into two local partial differential equations. One of these equations is equivalent to the one-dimensional Euler equation expressing momentum conservation for inviscid fluid flow. The second equation provides the constraint that helps us …
Methods For Generating Quasi-Exactly Solvable Potentials, Asim Gangopadhyaya, Avinash Khare, Uday P. Sukhatme
Methods For Generating Quasi-Exactly Solvable Potentials, Asim Gangopadhyaya, Avinash Khare, Uday P. Sukhatme
Physics: Faculty Publications and Other Works
We describe three different methods for generating quasi-exactly solvable potentials, for which a finite number of eigenstates are analytically known. The three methods are respectively based on (i) a polynomial ansatz for wave functions; (ii) point canonical transformations; (iii) supersymmetric quantum mechanics. The methods are rather general and give considerably richer results than those available in the current literature.
Quantum Mechanics Of Multi-Prong Potentials, Asim Gangopadhyaya, A Pagnamenta, Uday P. Sukhatme
Quantum Mechanics Of Multi-Prong Potentials, Asim Gangopadhyaya, A Pagnamenta, Uday P. Sukhatme
Physics: Faculty Publications and Other Works
We describe the bound state and scattering properties of a quantum mechanical particle in a scalar N-prong potential. Such a study is of special interest since these situations are intermediate between one and two dimensions. The energy levels for the special case of N identical prongs exhibit an alternating pattern of non-degeneracy and (N−1) fold degeneracy. It is shown that the techniques of supersymmetric quantum mechanics can be used to generate new solutions. Solutions for prongs of arbitrary lengths are developed. Discussions of tunneling in N-well potentials and of scattering for piecewise constant potentials are given. Since our treatment is …