Supersymmetric WKB Research Articles

Non-relativistic energy spectra of and diatomic molecules confined in a modified Scarf potential via supersymmetric WKB approach

In this research, the energy equation of the Schrodinger equation with the modified Scarf potential has been obtained using the supersymmetric WKB (SWKB) approach. The Pekeris approximation has been applied to enable the analytical solution. The energy equation was applied to determine the rovibrational states of two diatomic molecules such as O 2 + ( X 2 Π g ) and N 2 ( X 1 Σ g + ) . The average kinetic and potential energies were computed via the Hellman-Feynman theorem. The analytical result of the energy equation obtained coincides with the work reported in earlier literature where the authors had applied a different approach. The energy eigenvalues for the studied molecules were found to agree with the ones obtained using the Teitz-Hua molecular potential, the deformed modified Rosen-Morse (DMRM) potential, the Morse potential and the Rydberg-Klein-Rees data points. For the energy spectra, we found average absolute deviations of 0.129515% for O 2 + ( X 2 Π g ) and 1.834993% for N 2 ( X 1 Σ g + ) molecules respectively. The energy spectra increase as the quantum numbers increase up to the dissociation limit before decreasing. For the molecules, the atoms were found to possess an oscillatory-type motion where an increase in the mean kinetic energy resulted in a decrease in mean potential energy and vice-visa.

Non-relativistic bound state solutions with α-deformed Kratzer-type potential using the super-symmetric WKB method: application to theoretic-information measures

Non-relativistic bound state solutions with α-deformed Kratzer-type potential using the super-symmetric WKB method: application to theoretic-information measures

Improved energy spectra of the Klein–Gordon and Schrödinger equations under the Tietz potential by WKB and super-symmetric WKB methods

The relative convergence of the WKB and the Super-symmetric WKB (SWKB) approximations are studied with the hyper-radial Klein–Gordon and Schrodinger equations under the molecular Tietz potential. We used a Pekeris approximation to deal with the orbital centrifugal barrier energy. The variations of the energy levels with the potential parameters are investigated. The results revealed that the SWKB energy modeled for a hypothetical system is exact and nearly indistinguishable from the usual WKB energy. Both the WKB and SWKB approach give better results of the energy levels compared to the results obtained by other analytical methods such as the parametric Nikiforov–Uvarov (NU) and the improved asymptotic iteration (AIM) used with the Greene–Aldrich approximation for the centrifugal energy. For the s-wave case , the SKWB energy coincides with the NU and AIM obtained values whereas the WKB energy fluctuates slightly due to the effect of regularization of the orbital centrifugal barrier. Furthermore, the obtained energy eigenvalues for CO and LiH diatomic molecules are in excellent agreement with the results in the existing literature.

Numerical study of the SWKB condition of novel classes of exactly solvable systems

The supersymmetric WKB (SWKB) condition is supposed to be exact for all known exactly solvable quantum mechanical systems with the shape invariance. Recently, it was claimed that the SWKB condition was not exact for the extended radial oscillator, whose eigenfunctions consisted of the exceptional orthogonal polynomial, even the system possesses the shape invariance. In this paper, we examine the SWKB condition for the two novel classes of exactly solvable systems: one has the multi-indexed Laguerre and Jacobi polynomials as the main parts of the eigenfunctions, and the other has the Krein–Adler Hermite, Laguerre and Jacobi polynomials. For all of them, one can always remove the [Formula: see text]-dependency from the condition, and it is satisfied with a certain degree of accuracy.

The supersymmetric WKB formalism is not exact for all additive shape invariant potentials

Following the verification of the conjecture made by Comtet et al that the supersymmetry-inspired semiclassical method known as SWKB is exact for the conventional additive shape invariant potentials, it was widely believed that SWKB yields exact results for all additive shape invariant potentials. In this paper we present a concrete example of an additive shape invariant potential for which the SWKB method fails to produce exact results.

