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CRC Press eBooks

Time-dependent scattering theory

Dževad Belkić

Published Jan 15, 2020• 1 citation
Publisher

Abstract

In time-dependent scattering theory, the collision problem is described by assuming that a particle was localized outside the interaction domain where V = 0. After collision, there exists a definite probability that the particle is deflected from its impact direction at a certain scattering angle. After collision, there exists a definite probability that the particle is deflected from its impact direction at a certain scattering angle. This probability cannot be directly observed in experiments but is indirectly accessible to measurements through differential cross sections. The unitarity of the operator S is one of the most important properties of the S-matrix theory of scattering. Of course, the study of the S-matrix will greatly be facilitated by the knowledge that the S-operator belongs to a narrower class of the unitary operators.

DOI
10.1201/9780429146497-11

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In time-dependent scattering theory, the collision problem is described by assuming that a particle was localized outside the interaction domain where V = 0. After collision, there exists a definite probability that the particle is deflected from its impact direction at a certain scattering angle. After collision, there exists a definite probability that the particle is deflected from its impact direction at a certain scattering angle. This probability cannot be directly observed in experiments but is indirectly accessible to measurements through differential cross sections. The unitarity of the operator S is one of the most important properties of the S-matrix theory of scattering. Of course, the study of the S-matrix will greatly be facilitated by the knowledge that the S-operator belongs to a narrower class of the unitary operators.
Stationary states are unavoidable from the practical viewpoint, since all the computations of observables, such as cross sections and other experimentally measurable physical quantities, are carried out within timeindependent scattering theory. In a scattering experiment, one attempts to determine the momentum of the impact projectile as precisely as possible. However, in the standard formulation of scattering theory, one tries to reach the same ideal goal by identifying the initial state in the remote past with the plane wave. The problem of the asymptotic convergence of scattering states cannot be formulated in the same way for both proper and improper state vectors. The optical theorem is a convenient way of computing the total cross section Q, when one knows the value of the imaginary part of the scattering amplitude in the forward direction. The scattered particles are moved away from the incident beam for a certain amount which is proportional to the value of the total cross section Q.
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.
For a single nonrelativistic particle moving in a spherically symmetric potential, the existence of the Hilbert space wave operators and S operator is proved and phase shift formulas for these operators are deduced. The probability, P(Ω), for scattering into the solid angle Ω is obtained from the time dependent theory. The relation between P(Ω) and the R matrix of the standard plane wave formulation of scattering theory is established. For collimated incoming packets, it is shown that P(Ω) can be expressed as an energy average of the differential cross section.
The Quantum Mechanical Theory of Collis3ions. I»-In the present course we shall consider the quantum mechanical treatment of problems of particle interactions in which at least one of the particles is unbound and therefore able to travel large distances freely.In effect such particles are able to communicate the results of their interaction directly to macroscopic detection apparatus* The interpretation of the resulting data is in general the most direct way of forming conclusions about the interactions.A considerable variety of physical phenomena involves unbound particles» Among the simplest are scattering processes ;in particular the deflection of particles from a collinated beam.More general collision processes which induce reactions of various sorts also lie within the class we shall examine, Our chief concern willbe with the means of solving problems in which well-defined models have been postulated to describe the basic interactions.We shall restrict ourselves, in fact, to the treatment of fairly simple models, and shall discuss a number of approaches to the problems they raise.We begin by discussing the scattering of particles by a simple potential field, fixed in space» The problem of two free particles with a potential between them of course assumes this form in the center-of-mass system.By imagining that a beam of particles of constant strenght is at all times being scattered by the potential, we cay seek to describe the situation with the stationary-state Schrb'dinger equation.
If the lifetime of a collision is defined as the difference between the times in which a particle moves in free space and under an interaction from a scattering center to the outside (or in the reverse direction), this paper shows that exactly the same relation between the lifetime operator Q and the scattering matrix S as was found by Smith (1960) can be derived. This can be done in a natural way provided that the time operator is defined as [Formula: see text] and the manifold of the asymptotic outgoing (or incoming) states is chosen to be the set of corresponding eigenstates. The part of the time needed for the particle moving freely from the scattering center to R is equal to R/v.
The problem of a quantum particle moving in a narrow, bent two-dimensional channel is used to illustrate features of collisions in systems with semi-constrained structures. The collision theory of such systems is developed and, under certain general restrictions on the geometry, the following properties are rigorously established: (1) The (localized) Møller wave operator exist and the S operator is unitary in the scattering channels. (2) The singular spectrum is discrete in intervals away from the continuum thresholds. The discussion is based on a time-dependent approach.