Supersymmetric WKB solutions for pseudoharmonic and Mie-type potentials (original) (raw)
Supersymmetric WKB approach to scattering problems
Physics Letters A - PHYS LETT A, 1991
The supersymmetric WKB method is employed to examine scattering parameters in one and three dimensions. In the leading order approximation, we retrieve all exact results obtained previously by operator methods for various scattering amplitudes and their corresponding supersymmetric partners. It is further shown that the phase shifts for the spherically symmetric potentials can be computed with high accuracy without making necessary the Langer modification for the centrifugal term.
New WKB method in supersymmetry quantum mechanics
In this paper, we combine the perturbation method in supersymmetric quantum mechanics with the WKB method to restudy an angular equation coming from the wave equations for a Schwarzschild black hole with a straight string passing through it. This angular equation serves as a naive model for our investigation of the combination of supersymmetric quantum mechanics and the WKB method, and will provide valuable insight for our further study of the WKB approximation in real problems, like the one in spheroidal equations, etc.
Supersymmetric solutions of non-central potentials
Physics Letters A, 2000
Using the ideas of supersymmetry and shape invariance we show that the eigenvalues of a wide class of non-central potentials can be obtained algebraically in a simple and elegant manner. As an illustration we discuss the generalized coulomb and oscillator systems.
Supersymmetry in Quantum Mechanics
2001
An elementary introduction is given to the subject of Supersymmetry in Quantum Mechanics which can be understood and appreciated by any one who has taken a first course in quantum mechanics. We demonstrate with explicit examples that given a solvable problem in quantum mechanics with n bound states, one can construct new exactly solvable n Hamiltonians having n-1,n-2,...,0 bound states. The relationship between the eigenvalues, eigenfunctions and scattering matrix of the supersymmetric partner potentials is derived and a class of reflectionless potentials are explicitly constructed. We extend the operator method of solving the one-dimensional harmonic oscillator problem to a class of potentials called shape invariant potentials. It is worth emphasizing that this class includes almost all the solvable problems that are found in the standard text books on quantum mechanics. Further, we show that given any potential with at least one bound state, one can very easily construct one continuous parameter family of potentials having same eigenvalues and s-matrix. The supersymmetry inspired WKB approximation (SWKB) is also discussed and it is shown that unlike the usual WKB, the lowest order SWKB approximation is exact for the shape invariant potentials and further, this approximation is not only exact for large quantum numbers but by construction, it is also exact for the ground state. Finally, we also construct new exactly solvable periodic potentials by using the machinery of supersymmetric quantum mechanics.
Asian Journal of Physical and Chemical Sciences, 2020
In this paper, we applied the semi-classical quantization approximation method to solve the radial Schrödinger equation with a generalized Pseudoharmonic potential. The four turning points problem within the framework of the Wentzel-Kramers-Brillouin (WKB) method was transformed into two turning points and subsequently, the energy spectrum was obtained. Some special cases of the generalized Pseudoharmonic potential are presented. The WKB approximation approach reproduces the exact energy expression obtained with several analytical methods in the literature. The values of the energy levels for some selected diatomic molecules (N2, CO, NO, CH) obtained numerically are in excellent agreement with those from previous works in the literature
Journal of Physics A: Mathematical and General, 2006
The analytic solutions of the one-dimensional Schrödinger equation for the trigonometric Rosen-Morse potential reported in the literature rely upon the Jacobi polynomials with complex indices and complex arguments. We first draw attention to the fact that the complex Jacobi polynomials have non-trivial orthogonality properties which make them uncomfortable for physics applications.
Exact solution of Schrödinger equation for Pseudoharmonic potential
Journal of Mathematical Chemistry, 2008
Exact solution of Schrödinger equation for the pseudoharmonic potential is obtained for an arbitrary angular momentum. The energy eigenvalues and corresponding eigenfunctions are calculated by Nikiforov-Uvarov method. Wavefunctions are expressed in terms of Jacobi polynomials. The energy eigenvalues are calculated numerically for some values of ℓ and n with n ≤ 5 for some diatomic molecules.
Supersymmetric one-parameter strict isospectrality for the attractive potentials
Journal of Physics A: Mathematical and General, 1998
The Schrödinger equation with attractive δ potential has been previously studied in the supersymmetric quantum mechanical approach by a number of authors, but they all used only the particular superpotential solution. Here, we introduce a one-parameter family of strictly isospectral attractive δ function potentials, which is based on the general superpotential (general Riccati) solution, we study the problem in some detail and suggest possible applications.
Supersymmetric Quantum Mechanics
2010
Supersymmetric quantum mechanics (SUSY QM) is a powerful tool for generating new potentials with known spectra departing from an initial solvable one. In these lecture notes we will present some general formulae concerning SUSY QM of first and second order for one-dimensional arbitrary systems, and we will illustrate the method through the trigonometric Pöschl-Teller potentials. Some intrinsically related subjects, as the algebraic structure inherited by the new Hamiltonians and the corresponding coherent states will be analyzed. The technique will be as well implemented for periodic potentials, for which the corresponding spectrum is composed of allowed bands separated by energy gaps.
Supersymmetry and quantum mechanics
Physics Reports, 1995
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 ...
New shape-invariant potentials in supersymmetric quantum mechanics
Journal of Physics A: Mathematical and General, 1993
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, we 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.
Supersymmetric WKB and WKB s wave phase shifts for the Morse potential
Chemical Physics Letters, 1996
The supersymmetric WKB (SKWB) l = 0 phase shifts for the Morse potential up to O(h ° and h 2) are obtained analytically and compared numerically to the exact results and to the lowest-order (h °) WKB approximations with and without the Langer modification. The results show that the lowest-order (h °) SWKB phase shifts are of better overall quality than their WKB counterparts, while the expression up to O(h 2) is in excellent agreement with the exact phase shifts in the range of validity of the semiclassical methods.
Bulletin of the American Physical Society, 2010
In the context of some recent papers suggesting CT-symmetric QM in order to generalize PT-symmetric QM, in this paper we present an idea that there is quite compelling reasoning to argue in favour of supersymmetric extension of Klein-Gordon equation. Its numerical solutions in some simplest conditions are presented. Since the potential corresponding to this supersymmetric KGE is neither Coulomb, Yukawa, nor Hulthen potential [2a], then one can expect to observe a new type of matter, which may be called 'supersymmetric-meson'. Its presence may be expected in particular in the process of breaking of Coulomb barrier in low energy schemes. Further observation is of course recommended in order to refute or verify this proposition.
Eprint Arxiv Quant Ph 0703131, 2007
We present analytically the exact energy bound-states solutions of the Schrödinger equation in D-dimensions for a pseudoharmonic potential plus ring-shaped potential of the form V (r, θ) = D e r re − re r 2 + β cos 2 θ r 2 sin 2 θ by means of the conventional Nikiforov-Uvarov method. We also give a clear recipe of how to obtain an explicit solution to the radial and angular parts of the wave functions in terms of orthogonal polynomials. The total energy of the system is different from the pseudoharmonic potential because of the contribution of the angular part. The general results obtained in this work can be reduced to the standard forms given in the literature.