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.
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.
We derive and explain the key ideas behind a time-dependent formulation of quantum scattering theory, applicable generally to systems with a finite-range scattering potential. The scattering is initiated and probed by plane wave packets, which are localized just outside the range of the potential. The asymptotic limits of conventional scattering theory (initiation in the remote past; detection in the remote future) are not taken. Instead, the differential cross section (DCS) is obtained by projecting the scattered wave packet onto the probe plane wave packets. The projection also yields a time-dependent version of the DCS. Cuts through the wave packet, just as it exits the scattering potential, yield time-dependent and time-independent angular distributions that give a close-up picture of the scattering which complements the DCS. We have previously applied the theory to interpret experimental cross sections of chemical reactions [e.g., S. C. Althorpe, F. Fern\'andez-Alonso, B. D. Bean, J. D. Ayers, A. E. Pomerantz, R. N. Zare, and E. Wrede, Nature (London) 416, 67 (2002)]. This paper gives the derivation of the theory, and explains its relation to conventional scattering theory. For clarity, the derivation is restricted to spherical-particle scattering, though it may readily be extended to general multichannel systems. We illustrate the theory using a simple application to hard-sphere scattering.
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.
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.
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.
A new variant of the packet analysis to solve the tunneling time problem for the so-called completed elastic scattering (a potential barrier of an arbitrary form is located in the finite spatial interval) is presented. In particular, we propose an explanation of the "paradoxes" of wave packet tunneling, and elaborate a rule enabling one to extract the tunneling times for a particle from the analysis of tunneling the corresponding wave packet. It is shown that the quantum ensembles of transmitted and reflected particles are statistically independent not only after the scattering but also before this event. We find the corresponding counterparts for the transmitted and reflected packets, which describe separately both scattering channels at early times, and define the expectation values of delay times for each channel. Besides, to describe jointly all particles at the stage when they cannot be divided into two independent subensembles, we find the scattering time taking into account broadening the packets. We derive the condition to be satisfied for the completed scattering. The times above are defined in terms of the well-known quantum-mechanical averages for the position and wave-number operators.
In the scattering of three free incident particles, a finite cross section for the double scattering has to be defined differently from the processes other than the double scattering. That is, we want to maintain that, in order to obtain a definite quantity being connected to the experimental observation, a particular definition of the cross section has to be required for the double scattering owing to a particular time dependence of its scattering rate. Our definition differs from the ones given recently by others in a form depending on some macroscopic quantities.1), 2) In the previous paper (refered as I in the following)3), we have attained to a finite cross section by working on the basis of the wave packet formalism, while we want to give here a clear-cut derivation in a simple and familear form.