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International Journal of Advanced Trends in Computer Science and Engineering

Study of Schrodinger Equation with Quantum Deformation for Three-Dimensional Harmonic Oscillator plus Inverse Quadratic Potential by Hypergeometric Method

Dyah Ayu Dianawati

Published Dec 15, 2019• 4 citations
Publisher

Abstract

The Schrodinger equation with quantum deformation for three-dimensional Harmonic Oscillator plus Inverse Quadratic potential was obtained the solution by Hypergeometric method with substituting variable and parameter to reduce Schrodinger equation into second order Hypergeometric differential equation.The non-relativistic energy was obtained from condition a=-n and wave function was expressed by Hypergeometric function.The results were calculated and visualized by Matlab R2013a software.The energy spectrum and wave function depend on the quantum deformation and quantum number parameters.

DOI
10.30534/ijatcse/2019/17862019

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The Schrodinger equation with quantum deformation for three-dimensional Harmonic Oscillator plus Inverse Quadratic potential was obtained the solution by Hypergeometric method with substituting variable and parameter to reduce Schrodinger equation into second order Hypergeometric differential equation.The non-relativistic energy was obtained from condition a=-n and wave function was expressed by Hypergeometric function.The results were calculated and visualized by Matlab R2013a software.The energy spectrum and wave function depend on the quantum deformation and quantum number parameters.
Spatially quadratic deformation radial momentum of D-dimensional Klein-Gordon equation with three-dimensional Harmonic Oscillator and Inverse Quadratic potential was investigated using Hypergeometric Method. By using appropriate variable transformation, D-Dimensional Klein-Gordon equation for q deformed momentum was reduced into one order dimensional Schrodinger like equation with Pöschl-Teller like potential. The relativistic energy and wave function equations were obtained from the solution of Schrodinger like equation. Then, the relativistic energy and wave function were calculated and visualized using Matlab R2013 software, respectively. The q deformed parameter had the effect to the relativistic energy and wave function. The increase of q deformed parameter causes the increase of energy spectra and the decrease of amplitude.
The Hypergeometric method was used to obtain the solution of Schrodinger equation for radial quantum deformation on Coulomb potential. The Schrodinger equation was reduced into the general form of Hypergeometric function with variable and parameter substitutions. As a result, the energy was calculated from the energy equation and wave function was visualized by Matlab R2013a software. The decrease of energy values causes the increase of quantum number and quantum deformation parameters, while the wave functions have the depth of deep amplitude by the increase of quantum number and quantum number parameters.
The Hypergeometric method was used to describe radial equations by probeneus approaximation at a singular regular point. Schrodinger equations radial part was reduced into differential equations with variable substitution. By using q-deformed momentum parameter and Coulomb potential as well as variable substitution, the second order of Schrodinger equation with centrifugal approach for q-deformed momentum in Coulomb potential can be solved with Hypergeometric method. The analytical result of energy spectrum and wave function with quantum number and q-deformed variations, while the numerical result for energy spectrum can be calculated by using Matlab 2013 software.
Abstract Hypergeometric method was used to obtain the solution of Klein-Gordon equation with q-deformed quantum mechanics for Coulomb potential. By introduction the new variable and parameters in the Klein-Gordon equation, then Klein-Gordon equation reduced to the second order differential equation of Hypergeometric function. From the solution of the Hypergeometric differential equation, the energy and wave function equations were obtained analytically. The energy was numerically calculated by Matlab R2013a software and the first three wave function was obtained. The result shows that the decrease of quantum deformation caused the increase of the energy eigen value, and the increase of quantum number caused the increase of the energy eigen value too.
The Q-deformed D-dimensional Klein Gordon equation with Kratzer potential is solved by using Hypergeometric method in the case of exact spin symmetry. The linear radial momentum of D-dimensional Klein Gordon equation is disturbed by the presence of the quadratic radial posisiton. The Klein-Gordon D-dimensional equation is reduced to one-dimensional Schrodinger like equation with variable substitution. The solution of the D-dimensional Klein-Gordon equation is determined in the form of a general equation of the Hypergeometry function using the Kratzer potential variable and the quantum deformation variable. From this equation, relativistic energy and wave function are determined. In addition, the relativistic energy equation can be used to calculate numerical energy levels for diatomic particles (CO, NO, O2) using Matlab R2013a software. The results obtained show that the q-deformed quantum parameters, quantum numbers and dimensions affect the value of relativistic energy for zero-pin particles. The value of energy increases with increasing value of quantum number n, q-deformed parameters, and d-dimensional parameters. Of the three parameters, q-deformed parameter is the most dominant to give change in energy value; the increasing q-deformed parameter causes the energy value increases significantly compared to the d-dimensional parameter and quantum numbers n.
The Schrödinger solutions for a three-dimensional central potential system whose Hamiltonian is composed of a time-dependent harmonic plus an inverse harmonic potential are investigated. Because of the time-dependence of parameters, we cannot solve the Schrödinger solutions relying only on the conventional method of separation of variables. To overcome this difficulty, special mathematical methods, which are the invariant operator method, the unitary transformation method, and the Nikiforov-Uvarov method, are used when we derive solutions of the Schrödinger equation for the system. In particular, the Nikiforov-Uvarov method with an appropriate coordinate transformation enabled us to reduce the eigenvalue equation of the invariant operator, which is a second-order differential equation, to a hypergeometric-type equation that is convenient to treat. Through this procedure, we derived exact Schrödinger solutions (wave functions) of the system. It is confirmed that the wave functions are represented in terms of time-dependent radial functions, spherical harmonics, and general time-varying global phases. Such wave functions are useful for studying various quantum properties of the system. As an example, the uncertainty relations for position and momentum are derived by taking advantage of the wave functions.