Supersymmetry and SWKB Approach to Dirac Equation with Hyperbolic Scarf Potential

By using the supersymmetric quantum mechanics and shape invariance concept, we study the Dirac equation with the hyperbolic Scarf potential and the exact energy spectrum is obtained. Also, we calculate the bound state energy eigenvalues by using the supersymmetric WKB approximation approach so that we get the same results.

Molecular spinless energies of the improved Rosen-Morse potential energy model in D dimensions

Under the circumstance of equal scalar and vector potentials, we solve the Klein-Gordon equation with the improved Rosen-Morse potential energy model in D spatial dimensions. The relativistic bound state energy equation has been obtained by using the supersymmetric WKB approximation approach. For fixed vibrational quantum number and various rotational quantum numbers, the relativistic energies for the 33Σg+ state of the Cs2 molecule and the 51Δg state of the Na2 molecule increase as D increases. We observe that the behavior of the relativistic vibrational energies in higher dimensions remains similar to that of the three-dimensional system.

Molecular spinless energies of the modified Rosen–Morse potential energy model in higher spatial dimensions

We solve the Klein–Gordon equation with the modified Rosen–Morse potential energy model in D spatial dimensions. The bound state energy equation has been obtained by using the supersymmetric WKB approximation approach. We find that the inter-dimensional degeneracy symmetry exists for the molecular system represented by the modified Rosen–Morse potential. For fixed vibrational and rotational quantum numbers, the relativistic energies for the 61Πu state of the 7Li2 molecule and the X3Π state of the SiC radical increase as D increases. We observe that the behavior of the relativistic vibrational energies in higher dimensions remains similar to that of the three-dimensional system.

Bound state solutions of the Klein–Gordon equation with the improved expression of the Manning–Rosen potential energy model

Abstract By employing the improved Greene–Aldrich approximation scheme to deal with the centrifugal term, we solve approximately the Klein–Gordon equation with the improved expression of the Manning–Rosen empirical potential energy model. The bound state energy equation and the unnormalized radial wave functions have been approximately obtained by using the supersymmetric WKB approach and the function analysis method. The relativistic vibrational transition frequencies for the a 3 Σ u + state of 7Li2 molecule have been computed by using the Manning–Rosen potential model. The relativistic vibrational transition frequencies are in good agreement with the observed data.

Application of supersymmetric WKB method to cyclic shape invariant potentials

We examine the accuracy and the validity of the lowest order supersymmetric WKB (SWKB) formula for cyclic shape invariant potentials (CSIPs). For period-2 CSIPs, we show analytically that the SWKB formula can yield exact eigenenergies for either all the even states or all the odd states. Such alternate exactness of the SWKB formula is due to the fact that period-2 CSIPs can also be considered as a kind of translational shape invariant potential. For CSIPs with periods greater than 2, we note that the accuracy of the SWKB formula also demonstrates similar alternating patterns. However, as a consequence of the rapid oscillations in the potential at large distances, the SWKB quantization formula fails to produce highly accurate results even in the high-energy limit.

Analytical approximation to the solution of the Dirac equation with the Eckart potential including the spin–orbit coupling term

By using the supersymmetric WKB approximation approach and the functional analysis method, we solve approximately the Dirac equation with the Eckart potential for the arbitrary spin–orbit quantum number κ. The bound state energy eigenvalues and the associated two-component spinors of the Dirac particles are obtained approximately.

Bounded Solutions of the Dirac Equation with a PT-symmetric Kink-Like Vector Potential in Two-Dimensional Space-Time

The (1+1)-dimensional Dirac equation with a PT-symmetric kink-like vector potential is investigated. By using the basic concepts of the supersymmetric WKB formalism and the function analysis method, we solve exactly the Dirac equation and obtain the bound-state energy levels and two-component spinor components. The PT-symmetric kink-like potential is not Hermitian and absent of bound states in the context of non-relativistic Schrodinger equation, but it possesses two sets of real discrete relativistic energy spectra in the context of the Dirac theory. When the PT symmetry is spontaneously broken, two sets of real energy spectra come into complex conjugate.

