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arXiv:1701.00675 · Journal of Physics A Mathematical and Theoretical

Delay-time distribution in the scattering of time-narrow wave packets. (I)

Uzy Smilansky

Published Apr 12, 2017• 9 citations
arXivHTMLPDF

Abstract

Abstract This is the first of two subsequent publications where the probability distribution of delay-times in scattering of wave packets is discussed. The probability distribution is expressed in terms of the on-shell scattering matrix, the dispersion relation of the scattered beam and the wave packet envelope. In the monochromatic limit (poor time resolution) the mean delay-time coincides with the expression derived by Eisenbud and Wigner and generalized by Smith more than half a century ago. In the opposite limit, and within the semi-classical approximation, the resulting distribution coincides with the result obtained using classical mechanics or geometrical optics. The general expression interpolates smoothly between the two extremes. An application for the scattering of electromagnetic waves in networks of RF transmission lines will be discussed in the next paper to illustrate the method in an experimentally relevant context.

DOI
10.1088/1751-8121/aa6cc8

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Abstract This is the first of two subsequent publications where the probability distribution of delay-times in scattering of wave packets is discussed. The probability distribution is expressed in terms of the on-shell scattering matrix, the dispersion relation of the scattered beam and the wave packet envelope. In the monochromatic limit (poor time resolution) the mean delay-time coincides with the expression derived by Eisenbud and Wigner and generalized by Smith more than half a century ago. In the opposite limit, and within the semi-classical approximation, the resulting distribution coincides with the result obtained using classical mechanics or geometrical optics. The general expression interpolates smoothly between the two extremes. An application for the scattering of electromagnetic waves in networks of RF transmission lines will be discussed in the next paper to illustrate the method in an experimentally relevant context.
Abstract We apply the framework developed in the preceding paper in this series (Smilansky 2017 J. Phys. A: Math. Theor . 50 215301) to compute the time-delay distribution in the scattering of ultra short radio frequency pulses on complex networks of transmission lines which are modeled by metric (quantum) graphs. We consider wave packets which are centered at high wave number and comprise many energy levels. In the limit of pulses of very short duration we compute upper and lower bounds to the actual time-delay distribution of the radiation emerging from the network using a simplified problem where time is replaced by the discrete count of vertex-scattering events. The classical limit of the time-delay distribution is also discussed and we show that for finite networks it decays exponentially, with a decay constant which depends on the graph connectivity and the distribution of its edge lengths. We illustrate and apply our theory to a simple model graph where an algebraic decay of the quantum time-delay distribution is established.
The theory of radiation from a time-dependent state (wave packet) is developed, using Heitler's method. It is first applied to a two-component wave packet, and general expressions for the radiation rate and the spectral distribution are derived. They are evaluated to lowest order in the perturbing Hamiltonian, and a number of special cases are considered. The theory is then generalized to an N-component wave packet, and formulas for the radiation rate and spectral distribution are derived. The latter is briefly discussed within the context of an approximation method. The general conclusions are: the radiation rate from a wave packet has a periodic component; energy nonconserving transitions appear; interference terms tend to distort the Breit—Wigner shape of the spectral distribution.
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-body scattering is analyzed using a time-dependent formalism and wave packets of arbitrary shape as the initial wavefunctions. In particular, the case of single-channel resonant scattering is discussed in some detail. In addition to the usual amplitudes resulting from the potential term and the resonant term, a third amplitude is found which interferes with the other two. Another interesting result is that in this formalism the final state is not necessarily represented by spherical outgoing waves, but can be interpreted instead as two separate outgoing wave packets whose shape is different from the initial packets. The Wigner time delay is obtained as a natural consequence of the analysis, and the resonant cross sections are expressed as density matrices constructed from the wave packets.
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.