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Journal of Physics A Mathematical and General

Solvable model for quantum wavepacket scattering in one dimension

Juan Gonzalo Muga, José P. Palao

Published Nov 27, 1998• 15 citations
Publisher

Abstract

Quantum wavepacket collisions of a Lorentzian state with a separable potential are described analytically. This model problem illustrates many different aspects of time-dependent scattering: in particular, the effects associated with resonance poles (time delay, exponential decay) and other critical points in the complex momentum plane, the effects of complex absorbing potentials, as well as some unique aspects of short- and long-time behaviour.

DOI
10.1088/0305-4470/31/47/012

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Quantum wavepacket collisions of a Lorentzian state with a separable potential are described analytically. This model problem illustrates many different aspects of time-dependent scattering: in particular, the effects associated with resonance poles (time delay, exponential decay) and other critical points in the complex momentum plane, the effects of complex absorbing potentials, as well as some unique aspects of short- and long-time behaviour.
Delay time is usually discussed in terms of a wavepacket going through a potential barrier and its time of reappearance on the other side of it, as compared with the corresponding problem in free motion. Wigner stressed though that an interaction should be described by an matrix and a particular case would be that of a single resonance. Long ago a dynamical model was developed for this case in which the interaction takes place at a point through boundary conditions. It is used here to discuss the passage to x > 0 of a plane wave initially constrained to the interval . Comparing our result with the corresponding problem of free-particle motion, at the classical time of flight of the particles, we obtain information on delay times for wide and narrow resonances. Note that our approach is different from the one using stationary states, i.e. when the time , originally introduced by Wigner. For completeness we also discuss the latter for our problem in the concluding section and compare the results with those obtained in this paper.
We consider a very straightforward extension of the usual self-adjoint techniques to solve the quantum mechanical problem of contact interaction potentials (i.e., infinite walls with different boundary conditions and point interactions) moving with constant velocity. From these calculations we obtain general expressions that describe the scattering of wavepackets. Furthermore, we also discuss the cases where the potentials oscillate with constant velocity or have time dependent strengths. Finally, we show, with an explicit example, that the prescription used is not the only possibility to implement self-adjoint extensions for contact interactions. Thus, in principle different models can be realized from such moving systems. Throughout the work we propose some simple physical applications of our results.
Scattering theory studies the behaviour of (quantum) systems over time and length scales which are very large, compared with the time or length that are characteristic of the interaction which affects them. The chapter begins with an outline of the basic problems of scattering theory, i.e. the existence of scattering states, their uniqueness, weak asymptotic completeness the existence of the scattering transformation, S , the reduction of S due to symmetries, the analyticity of S , asymptotic completeness and the existence of wave operators. After considering the Schrödinger equation for stationary states, the integral equation for scattering problems is derived and studied. The Born approximation and the Born series are presented, and the conditions which ensure convergence of the Born series are also discussed. Further topics are partial wave expansion, its convergence and the uniqueness of the solution satisfying the asymptotic condition, the Levinson theorem. Lastly, in the first part of the chapter, scattering theory from singular potentials is introduced, with emphasis on the polydromy of the wave function, following early work in the literature, and the general problems in the theory of resonances are studied. In the second part, we examine in detail a separable potential model, the occurrence of bound states in the completeness relationship, an excitable potential model and the unitarity of the Möller wave operator. Lastly, we study the survival amplitude associated with quantum decay transitions. Aims and problems of scattering theory Scattering theory is the branch of physics that is concerned with interacting systems on a scale of time and/or distance that is large compared with the scale of the interaction itself, and it provides the most powerful tool for studying the microscopic nature of the world.
One-dimensional time-independent scattering problems are investigated in the framework of the quantum Hamilton-Jacobi formalism. The equation for the local approximate quantum trajectories near the stagnation point of the quantum momentum function is derived, and the first derivative of the quantum momentum function is related to the local structure of quantum trajectories. Exact complex quantum trajectories are determined for two examples by numerically integrating the equations of motion. For the soft potential step, some particles penetrate into the nonclassical region, and then turn back to the reflection region. For the barrier scattering problem, quantum trajectories may spiral into the attractors or from the repellers in the barrier region. Although the classical potentials extended to complex space show different pole structures for each problem, the quantum potentials present the same second-order pole structure in the reflection region. This paper not only analyzes complex quantum trajectories and the total potentials for these examples but also demonstrates general properties and similar structures of the complex quantum trajectories and the quantum potentials for one-dimensional time-independent scattering problems.