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Journal of Physics A Mathematical and General

Classical properties of quantum scattering

Władysław Żakowicz

Published Apr 2, 2003• 7 citations
Publisher

Abstract

Quantum elastic potential scattering of a particle is re-examined taking into account exact solutions of the corresponding Schrödinger equation. In addition to the scattering of stationary plane waves and stationary finite-width wave beams, nonstationary wave packets having finite duration times are studied and some corresponding examples are presented. The role of interference between the scattered wave and the advancing incident beam is studied. Several two-dimensional scattering problems, involving axially symmetric, generic examples of nonuniform attractive and repulsive potentials, are discussed in more detail. This discussion concentrates on finding proper conditions when the solutions of the Schrödinger equation may resemble the corresponding solutions of the classical Newton equation. Examples are shown where such similarities occur.

DOI
10.1088/0305-4470/36/15/316

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Quantum elastic potential scattering of a particle is re-examined taking into account exact solutions of the corresponding Schrödinger equation. In addition to the scattering of stationary plane waves and stationary finite-width wave beams, nonstationary wave packets having finite duration times are studied and some corresponding examples are presented. The role of interference between the scattered wave and the advancing incident beam is studied. Several two-dimensional scattering problems, involving axially symmetric, generic examples of nonuniform attractive and repulsive potentials, are discussed in more detail. This discussion concentrates on finding proper conditions when the solutions of the Schrödinger equation may resemble the corresponding solutions of the classical Newton equation. Examples are shown where such similarities occur.
Time-dependent scattering of finite duration wave packets on a rigid cylinder is studied using exact solutions of the corresponding Schrodinger equation. The similarity between the time evolution of the total wavefunction and evolution of the corresponding classical ensemble of particles is exposed. Special attention is paid to a discussion of the differences between the total wavefunction dynamics and the dynamics of the scattered part of the wavefunction, the part which is commonly used for the evaluation of scattering cross-sections. This discussion emphasizes the necessity of using the total wavefunction in the description of scattering.
Two different formulations of the problem of time-delay caused by scattering from an oscillating center of force are studied. In the first one, appropriate for local interactions, a semi-classical time-dependent solution of the Schrödinger equation is obtained by writing the wave function as a solution of two coupled equations one of which is the Hamilton–Jacobi equation. These equations are then solved using the narrow wave packet approximation. The classical motion of the center of this wave packet is found for two specific cases, a square well and a Morse potential. In the second formulation a time-independent many-channel solution of the Schrödinger equation is obtained for an exactly solvable quantum-mechanical problem with a potential consisting of a separable attractive part which oscillates and a fixed rigid wall. The results show that the time-delay is extremely sensitive to the values of the amplitude and the frequency of oscillations, and also to the energy of the incident particle.
Two methodological difficulties of the quantum theory of collisions are considered. The first is the undesirable interference of the incident and scattered waves in the stationary approach to scattering. The second concerns the nonstationary approach to the theory of collisions of the type a + b → c + d. In order to calculate the cross section one uses the matrix element 〈cd|S|ab〉 of the S-matrix. The element is proportional to δ-function expressing the energy conservation. The corresponding probability |〈cd|S|ab〉|2 contains δ2, which is mathematically meaningless. The known regular way to overcome the difficulty seems to be unsatisfactory. In this paper, both the problems are resolved using wave packets of incident particles.
In this chapter, the Schrödinger equation is “derived” for particles that can be described by de Broglie waves. The Schrödinger equation is very different from the corresponding equation of motion in classical mechanics. In order to illustrate the fundamental differences between the two theories, one of the simplest problems of particle dynamics is solved in both Newtonian and quantum mechanics. This simple example also helps to show that quantum mechanics is the fundamental theory and classical mechanics is an approximation, a remarkably good approximation, when considering macroscopic objects. The solution of the Schrödinger equation is presented for a particle inside a box and the quantization condition is derived. The amazing possibility of quantum tunneling when a particle is incident on a barrier of height larger than the energy of the incident particle is also discussed. Finally the three-dimensional Schrödinger equation is solved for the hydrogen atom.
The scattering of a particle wave packet by a static potential of finite range is considered. In such an interaction, the incident packet enters the potential region, is scattered, and departs, changed in a way which depends on the nature and strength of the interaction. Intuitively there is a region of space and an interval of time in which the interaction occurs. With the aid of the Schr\"odinger equation, this situation is discussed, and a function is derived which can be interpreted as describing the distribution of the interaction in space and time. The properties of the function are investigated, and arguments are given which support this interpretation. With the distribution, the space-time properties of the interaction are explored. Finally, it it is suggested that the distribution function, appropriately extended, may be a useful tool in studying other aspects of particle interactions.
In this paper we reexamine, and analyze solutions to, the recently derived time-independent wave packet-Schrödinger (TIWS) and time-independent wave packet-Lippmann–Schwinger (TIWLS) equations. These equations are so named because they are inhomogeneous, with the inhomogeneity being the initial L2 wave packet from an underlying time-dependent treatment of the dynamics. We explicitly show that a particular solution of the homogeneous Schrödinger equation can be constructed out of two particular solutions of the inhomogeneous TIW equation satisfying causal and anticausal boundary conditions. The structure of this solution of the homogeneous equation is shown to depend sensitively on the nature of the initial wave packet inhomogeneity, but, as we demonstrate, correct scattering information can be obtained even when the initial wave packet is nonzero only in the target region. It thus becomes possible to carry out quantum scattering calculations in which one need not propagate the wave packet from the noninteracting to the interacting region. The method is illustrated by calculations for two 1-D scattering problems, namely the transmission of an electron through a single barrier and through a double barrier. The latter is especially challenging because of the occurrence of long-lived resonances with the electron trapped inside the double barrier. In addition, we show that the method can still make use of absorbing potentials to decrease grid size, and we show how the formalism can be used to treat inelastic and reactive scattering, and radial scattering variables, thereby constituting a general approach to time-independent wave packet quantum scattering.