Physical Review A

Comment on “Solution of the Schrödinger equation for the time-dependent linear potential”

Hacene Bekkar, Farid Benamira, Mustapha Maamache

Published Jul 8, 2003• 44 citations

Abstract

We present the correct way to obtain the general solution of the Schr"odinger equation for a particle in a time-dependent linear potential following the approach used in the paper of Guedes [Phys. Rev. A 63, 034102 (2001)]. In addition, we show that, in this case, the solutions (wave packets) are described by the Airy functions.

DOI: 10.1103/physreva.68.016101

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Comment on “Solution of the Schrödinger equation for the time-dependent linear potential”

Jarosław H. Bauer

It is shown that the wave function recently found by Guedes [Phys. Rev. A 63, 034102 (2001)] is a special case of the well-known Volkov solution to the time-dependent Schr\"odinger equation describing a nonrelativistic charged particle moving in an electromagnetic field.

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Reply to “Comment on ‘Solution of the Schrödinger equation for the time-dependent linear potential’ ”

I. Guedes

In this reply I discuss further the misconception in my paper [Phys. Rev. A 63, 034102 (2001)] related to the solution of the Schr\"odinger equation for a particle in a time-dependent linear potential.

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Comment on "Solutions of the Schrödinger equation for the time-dependent linear potential''

Jian Qi Shen

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Complete solution of the Schrödinger equation for the time-dependent linear potential

Mang Feng

The complete solutions of the Schr\"odinger equation for a particle with time-dependent mass moving in a time-dependent linear potential are presented. One solution is based on the wave function of the plane wave, and the other is in the form of the Airy function. A comparison is made between the present solution and former ones to show the completeness of the present solution.

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

Jian Qi Shen

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Linear Potentials, Airy Wave Packets and Airy Transform

G. Dattoli, Жуковский Константин Владимирович

The solution of the Schrodinger equation with a linear potential is considered. We use algebraic methods to obtain the explicit form of the solution for the explicitly time dependent Hamiltonian and discuss the general conditions which allow us to get solutions in terms of the Airy functions, yelding non spreading wave packets. We analyze the relevant physical meaning of these solutions and give examples of their applications. We discuss the analogy between Airy and Gauss-Weirestrass transform.

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Physical Review A

Comment on “Solution of the Schrödinger equation for the time-dependent linear potential”

Hacene Bekkar, Farid Benamira, Mustapha Maamache

Published Jul 8, 2003• 44 citations

Publisher

Abstract

DOI: 10.1103/physreva.68.016101

Related papers

Comment on “Solution of the Schrödinger equation for the time-dependent linear potential”

Jarosław H. Bauer

Physical Review AMar 1, 200233 citations

Reply to “Comment on ‘Solution of the Schrödinger equation for the time-dependent linear potential’ ”

I. Guedes

Physical Review AJul 8, 200318 citations

Comment on "Solutions of the Schrödinger equation for the time-dependent linear potential''

Jian Qi Shen

We show that the solution obtained by Bekkar {\it et al.} in their comment [Phys. Rev. A {\bf 68}, 016101 (2003)] on Guedes's work of solving the quantum system with a time-dependent linear potential is still {\it not} the {\it general} one of the Schrödinger equation. It is concluded that Bekkar {\it et al.}'s solution (corresponding to the linear Lewis-Riesenfeld invariant) and our solution (corresponding to the quadratic-form Lewis-Riesenfeld invariant) presented here will constitute together a complete set of solutions (general solutions) of the time-dependent Schrödinger equation of the system under consideration.We show that the solution obtained by Bekkar {\it et al.} in their comment [Phys. Rev. A {\bf 68}, 016101 (2003)] on Guedes's work of solving the quantum system with a time-dependent linear potential is still {\it not} the {\it general} one of the Schrödinger equation. It is concluded that Bekkar {\it et al.}'s solution (corresponding to the linear Lewis-Riesenfeld invariant) and our solution (corresponding to the quadratic-form Lewis-Riesenfeld invariant) presented here will constitute together a complete set of solutions (general solutions) of the time-dependent Schrödinger equation of the system under consideration.

arXiv:quant-ph/0311002Nov 1, 20030 citations

Complete solution of the Schrödinger equation for the time-dependent linear potential

Mang Feng

arXiv:quant-ph/0105145Physical Review AAug 20, 2001105 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)].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/0310179Oct 30, 20034 citations

Linear Potentials, Airy Wave Packets and Airy Transform

G. Dattoli, Жуковский Константин Владимирович

arXiv:1010.1678Oct 8, 20102 citations

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)].