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.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 DASARJul 2, 20091 citation

−symmetric harmonic oscillators

Miloslav Znojil

arXiv:quant-ph/9905020Physics Letters AAug 1, 1999272 citations

A New Framework for the Determination of the Eigenvalues and Eigenfunctions of the Quantum Harmonic Oscillator

Francis Tabi Oduro, Amos Odoom

This study was designed to obtain the energy eigenvalues and the corresponding Eigenfunctions of the Quantum Harmonic oscillator through an alternative approach. Starting with an appropriate family of solutions to a relevant linear di erential equation, we recover the Schr¨odinger Equation together with its eigenvalues and eigenfunctions of the Quantum Harmonic Oscillator via the use of Gram Schmidt orthogonalization process in the usual Hilbert space. Significantly, it was found that there exists two separate sequences arising from the Gram Schmidt Orthogonalization process; one in respect of the even eigenfunctions and the other in respect of the odd eigenfunctions.This study was designed to obtain the energy eigenvalues and the corresponding Eigenfunctions of the Quantum Harmonic oscillator through an alternative approach. Starting with an appropriate family of solutions to a relevant linear di erential equation, we recover the Schr¨odinger Equation together with its eigenvalues and eigenfunctions of the Quantum Harmonic Oscillator via the use of Gram Schmidt orthogonalization process in the usual Hilbert space. Significantly, it was found that there exists two separate sequences arising from the Gram Schmidt Orthogonalization process; one in respect of the even eigenfunctions and the other in respect of the odd eigenfunctions.

Journal of Mathematics ResearchOct 27, 20211 citation

Harmonic Oscillator

Alto Osada, Rekishu Yamazaki, Atsushi Noguchi

Lecture notes in physicsJan 1, 20220 citations

On the quantum-mechanical singular harmonic oscillator

Francisco Marcelo Fernández

We obtain the eigenvalues and eigenfunctions of the singular harmonic oscillator $V(x)=α/(2x^2)+x^2/2$ by means of the simple and straightforward Frobenius (power-series) method. From the behaviour of the eigenfunctions at origin we derive two branches for the eigenvalues for negative values of $α$ . We discuss the well known fact that there are square-integrable solutions only for some allowed discrete values of the energy.We obtain the eigenvalues and eigenfunctions of the singular harmonic oscillator $V(x)=α/(2x^2)+x^2/2$ by means of the simple and straightforward Frobenius (power-series) method. From the behaviour of the eigenfunctions at origin we derive two branches for the eigenvalues for negative values of $α$. We discuss the well known fact that there are square-integrable solutions only for some allowed discrete values of the energy.

arXiv:2112.03693Dec 5, 20211 citation

We obtain the eigenvalues and eigenfunctions of the singular harmonic oscillator

V(x)=α/(2x^2)+x^2/2

by means of the simple and straightforward Frobenius (power-series) method. From the behaviour of the eigenfunctions at origin we derive two branches for the eigenvalues for negative values of

α

. We discuss the well known fact that there are square-integrable solutions only for some allowed discrete values of the energy.