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arXiv:1805.00586 · Europhysics Letters (EPL)

A novel SUSY energy bound-states treatment of the Klein-Gordon equation with PT-symmetric and q-deformed parameter Hulthén potential

Metin Aktaş

Published Jan 1, 2018• 8 citations

Abstract

In this study, we focus on investigating the exact relativistic bound-state spectra for supersymmetric, PT -supersymmetric and non -Hermitian versions of the q -deformed parameter Hulthén potential. The Hamiltonian hierarchy mechanism, namely the factorization method , is adopted within the framework of SUSYQM. This algebraic approach is used in solving the Klein-Gordon equation with the potential cases. The results obtained analytically by executing the straightforward calculations are in consistent forms for certain values of q . Achieving the results may have a particular interest for such applications. That is, they can be involved in determining the quantum structural properties of molecules for ro-vibrational states, and optical spectra characteristics of semiconductor devices with regard to the lattice dynamics. They are also employed to construct the broken or unbroken case of the supersymmetric particle model concerning the interaction between the elementary particles.

DOI: 10.1209/0295-5075/121/10005

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The relativistic wave equations determine the dynamics of quantum fields in the context of quantum field theory. One of the conventional tools for dealing with the relativistic bound state problem is the Klein-Fock-Gordon equation. In this work, using a developed scheme, we present how to surmount the centrifugal part and solve the modified Klein-Fock-Gordon equation for the linear combination of Hulthén and Yukawa potentials. In particular, we show that the relativistic energy eigenvalues and corresponding radial wave functions are obtained from supersymmetric quantum mechanics by applying the shape invariance concept. Here, both scalar potential conditions, which are whether equal and nonequal to vector potential, are considered in the calculation. The energy levels and corresponding normalized eigenfunctions are represented as a recursion relation regarding the Jacobi polynomials for arbitrary l states. Beyond that, a closed form of the normalization constant of the wave functions is found. Furthermore, we state that the energy eigenvalues are quite sensitive with potential parameters for the quantum states. The nonrelativistic and relativistic results obtained within SUSY QM overlap entirely with the results obtained by ordinary quantum mechanics, and it displays that the mathematical implementation of SUSY quantum mechanics is quite perfect.

In this paper, the bound state solution of the modified Klein–Fock–Gordon equation is obtained for the Hulthén plus ring-shaped-like potential by using the developed scheme to overcome the centrifugal part. The energy eigenvalues and corresponding radial and azimuthal wave functions are defined for any [Formula: see text] angular momentum case on the conditions that scalar potential is whether equal and nonequal to vector potential, the bound state solutions of the Klein–Fock–Gordon equation of the Hulthén plus ring-shaped-like potential are obtained by Nikiforov–Uvarov (NU) and supersymmetric quantum mechanics (SUSY QM) methods. The equivalent expressions are obtained for the energy eigenvalues, and the expression of radial wave functions transformations to each other is revealed owing to both methods. The energy levels and the corresponding normalized eigenfunctions are represented in terms of the Jacobi polynomials for arbitrary [Formula: see text] states. A closed form of the normalization constant of the wave functions is also found. It is shown that the energy eigenvalues and eigenfunctions are sensitive to [Formula: see text] radial and [Formula: see text] orbital quantum numbers.

The Dirac equation of q-deformed hyperbolic Manning Rosen potential in D dimension was solved by using Supersymmetric Quantum Mechanics (SUSY QM). The D dimensional relativistic energy spectra were obtained by using SUSY QM and shape invariant properties and D dimensional wave functions of q-deformed hyperbolic Manning Rosen potential were obtained by using the SUSY raising and lowering operators. In the nonrelativistic limit, the relativistic energy spectra for exact spin symmetry case reduced into nonrelativistic energy spectra and so for the wave functions. In the classical regime, the partition function, the vibrational specific heat, and the vibrational mean energy of some diatomic molecules were calculated from the non-relativistic energy spectra with the help of error function and imaginary error function.