Analytical Solution of Dirac Equation for q-Deformed Hyperbolic Manning-Rosen Potential in D Dimensions using SUSY QM and its Thermodynamics'Application
C. Cari, Suparmi Suparmi, Mohtar Yunianto, Beta Nur Pratiwi
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.
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.
The approximate analytical solution of Schrodinger equation for Q-Deformed Rosen-Morse potential was investigated using Supersymmetry Quantum Mechanics (SUSY QM) method. The approximate bound state energy is given in the closed form and the corresponding approximate wave function for arbitrary l-state given for ground state wave function. The first excited state obtained using upper operator and ground state wave function. The special case is given for the ground state in various number of q. The existence of Rosen-Morse potential reduce energy spectra of system. The larger value of q, the smaller energy spectra of system.
ABSTRACTThe objectives of this research are to determine energy spectra and wavefunction of schrodinger equation for q-deformed Rosen Morse plus non centraltrigonometric q-deformed Posch Teller potential and hyperbolic q-deformed Scarfplus Non central q-deformed Rosen Morse Trigonometric Potential using SUSY QMApproach. Actually, SUSY QM propose that the boson can change into fermion viceversa. However, the evidence of this phenomena does not exist in this world. Wittenproposed that SUSY must be broken. This SUSY theoretically can be solve most ofproblems in Quantum Mechanics. After the energy spectra and wave functionfounded, we will plot a graph simulation to understand the effect of potentials inenergy spectra and wave function of system.Keywords: Schrodinger Equation, q-Deformed Non-Central Shape InvariantPotential, and Supersymmetry of Quantum MechanicsABSTRAKTujuan dari penelitian ini adalah untuk menentukan spektrum energi danfungsi gelombang persamaan Schrodinger untuk q-deformasi Rosen Morse ditambahnon sentral trigonometri q-deformasi Poschl Teller potensial dan hiperbolik q-cacatScarf ditambah Non pusat q-deformasi Rosen Morse trigonometri Potensimenggunakan SUSY Pendekatan QM. Sebenarnya SUSY QM mengusulkan bahwaboson bisa berubah menjadi fermion sebaliknya. Namun, bukti fenomena ini tidakada di dunia ini. Witten mengusulkan agar SUSY harus rusak. SUSY ini secarateoritis dapat dipecahkan sebagian besar masalah di Kuantum Mekanika. Setelahspektrum energi dan gelombang fungsi didirikan, kita akan plot simulasi grafik untukmemahami efek dari potensi dalam spektrum energi dan fungsi gelombang sistem.Kata kunci: Schrodinger Equation, q-Deformasi non-Central Shape InvariantPotensial, dan Supersymmetri of Kuantum Mechanics
The 4-Dimensional Schrodinger's Equation is solved using the Supersymmetric Quantum Mechanics (SUSY QM) to obtain the first three energy spectrum and the ground state wave functions both in radial and angular parts. The non-central potential consisted of radial Scarf potential, and mix trigonometry potential such us Poschl-Teller, Rosen Morse, and Scarf for the angular part were used in this study. The energy equation of the 4-dimensional Schrodinger equation system was used to analyze some thermodynamic properties (vibrational mean energy and specific heat) of the system.
The exact solutions of the Schrodinger equations (SE) in the D-dimensional coordinate system have attracted the attention of many theoretical researchers in branches of quantum physics and quantum chemistry. The energy eigenvalues and the wave function are the solutions of the Schrodinger equation that implicitly represents the behavior of a quantum mechanical system. This study aimed to obtain the eigenvalues, wave functions, and thermodynamic properties of the 6-Dimensional Schrodinger equation under Double Ring-Shaped Oscillator (DRSO) and Manning-Rosen potential. The variable separation method was applied to reduce the one 6-Dimensional Schrodinger equation depending on radial and angular non-central potential into five onedimensional Schrodinger equations: one radial and five angular Schrodinger equations. Each of these onedimensional Schrodinger equations was solved using the SUSY QM method to obtain one eigenvalue and one wave function of the radial part, five eigenvalues, and five angular wave functions angular part. Some thermodynamic properties such, the vibrational mean energy 𝑈, vibrational specific heat 𝐶, vibrational free energy 𝐹, and vibrational entropy 𝑆, were obtained using the radial energy equations. The results showed that except the 𝑛𝑙1, all increment of angular quantum number decreases the energy values. Increments of all potential parameter increase the energy values. Increment of angular quantum number and potentials parameter increases the amplitude and shifts the wave functions to the left. However, the increment of 𝑛𝑙1, 𝛼, 𝜎, and 𝜌 decrease the amplitude and shift wavefunctions to the right. Moreover, the vibrational mean energy 𝑈 and free energy 𝐹 increased as the increasing value of potentials parameters, where the ω parameter has the dominant effect than the other parameters. The vibrational specific heat 𝐶 and entropy 𝑆 affected only by the 𝜔 parameter, where 𝐶 and 𝑆 decreased as the increase of 𝜔.
Using the supersymmetric quantum mechanics (SUSY) approach, we obtained analytical approximate solutions to the Dirac equation with an improved Tietz diatomic potential function under spin and pseudospin symmetries. The energy eigenvalues equation and the corresponding wave spinor for the two symmetries have been explicitly obtained. The non relativistic limit of the spin symmetry has been obtained. The solution of Morse, Deng-Fan, and Rosen-Morse were obtained by changing the numerical value of q We have also obtained the thermodynamic properties of a system under the improved Tietz potential function.
In this research work, within the framework of relativistic and nonrelativistic noncommutative quantum mechanics, the deformed Klein–Gordon and Schrödinger equations were solved with the modified equal vector scalar Manning-Rosen potential that has been of significance interest in recent years using Bopp's shift method and standard perturbation theory in the first-order in the noncommutativity parameters in 3-dimensions noncommutative quantum mechanics. By employing the improved approximation of the centrifugal term, the relativistic and nonrelativistic bound state energies were obtained for some diatomic molecules such as (HCl, CH, LiH, CO, NO, O2, I2, N2, H2, and Ar2). The obtained energy eigenvalues appear as a function of the generalized Gamma function, the parameters of noncommutativity, and the parameters of studied potential, in addition to the atomic quantum numbers . In both relativistic and nonrelativistic problems, we show that the corrections on the spectrum energy are smaller than the main energy in the ordinary cases of RQM and NRQM. A straightforward limit of our results to ordinary quantum mechanics shows that the present result is consistent with what is obtained in the literature. We have seen that the improved approximation of the centrifugal term is better than the other approximations in finding the approximate analytical solutions of the Klein-Gordon and Schrödinger equations for the modified Manning–Rosen potential in RNCQM and NRNCQM.