Generating of Conditionally Exactly Soluble Potentialsby Method of Supersymmetric Quantum Mechanics

This paper suggests a systematic method based on supersymmetric quantummechanics for generating conditionally exactly soluble potentials, and usesthe variational supersymmetric WKB method to obtain the approximate values ofthe energy spectrum of the whole class.

Supersymmetric WKB Approximation of Anharmonic PotentialV(r) = ar2+br−4+cr−6

This article uses the supersymmetric WKB approximation to obtain theapproximate energy levels and wave functions of the anharmonic potentialV(r) = ar2+br−4+cr−6 in order to tesify the correctness between[Phys. Lett. A 170 (1992) 335] and the paper written by M. Landtman[Phys. Lett. A 175 (1993) 147].

SUPERSYMMETRIC WKB TUNNELING THROUGH TRIPLE FINITE SQUARE BARRIER AND QUALITY OF SWKB QUANTIZATION CONDITION

Tunneling through a triple finite square barrier is examined in the light of WKB and supersymmetric WKB (SWKB) method. We have studied the quasi-bound states using lowest order WKB and SWKB quantization condition. The resonance energies have a doublet structure which represents a twofold quasi-degeneracy as expected from the symmetry of the three barriers enclosing two identical trapping regions. For very closely spaced low-lying resonance doublets, we find that the SWKB method gives much improved results than the conventional WKB method.

SUSYQM and SWKB Approaches to the Relativistic Equations with Hyperbolic Potential V0 tanh2 (r/d)

We apply supersymmetric quantum mechanics (SUSYQM) and shape invariance to investigate the Klein-Gordon and Dirac equations with the hyperbolic potential V0tanh2 (r/d). We find that the master equations for the Klein-Gordon and Dirac equations (radial wave function F(r)) with this potential for the S wave are unified as a same second order differential equation. The energy levels are obtained by SUSYQM and supersymmetric WKB (SWKB) approaches. We also apply the traditional hypergeometric differential equation approach to solve the master equation and obtain the exact solutions. We show that the vibrational quantum number n for the energy levels has to be an odd integer only through studying the properties of the eigenfunctions expressed by the Gegenbauer polynomials. Finally, we study the harmonic limit of this system.

Supersymmetry and SWKB approach to the Dirac equation with a Coulomb potential in [formula omitted] dimensions

By using the supersymmetric quantum mechanics and shape invariance, the ( 2 + 1 ) -dimensional Dirac equation with a Coulomb potential is investigated. The solutions of the bound states expressed by the confluent hypergeometric functions are analytically obtained. The energy levels are also given by means of both supersymmetric quantum mechanics and supersymmetric WKB approach.

The supersymmetric WKB approximation of ring-shaped oscillator potential in r and dimensions with spherical polar coordinates

According to calculation of the energy spectrum of ring-shape oscillator potential by using the supersymmetric WKB approximation, it is shown that the energy spectrum of some noncentral separable potentials can be exactly obtained in r and θ dimensions by above method.

A Class of Potentials for Which Exact Semiclassical Quantization Can Be Achieved

We consider a class of potentials for which the exact semiclassical quantization is achieved by a certain modification of the quantization condition. A list of potentials for which the new quantization condition is exact coincides with the list of potentials for which the spectrum is determined by the factorization method. We construct a one-parameter family of quantization conditions including the supersymmetric WKB condition as a special case. The new condition allows considering the interrelations between different modifications of the leading approximation and their validity ranges and also allows developing new approximate methods for calculating spectra.

The supersymmetric WKB approximation for Hartmann potential

Using the method of supersymmetric WKB approximation, the energy spectrum of some noncentral separable potentials can be exactly obtained in r and θ dimensions. We take the Hartmann potential as an important example for its validation and the result is consistent with that obtained by using the supersymmetric quantum mechanics and shape invariance.