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

Physica Scripta

Time-dependent Schrödinger equation with non-central potentials

N. Ferkous, Abdelhafid Bounames, Mustapha Maamache

Published Aug 7, 2013• 6 citations
Publisher

Abstract

Using the Lewis–Riesenfeld theory, we show that the time-dependent Schrödinger equation for non-central potentials with an arbitrary angular function U ( θ ) is analytically solvable. As a special case, we derive the exact solution for the double ring-shaped generalized non-central oscillator with time-dependent mass and frequency. The time-independent case, studied in the literature, is recovered.

DOI
10.1088/0031-8949/88/03/035001

Related papers

Time-dependent Dunkl–Schrödinger equation with an angular-dependent potential

B. C. Lütfüoğlu, Amar Benchikha, B. Hamil +1 more

In this paper, we investigate the analytical solution of the time-dependent Schrödinger equation for a harmonic oscillator with time-dependent mass and frequency, coupled with angular-dependent potential energy by utilizing the Dunkl derivatives. To obtain the solution, we employ the Lewis–Riesenfeld invariant methodology. Our approach broadens the scope of quantum mechanical analyses, offering exact solutions and new insights into dynamic quantum systems under varying conditions.

arXiv:2408.02021•Modern Physics Letters A•Feb 28, 2025•15 citations

Solutions to the time-dependent Schrodinger equation in the continuous spectrum case

Mustapha Maamache, Y. Saadi

We generalize the Lewis-Riesenfeld technique of solving the time-dependent Schrodinger equation to cases where the invariant has continuous eigenvalues. An explicit formula for a generalized Lewis-Riesenfeld phase is derived in terms of the eigenstates of the invariant. As an illustration the generalized phase is calculated for a particle in a time-dependent linear potential.

arXiv:0804.4289•Apr 27, 2008•0 citations

Exact Solutions for Time-Dependent Schrödinger Equation in Presence of the Pöschl-Teller Double-Ring Shaped Harmonic Potential

Hadi Sobhani, HASSAN HASSANABADI

In this paper, time-dependent Schrödinger equation is considered with the Pöschl-Teller double-ring shaped harmonic potential.Then, for investigation of such a system, the Lewis-Riesenfeld dynamical invariant method is presented and shown that how it works.After that, the form and eigenfunctions for the Lewis-Riesenfeld dynamical invariant are derived and the exact solutions are studied.Then the physical interpretations of derived results are discussed.

Acta Physica Polonica A•Jul 1, 2019•3 citations

QUANTUM STATES OF A GENERALIZED TIME-DEPENDENT INVERTED HARMONIC OSCILLATOR

I. A. PEDROSA, I. GUEDES

We discuss the extension of the Lewis and Riesenfeld method of solving the time-dependent Schrödinger equation to cases where the invariant has continuous eigenvalues and apply it to the case of a generalized time-dependent inverted harmonic oscillator. As a special case, we consider a generalized inverted oscillator with constant frequency and exponentially increasing mass.

arXiv:quant-ph/0307084•International Journal of Modern Physics B•Apr 10, 2004•29 citations

Solutions of the Schrödinger equation for the time-dependent linear potential

Jian Qi Shen

By making use of the Lewis-Riesenfeld invariant theory, the solution of the Schrödinger equation for the time-dependent linear potential corresponding to the quadratic-form Lewis-Riesenfeld invariant $I_{\rm q}(t)$ is obtained in the present paper. It is emphasized that in order to obtain the general solutions of the time-dependent Schrödinger equation, one should first find the complete set of Lewis-Riesenfeld invariants. For the present quantum system with a time-dependent linear potential, the linear $I_{\rm l}(t)$ and quadratic $I_{\rm q}(t)$ (where the latter $I_{\rm q}(t)$ cannot be written as the squared of the former $I_{\rm l}(t)$, {\it i.e.}, the relation $I_{\rm q}(t)= cI_{\rm l}^{2}(t)$ does not hold true always) will form a complete set of Lewis-Riesenfeld invariants. It is also shown that the solution obtained by Bekkar {\it et al.} more recently is the one corresponding to the linear $I_{\rm l}(t)$, one of the invariants that form the complete set. In addition, we discuss some related topics regarding the comment [Phys. Rev. A {\bf 68}, 016101 (2003)] of Bekkar {\it et al.} on Guedes's work [Phys. Rev. A {\bf 63}, 034102 (2001)] and Guedes's corresponding reply [Phys. Rev. A {\bf 68}, 016102 (2003)].

arXiv:quant-ph/0310179•Oct 30, 2003•4 citations

WAVE FUNCTIONS OF TIME-DEPENDENT TWO COUPLED OSCILLATORS BY LEWIS–RIESENFELD METHOD

Hong-Yi Fan, Zhonghua Jiang

For the two time-dependent coupled oscillators model we derive its time-dependent invariant in the context of Lewis–Riesenfeld invariant operator theory. It is based on the general solutions to the Schrödinger equation which is obtained and turns out to be the superposition of the generalized atomic coherent states in the Schwinger bosonic realization. The energy eigenvectors and eigenvalues of the corresponding time-independent Hamiltonian are also obtained as a by-product.

International Journal of Modern Physics B•Apr 4, 2006•2 citations
Using the Lewis–Riesenfeld theory, we show that the time-dependent Schrödinger equation for non-central potentials with an arbitrary angular function U ( θ ) is analytically solvable. As a special case, we derive the exact solution for the double ring-shaped generalized non-central oscillator with time-dependent mass and frequency. The time-independent case, studied in the literature, is recovered.
By making use of the Lewis-Riesenfeld invariant theory, the solution of the Schrödinger equation for the time-dependent linear potential corresponding to the quadratic-form Lewis-Riesenfeld invariant Iq(t)I_{\rm q}(t)Iq​(t) is obtained in the present paper. It is emphasized that in order to obtain the general solutions of the time-dependent Schrödinger equation, one should first find the complete set of Lewis-Riesenfeld invariants. For the present quantum system with a time-dependent linear potential, the linear Il(t)I_{\rm l}(t)Il​(t) and quadratic Iq(t)I_{\rm q}(t)Iq​(t) (where the latter Iq(t)I_{\rm q}(t)Iq​(t) cannot be written as the squared of the former Il(t)I_{\rm l}(t)Il​(t), {\it i.e.}, the relation Iq(t)=cIl2(t)I_{\rm q}(t)= cI_{\rm l}^{2}(t)Iq​(t)=cIl2​(t) does not hold true always) will form a complete set of Lewis-Riesenfeld invariants. It is also shown that the solution obtained by Bekkar {\it et al.} more recently is the one corresponding to the linear Il(t)I_{\rm l}(t)Il​(t), one of the invariants that form the complete set. In addition, we discuss some related topics regarding the comment [Phys. Rev. A {\bf 68}, 016101 (2003)] of Bekkar {\it et al.} on Guedes's work [Phys. Rev. A {\bf 63}, 034102 (2001)] and Guedes's corresponding reply [Phys. Rev. A {\bf 68}, 016102 (2003)].