Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1009.0250 · Journal of Physics A Mathematical and Theoretical

Remarks on the solution of the position-dependent mass Schrödinger equation

Ramazan Koç, Seda Sayın

Published Oct 19, 2010• 18 citations
arXivHTMLPDF

Abstract

An approximate method is proposed to solve the position-dependent mass (PDM) Schrödinger equation. The procedure suggested here leads to the solution of the PDM Schrödinger equation without transforming the potential function to the mass space or vice versa. The method based on the asymptotic Taylor expansion of the function produces an approximate analytical expression for eigenfunction and numerical results for eigenvalues of the PDM Schrödinger equation. The results show that the PDM and constant mass Schrödinger equations are not isospectral. The calculations are carried out with the aid of a computer system of symbolic or numerical calculation by constructing a simple algorithm.

DOI
10.1088/1751-8113/43/45/455203

Related papers

Remarks on the Solution of the Position Dependent Mass (PDM)

Ramazan Koç, Seda Sayõn

An approximate method is proposed to solve position dependent mass Schrodinger equation. The procedure suggested here leads to the solution of the PDM Schrodinger equation without trans- forming the potential function to the mass space or vice verse. The method based on asymptotic Taylor expansion of the function, produces an approximate analytical expression for eigenfunction and numerical results for eigenvalues of the PDM Schrodinger equation. The results show that PDM and constant mass Schrodinger equations are not isospectral. The calculations are carried out with the aid of a computer system of symbolic or numerical calculation by constructing a simple algorithm.

•Jan 1, 2010•0 citations

The coordinate transformation and the exact solutions of the Schrödinger equation with position-dependent effective mass

Ju Guo-Xing, Cai Chang-Ying, Xiang, Yang +1 more

Using the coordinate transformation method, we solve the one-dimensional Schrödinger equation with position-dependent mass(PDM). The explicit expressions for the potentials, energy eigenvalues and eigenfunctions of the systems are given. The eigenfunctions can be expressed in terms of the Jacobi, Hemite and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term $V_m$ which produced from the dependence of mass on the coordinate, compared with that for the systems of constant mass. The properties of $V_m$ for several mass functions are discussed.

arXiv:quant-ph/0601004•Dec 31, 2005•1 citation

Analytical Solution of Schrödinger Equation of the Harmonic Oscillator System with Position Dependent Mass Using Transformation Method

Tis Sutisna

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.

Jurnal ILMU DASAR•Jul 2, 2009•1 citation

Coordinate Transformation and Exact Solutions of Schrödinger Equation with Position-Dependent Effective Mass

Ju Guo-Xing, Cai Chang-Ying, Xiang Yang +1 more

Using the coordinate transformation method, we solve the one-dimensional Schrödinger equation with position-dependent mass. The explicit expressions for the potentials, energy eigenvalues, and eigenfunctions of the systems are given. The eigenfunctions can be expressed in terms of the Jacobi, Hermite, and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term V m , which is produced from the dependence of mass on the position, compared with those for the systems of constant mass. The properties of V m for several mass functions are discussed.

Communications in Theoretical Physics•Jun 1, 2007•13 citations

Analytical Solution of the Schrödinger Equation with an Exponential Type Mass Depending on the Spatial Variable

Edouard Diouf

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).

European Journal of Theoretical and Applied Sciences•Jul 5, 2023•2 citations

Semiclassical Method to Schrödinger Equation with Position-Dependent Effective Mass

Gang Chen, Pei-cai Xuan, Zidong Chen

In this paper, two novel semiclassical methods including the standard and supersymmetric WKB quantization conditions are suggested to discuss the Schrödinger equation with position-dependent effective mass. From a proper coordinate transformation, the formalism of the Schrödinger equation with position-dependent effective mass is mapped into isospectral one with constant mass and therefore for a given mass distribution and physical potential function the bound state energy spectrum can be determined easily by above method associated with a simple integral formula. It is shown that our method can give the analytical results for some exactly-solvable quantum systems.

Communications in Theoretical Physics•Jul 1, 2006•0 citations
An approximate method is proposed to solve the position-dependent mass (PDM) Schrödinger equation. The procedure suggested here leads to the solution of the PDM Schrödinger equation without transforming the potential function to the mass space or vice versa. The method based on the asymptotic Taylor expansion of the function produces an approximate analytical expression for eigenfunction and numerical results for eigenvalues of the PDM Schrödinger equation. The results show that the PDM and constant mass Schrödinger equations are not isospectral. The calculations are carried out with the aid of a computer system of symbolic or numerical calculation by constructing a simple algorithm.
An approximate method is proposed to solve position dependent mass Schrodinger equation. The procedure suggested here leads to the solution of the PDM Schrodinger equation without trans- forming the potential function to the mass space or vice verse. The method based on asymptotic Taylor expansion of the function, produces an approximate analytical expression for eigenfunction and numerical results for eigenvalues of the PDM Schrodinger equation. The results show that PDM and constant mass Schrodinger equations are not isospectral. The calculations are carried out with the aid of a computer system of symbolic or numerical calculation by constructing a simple algorithm.
Using the coordinate transformation method, we solve the one-dimensional Schrödinger equation with position-dependent mass(PDM). The explicit expressions for the potentials, energy eigenvalues and eigenfunctions of the systems are given. The eigenfunctions can be expressed in terms of the Jacobi, Hemite and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term VmV_mVm​ which produced from the dependence of mass on the coordinate, compared with that for the systems of constant mass. The properties of VmV_mVm​ for several mass functions are discussed.
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.
Using the coordinate transformation method, we solve the one-dimensional Schrödinger equation with position-dependent mass. The explicit expressions for the potentials, energy eigenvalues, and eigenfunctions of the systems are given. The eigenfunctions can be expressed in terms of the Jacobi, Hermite, and generalized Laguerre polynomials. All potentials for these solvable systems have an extra term V m , which is produced from the dependence of mass on the position, compared with those for the systems of constant mass. The properties of V m for several mass functions are discussed.
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).
In this paper, two novel semiclassical methods including the standard and supersymmetric WKB quantization conditions are suggested to discuss the Schrödinger equation with position-dependent effective mass. From a proper coordinate transformation, the formalism of the Schrödinger equation with position-dependent effective mass is mapped into isospectral one with constant mass and therefore for a given mass distribution and physical potential function the bound state energy spectrum can be determined easily by above method associated with a simple integral formula. It is shown that our method can give the analytical results for some exactly-solvable quantum systems.