Exact Green's function for the step and square-barrier potentials.

The space-time propagator contains all the information about a quantum system. Although in several specific problems one is interested only in the spectrum and eigenfunctions, and these are more easily obtained by a direct solution of the time-independent Schrodinger equation, a number of very important questions have been raised recently in which the propagator enters in a crucial way. One of these questions concerns the problem of dissipation in quantum tunneling. The pioneering work of Caldeira and l.eggett [l] introduced a formalism in which a subsystem in a metastable state is coupled to a bath of oscillators and the equilibrium density operator is calculated as a path integral [2]. Taking the trace over the bath variables results in a reduced density for the subsystem from which the tunneling rate can be extracted in the semiclassical limit. A second approach, particularly suited to treat Brownian motion in the classically accessible region, was also discussed by the abovementioned authors [3]. In that case, the time-dependent propagator for the "universe" (subsystem plus oscillators) is written in terms of the uncoupled propagators using the Feynman-Vernon theory [4]. As emphasized in Ref. [3], this method cannot be applied directly to the study of tunneling through a barrier. The main reason for this restriction is that the time-dependent propagator for scattering (unbounded) problems is very hard to obtain. If this difficulty could be overcome the effects of dissipation in quantum tunneling could be studied in a very detailed fashion. The two simplest problems where these effects could be observed are the one-dimensional step and square-barrier potentials. However, although these are very simple systems, only very recently [5] the spacetime propagator for the first case was derived in terms of integrals of simpler propagators. Besides, as far as the author knows, the space-time propagator for the square barrier has not yet been computed. The space-time propagator K(x,x', t) is related to the space-energy propagator, or Green's function, G(x, x', E) by a Fourier transform. In this paper we show that, although the space-time representation of the Green's function for the step and space-barrier potentials may be very complicated, its space-energy representation can be calculated in closed (and very simple) form. In a recent paper [6] G(x, x', E) was partially computed for the square barrier and some semiclassical results concerning group velocity and tunneling time were derived. In this paper we present a complete and exact calculation of the space-energy Green's function for both the step and square-barrier potentials. We also generalize some of these results for general symmetric potentials of finite range. The expressions for G(x,x', E) turn out to be very simple and allow for interpretations in terms of classical paths. Therefore, the basic motivation for the present calculation is the possibility of coupling the square-barrier potential to other degrees of freedom, such as free particles, and eventually to study the limit of many such freedoms. This study, now under current investigation, has become possible due to the very simple structure of the Green's function. As a final remark we should point out that the use of square-barrier and square-well potentials as simple models for more realistic physical problems has also a long history in the theory of heterostructures in solidstate physics [7] and the development of techniques to perform time propagation of wave packets has become of great importance [8]. We discuss the advantages of using the Green's function as an intermediate step for the time propagation in the Conclusion. The calculations presented here are based on the spectral decomposition of G(x, x', E) They ar.e lengthy and sometimes tedious. Therefore, this paper is organized in such a way that the results and discussions come before the calculations. In Sec. II we review the basic definitions of time and energy representations of the Green's function and their connections. In Sec. III we compute the energy Green's function for the step potential. The results of this computation are given by Eqs. (3.3)—(3.6). In Sec. IV we do the same calculations for the square barrier, with results given by Eqs. (4.3)—(4.6). In Sec. V we generalize some results of Sec. IV for symmetric potentials of finite range and Sec. VI is devoted to some concluding remarks.