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Progress of Theoretical Physics Supplement

The Wave Packet Interpretation of the Scattering

Tatuya Sasakawa

Published Jan 1, 1959• 8 citations

Abstract

In the present paper, we see the manner of the propagation of the wave packet in the presence of a scatterer and we see the form of the scattered packet. The form of the scattered packet at the detector is closely related to the form of the energy spectra. The general form of the energy spectra is given in connection with the wave packet nature of the scattered beam. Of course, this energy spectra tends to the conventional Lorentzian type in the stationary limit. In Part I of the present paper, the relation between the form of the scattered packet and the cross section is discussed in detail. In Part II, we see the relation between the average procedure of the cross section and the optical model, in terms of the wave packet. In general, the scattered packet draggles the tail of life 1/Γ, Γ being the level width, as well as fluctuations appearing in the transient state. If we average the cross section over many sharp resonances, these fluctuations and the tail in the scattered packet are washed out. Then, it seems as if a part of the initial beam were absorbed by this washing procedure. The remaining part is the shape elastic scattering and the dust which is washed out constitutes the compound elastic scattering. If the compound system is excited to organize the single particle mode as well as the compound nucleus mode, the fluctuation and the tail characteristic of the single particle mode is not washed out by the average procedure, provided that the energy range of the procedure is much smaller than the single particle level distance. Then in the shape elastic scattering obtained after the average procedure, only the feature of the single particle mode with the partial seeming absorption remains. This accounts for the optical model.

DOI: 10.1143/ptps.11.69

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In the present paper, we see the manner of the propagation of the wave packet in the presence of a scatterer and we see the form of the scattered packet. The form of the scattered packet at the detector is closely related to the form of the energy spectra. The general form of the energy spectra is given in connection with the wave packet nature of the scattered beam. Of course, this energy spectra tends to the conventional Lorentzian type in the stationary limit. In Part I of the present paper, the relation between the form of the scattered packet and the cross section is discussed in detail. In Part II, we see the relation between the average procedure of the cross section and the optical model, in terms of the wave packet. In general, the scattered packet draggles the tail of life 1/Γ, Γ being the level width, as well as fluctuations appearing in the transient state. If we average the cross section over many sharp resonances, these fluctuations and the tail in the scattered packet are washed out. Then, it seems as if a part of the initial beam were absorbed by this washing procedure. The remaining part is the shape elastic scattering and the dust which is washed out constitutes the compound elastic scattering. If the compound system is excited to organize the single particle mode as well as the compound nucleus mode, the fluctuation and the tail characteristic of the single particle mode is not washed out by the average procedure, provided that the energy range of the procedure is much smaller than the single particle level distance. Then in the shape elastic scattering obtained after the average procedure, only the feature of the single particle mode with the partial seeming absorption remains. This accounts for the optical model.

The elastic scattering of spinless particles from a spherically symmetric stationary target is considered. The main idea is to describe the incident beam of particles as a certain statistical ensemble of one-particle wave packets, and to define the differential cross section in terms of the average scattering of the particles in this ensemble. The physical significance of this mode of description is discussed, and arguments are presented as to why it should be regarded as a realistic description of typical scattering experiments. The discussion is general, and not restricted to potential scattering. It is based on the notion of the S matrix, and the most basic ideas of the S matrix are discussed. A simple derivation of the partial wave expansion is given within the framework of S matrix theory. The aim of this paper is mainly pedagogical, and the final results obtained are old and well established.

A wave-packet formalism is presented for obtaining the differential cross section for the elastic scattering of a single particle by a potential. The method avoids difficulties inherent in the more common formulations such as interference between incident and scattered waves, box normalization, and density of final states. The time sequence of events in the scattering process is pictured in an intuitively appealing manner. The initial form of the wave packet is specified explicitly, and the time development of the wave packet is determined from the time-translation operator. The asymptotic form of the wave packet is investigated after a sufficiently long time lapse, and the cross section is obtained from a calculation of particle flux densities.

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.

Chapter 18 is entirely devoted to the quantum theory of scattering, which is normally covered in more detail in a course on quantum field theory. We introduce the main concepts, including scattering amplitude, differential and total cross sections, and form factors, and derive the main formulas for elastic scattering, in a nonrelativistic framework; this allows us to appreciate how the information obtained from scattering experiments is used to explore the intricacies of quantum particles that are otherwise inaccessible. The Born approximation is studied and applied to scattering by a periodic potential, a weak potential, and Rutherford scattering. The partial wave expansion is derived and applied to obtain the optical theorem. The chapter concludes with a complementary section on resonant scattering.

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.