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Physics Letters B

Davydov–Chaban Hamiltonian in presence of time-dependent potential

Hadi Sobhani, HASSAN HASSANABADI

Published Jun 22, 2016• 20 citations
Publisher

Abstract

In this article, we have investigated collective effects of atomic nuclei in presence of a time-dependent potential in Davydov–Chaban Hamiltonian. Since such potential has an explicit time-dependency, in order to obtain the wave function of considered system, we should face with time-dependent Schrödinger equation. Obtaining the wave function could be possible using Lewis–Riesenfeld dynamical invariant method. Appropriate dynamical invariant has been constructed after determining the wave functions and values, the wave function will obtain.

DOI
10.1016/j.physletb.2016.05.048

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In this article, we have investigated collective effects of atomic nuclei in presence of a time-dependent potential in Davydov–Chaban Hamiltonian. Since such potential has an explicit time-dependency, in order to obtain the wave function of considered system, we should face with time-dependent Schrödinger equation. Obtaining the wave function could be possible using Lewis–Riesenfeld dynamical invariant method. Appropriate dynamical invariant has been constructed after determining the wave functions and values, the wave function will obtain.
In this paper I have drawn out the steps to be followed in order to derive the exact Schr\"odinger wave function for a particle in a general one-dimensional time-dependent linear potential. To this end I have used the so-called Lewis and Riesenfeld invariant method, which is based on finding an exact quantum-mechanical invariant in whose eigenstates the exact quantum states are found. In particular, I have obtained the wave functions of a particle in the linear potential well, driven by a monochromatic electric field.
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)].
In this article, a quantum many-body system is considered. Then two time-dependent interactions have been added to the system. Changing of them is assumed in general form. After that, by using algebraic method, time evolution of this many-body system has been investigated. In order to study the time evolution, Lewis-Riesenfeld dynamical invariant and time evolution operator method have been used. Appropriate dynamical invariants are constructed and their Eigenvalues are derived as well as appropriate time evolution operators are constructed. These calculations have been done in general form so there are no limiting assumptions on changing of time-dependent functions.