Abstract Potential scattering is the scattering of two collision partners whose mutual interaction is described by a static potential. This potential depends in general on the structure of the collision partners, in particular, on their size and shape (usually taken as spherical), but once the potential is specified and the centre-of-mass motion is separated out, the scattering problem in the remaining relative coordinate reduces to the description of the motion of a point-particle with a mass equal to the reduced mass of the two collision partners in the static field of the interaction potential. Only elastic scattering can be described in this way.
Abstract Potential scattering is the scattering of two collision partners whose mutual interaction is described by a static potential. This potential depends in general on the structure of the collision partners, in particular, on their size and shape (usually taken as spherical), but once the potential is specified and the centre-of-mass motion is separated out, the scattering problem in the remaining relative coordinate reduces to the description of the motion of a point-particle with a mass equal to the reduced mass of the two collision partners in the static field of the interaction potential. Only elastic scattering can be described in this way.
Aims and problems of quantum scattering theory Scattering theory is the branch of physics concerned with interacting systems on a scale of time and/or distance that is large compared with the scale of the interaction itself, and it provides the most powerful tool for studying the microscopic nature of the world. In quantum mechanics, a typical scattering process involves a beam of particles prepared with a given energy and with a more or less defined linear momentum. One then studies either the collision with a similar wave packet, or the collision of the given wave packet against a fixed target. In particular, for a two-body elastic scattering problem with conservative forces, we work in the centre of mass frame, hence reducing the original problem to the analysis of a particle in an external field of forces. Indeed, in Section 7.2 we already used such a technique in the investigation of the hydrogen atom. The crucial difference with respect to Section 7.2 is that, in scattering problems, the continuous spectrum and its perturbations are studied, whereas the Balmer formula has to do with bound states. In the following section we outline the basic aspects of what is called time-dependent scattering. Time-dependent scattering The physical situation we would like to study is as follows: a beam of particles is prepared in a state that is approximately free and allowed to evolve towards the scatterer. After the collision we detect the scattered beam far away from the interaction region.
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.
In this chapter we discuss the quantum theory of scattering of incident particles from the target where an incident beam has point particles, scattering is elastic, the scattering potential is time independent, there is no multiple scattering, and the interaction between an incident particle and a scattering center is described by the potential that depends only on relative positions of particles. Scattering cross-section and frame of reference, asymptotic expansion of wave function and formal expression for scattering amplitude, partial wave analysis, integral equation, and the Born approximation are covered in the chapter, in a broader sense. Free particle and asymptotic solutions, scattering amplitude and phase shift, optical theorem, scattering length, phase shift and its interpretation, scattering by a square well potential and by hard sphere potential, and interpretation of the phase shift and its expression are explained with the use of partial wave analysis. Scattering from bare and screened potentials within the Born approximation, validity of the Born approximation, and the transformation from the center of a mass coordinate system to a laboratory coordinate system are written towards the end of the chapter.
The concept of a generalized scattering amplitude is introduced and applied to quantum potential scatterings. After an outgoing spherical wave is factored out from a wave function, the remaining wave envelope is defined as the generalized scattering amplitude. The transformed Schr\"odinger equation, in terms of the generalized scattering amplitude, can be solved numerically by using a finite-difference method over the entire scattering domain. Example problems of scatterings by a spherical potential well and the (${\mathit{e}}^{\mathrm{\ensuremath{-}}}$,H) static field interaction potential are solved to validate the theoretical formulation and numerical method. The far-field solutions for the ordinary scattering amplitude and differential cross section agree very well with those obtained from the partial-wave analysis. The radial profiles of the generalized scattering amplitude and particle density function over the entire scattering region are also presented, and their properties are discussed. These results demonstrate that the method can be used to yield a complete and accurate solution to scattering problems involving arbitrary interaction potentials.
Abstract In the preceding chapters we have given a detailed discussion of quaternionic scattering theory for the case of a single particle incident on a fixed external scattering potential. In complex quantum mechanics the external potential problem is of course only an idealized model, obtained from the physically more realistic case of multiparticle scattering when all particles except the incoming projectile are very heavy, or, in the case of the space-translationinvariant two-body problem, as the exact formulation of the internal motion after separation of the center of mass motion.
This chapter studies applications drawn from scattering theory. A powerful and commonly used way to explore the interaction between particles is to study the way they scatter off each other. In the scattering problems considered here, motions are non-relativistic and particles are conserved: two particles move together, interact, and then move apart again. It is assumed that the range of the interaction is finite, so when the particles are well separated they move freely. In a scattering experiment, one imagines that the particles approach each other as wave packets with fairly definite momenta and positions. The motion is initially free, because the particles are separated by great distances compared to the range of their interaction. As the wave packets move together, the particles interact through a potential V that is some function of the particle separation. The wave packets then move apart in a scattering pattern that is determined by the interaction potential. The chapter simplifies the partial wave analysis by concentrating on s-wave scattering; this allows an easy treatment of interesting effects such as resonances and absorption.