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Acta Physica Polonica A

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

Hadi Sobhani, HASSAN HASSANABADI

Published Jul 1, 2019• 3 citations
Publisher

Abstract

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.

DOI
10.12693/aphyspola.136.17

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arXiv:quant-ph/0309174•Physical Review A•Jan 5, 2005•34 citations
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.
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)].
We reexamine the general solution of a Schr\"odinger equation in the presence of a time-dependent linear potential in configuration space based on the Lewis-Riesenfeld framework. For comparison, we also solve the problem in momentum space and then Fourier transform the solution to get the general wave function. Appropriately choosing the weight function in the latter method, we can obtain the same wave function as the former method. It is found that a non-Hermitian time-dependent linear invariant can be used to obtain Gaussian-type wave-packet solutions of the time-dependent system. This operator is a specific linear combination of the initial momentum and initial position operators. This fact indicates that the constants of integration such as the initial position and initial momentum that determine the classical motion play important roles in the time-dependent quantum system. The eigenfunction of the linear invariant is interpreted as a wave packet with a ``center of mass'' moving along the classical trajectory, while the ratio between the coefficients of the initial position and initial momentum determines the width of the wave packet.