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Prolate Spheroidal Wave Function as Exact Solution of the Schrödinger Equation

J. Akande, Kolawolé Kêgnidé Damien Adjaï, L. H. Koudahoun, Y. J. F. Kpomahou, M. D. Monsia

Published Jan 1, 2017• 2 citations
OpenAlex

Abstract

In quantum mechanics, the wave function and energy are required for the complete characterization of fundamental properties of a physical system subject to a potential energy. It is proved in this work, the existence of a Schrodinger equation with position-dependent mass having the prolate spheroidal wave function as exact solution, resulting from a classical quadratic Lienard-type oscillator equation. This fact may allow the extension of the current one-dimensional model to three dimensions and increase the understanding of analytical features of quantum systems.

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In quantum mechanics, the wave function and energy are required for the complete characterization of fundamental properties of a physical system subject to a potential energy. It is proved in this work, the existence of a Schrodinger equation with position-dependent mass having the prolate spheroidal wave function as exact solution, resulting from a classical quadratic Lienard-type oscillator equation. This fact may allow the extension of the current one-dimensional model to three dimensions and increase the understanding of analytical features of quantum systems.
In the present work, we proceed to study the Schrödinger equation with dependent mass position. From the resulting partial differential equations, we obtain exact analytical solutions governed by Bessel functions. The exact solution of Schrödinger's equation for a particle with dependent position of the mass (PDM) is a "half-harmonic potential" defined in a Hilbert space. The harmonic oscillator is carried by the wave function ψ(x) through the Bessel function. The magnitude of ψ(x) increases dramatically as the values of the spatial coordinate become larger and larger. This growth is all the more marked as the mass parameter increases. It appears that the wave function has the same asymptotic behavior as the solution of Airy's equation of the second kind. Their oscillations have the same amplitudes near infinity. The mass parameter has a great importance and influence on the mass and also on the wave potential. It is a control parameter in the Schrödinger equations (PDM).
Abstract In this work, we find the exact solution of Schrödinger wave equation for position dependent mass with pseudoharmonic oscillator using extended Nikiforov-Uvarov method. We obtained the energy-eigen equation presented in a closed and compact form and used the result to study both superstatistics and thermodynamic properties by first determining the partition function of the system. The unnormalized wave function was obtained and expressed in terms of Confluent Heun function. Using the resulting energy eigen equation, the numerical computation were computed for varying masses for with fixed physical constant potential parameter λ. The numerical result shows that the bound state energies increase with quantum states but decreases with the dependent mass m(x). The thermodynamics and superstatistics plots are also reported.
The Schrödinger equation with a position-dependent mass (SEPDM) is employed in many areas of quantum physics. Exact solutions for the SEPDM lie at the center of interest of the professional public because it helps us to understand the behavior of quantum particles in the cases in which their mass varies spatially. For this purpose, we used the mass function represented by a quartic polynomial and a quadratic potential function, which extends the current class of exact solutions of the SEPDM. The exact analytical solution of the problem is expressed as a linear combination of local Heun functions. Heun’s equation contains many parameters, resulting in its general nature. We studied how limit changes in some of these parameters will affect the solution of the SEPDM. The obtained solutions are particularly suitable for the transfer matrix method and solutions of scattering problems; this is demonstrated by the calculation of bound states.
The Schrodinger equation with position-dependent mass (PDM) becomes one of interesting subjects in the study of quantum systems, because of its wide applications in many physical problems. Meanwhile, harmonic oscillator becomes important model in most physical problems. In this paper, analytical solutions of the PDM Schrodinger equation, i.e. energy eigenvalues En and eigenfunctions , of a one dimensional harmonic oscillator system using transformation method has been studied. The results showed that there was no difference between the energy eigenvalues, En of the harmonic oscillator systems with and without PDM, whereas it is not for the eigenfunctions,  of these two harmonic oscillator systems. This difference was due to the presence of dimensionless mass function  and the changes of x becomes  variable for the eigenfunctions of the harmonic oscillator system with PDM.  Moreover, the differences in the eigenfunctions will affect the dynamical properties of the quantum systems. It can be concluded that the harmonic oscillator system with PDM is the generalization of the harmonic oscillator system with constant mass.
This communication has been dedicated to quantum dynamics of particle whose mass depends on coordinate. We considered one dimensional model which admits to obtain the exact solution of wave equation. The position dependent mass was represented as the periodic function. Inside of period the mass varies accordingly to the inverse proportionality. We have found wave functions in their explicit form as well as energy eiguen values. It has been shown that the energy spectrum manifests properties typical for periodic nanostructures.