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Physical Review A

Nonstationary theory of wave-packet potential scattering

Dmitri Bitouk, Mikhail Vladimirovich Fedorov

Published Aug 1, 1998• 4 citations
Publisher

Abstract

Potential scattering of free-electron wave packets is considered in the framework of the nonstationary quantum-mechanical theory. To characterize the efficiency of this process, the average angle of scattering is calculated in the lowest (first and second) orders of perturbation theory in the potential. In the approximation of a well-localized, narrow, and nonspreading wave packet, the first-order term is shown to agree with the classical angle of scattering, linear in the potential. This term is shown to vanish at a very large size of the initial wave packet (much larger than the size of a target). An analogy between the potential scattering of wave packets and electron-light ponderomotive scattering is discussed.

DOI
10.1103/physreva.58.1195

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Potential scattering of free-electron wave packets is considered in the framework of the nonstationary quantum-mechanical theory. To characterize the efficiency of this process, the average angle of scattering is calculated in the lowest (first and second) orders of perturbation theory in the potential. In the approximation of a well-localized, narrow, and nonspreading wave packet, the first-order term is shown to agree with the classical angle of scattering, linear in the potential. This term is shown to vanish at a very large size of the initial wave packet (much larger than the size of a target). An analogy between the potential scattering of wave packets and electron-light ponderomotive scattering is discussed.
Scattering of electron wave packets by large-size targets is considered. In the ``no-spreading'' approximation the general expression for the electron distribution function after scattering is found. An alternative regime of scattering, different from both the first-order Born approximation and classical limit, is found to exist in the case of narrow (compared to a target) nonspreading wave packets. In this regime, the average angle of scattering is shown to differ from the classical one and, counterintuitively, the difference is shown to be greater, the narrower the wave packets are. Physical reasons of this effect are discussed.
Investigation of the large angle scattering behaviour of the narrow wave packet approximation is made. First an expression is derived for the scattering amplitude in the narrow wave packet approximation when the potential is arbitrary. Then the same expression is derived for the case when a potential is spherically symmetric. In many aspects the approximation behaves as Jeffrey's approximation. Since the narrow wave packet approximation breaks down in the vicinity of the singularity of a potential, the validity of the approximation for the case when the potential is short range and has a singularity near the origin is investigated.
Two methodological troubles 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+da+b\to c+da+b→c+d. In order to calculate the cross section one uses the matrix element $$ of the SSS-matrix. The element is proportional to δδδ-function expressing the energy conservation. The corresponding probability ∣<cd∣S∣ab>∣2|< cd|S|ab>|^2∣<cd∣S∣ab>∣2 contains δ2δ^2δ2 which is mathematically senseless. The known regular way to overcome the difficulty seems to be unsatisfactory. In this paper, both the troubles are resolved using wave packets of incident particles.
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.
We discuss the propagation dynamics of nonspreading wave packets. We decompose the Hamiltonian into two parts. The first part is such that wave packets is its instantaneous eigenstate and is therefore irrelevant to the propagation of the packet. The second part is shown to be the effective Hamiltonian governing the motion of the packet both classically and quantum mechanically. Thus, analogous to Ehrenfest's theorem, nonspreading wave packets offer another view point directly connecting quantum mechanics and classical mechanics. This analysis also works for non-square-integrable packets, such as Airy packets.