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Communications in Theoretical Physics

A New Formula to Obtain Exact Green's Functions of Time-Dependent Schrödinger Equation

Axel Schulze‐Halberg

Published May 15, 2004• 1 citation
Publisher

Abstract

We obtain a new relation between Green's functions of the time-dependent Schrödinger equation for stationary potentials and Green's functions of the same equation for certain time-dependent potentials. The relation obtained here emerges very easily from a transformation introduced by Ray [J.R. Ray, Phys. Rev. A26 (1982) 729] and generalizes former work of Dodonov, et al. [V.V. Dodonov, V.I. Man'ko, and D.E. Nikonov, Phys. Lett. A162 (1992) 359.]

DOI
10.1088/0253-6102/41/5/723

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Time-Dependent Perturbation Theory

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In all the applications of the formalism of quantum theory, we have considered mainly time-independent potential functions, V ( r , t ) = V ( r ). Such potentials made it easier to solve exactly the Schrodinger's equation or by solving the Schrodinger's equation approximately by using perturbation theory. The solution of the Schrodinger equation for time-independent potentials is obtained by using method of separable variables. The solution of time-independent Schrodinger gives stationary states, and the time dependence of the wave function is obtained by adding a phase factor to the stationary state. The schematic that follows describes such an approach ( Figure 7.1 ).

CRC Press eBooks•Oct 10, 2018•1 citation
We obtain a new relation between Green's functions of the time-dependent Schrödinger equation for stationary potentials and Green's functions of the same equation for certain time-dependent potentials. The relation obtained here emerges very easily from a transformation introduced by Ray [J.R. Ray, Phys. Rev. A26 (1982) 729] and generalizes former work of Dodonov, et al. [V.V. Dodonov, V.I. Man'ko, and D.E. Nikonov, Phys. Lett. A162 (1992) 359.]
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 all the applications of the formalism of quantum theory, we have considered mainly time-independent potential functions, V ( r , t ) = V ( r ). Such potentials made it easier to solve exactly the Schrodinger's equation or by solving the Schrodinger's equation approximately by using perturbation theory. The solution of the Schrodinger equation for time-independent potentials is obtained by using method of separable variables. The solution of time-independent Schrodinger gives stationary states, and the time dependence of the wave function is obtained by adding a phase factor to the stationary state. The schematic that follows describes such an approach ( Figure 7.1 ).