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.
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.
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.
Scattering theory studies the behaviour of (quantum) systems over time and length scales which are very large, compared with the time or length that are characteristic of the interaction which affects them. The chapter begins with an outline of the basic problems of scattering theory, i.e. the existence of scattering states, their uniqueness, weak asymptotic completeness the existence of the scattering transformation, S , the reduction of S due to symmetries, the analyticity of S , asymptotic completeness and the existence of wave operators. After considering the Schrödinger equation for stationary states, the integral equation for scattering problems is derived and studied. The Born approximation and the Born series are presented, and the conditions which ensure convergence of the Born series are also discussed. Further topics are partial wave expansion, its convergence and the uniqueness of the solution satisfying the asymptotic condition, the Levinson theorem. Lastly, in the first part of the chapter, scattering theory from singular potentials is introduced, with emphasis on the polydromy of the wave function, following early work in the literature, and the general problems in the theory of resonances are studied. In the second part, we examine in detail a separable potential model, the occurrence of bound states in the completeness relationship, an excitable potential model and the unitarity of the Möller wave operator. Lastly, we study the survival amplitude associated with quantum decay transitions. Aims and problems of scattering theory Scattering theory is the branch of physics that is 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 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.
As we have seen in Section 8.4, the basis of our perception of an objective reality is statistical mechanics and field theory. These are routinely used in equilibrium and nonequilibrium statistical thermodynamics (see, for example, Calzetta & Hu [59]), which govern the macroscopic laws of everyday life. Thus the quantum theory of fields must be the fundamental description of Nature. Therefore, the simpler quantum mechanics of particles is necessarily a derived description. This chapter discusses the extent to which a particle picture of matter and radiation is appropriate. In relativistic quantum field theory, particles appear only as asymptotic excitations of the fields. The quantum particle concept therefore makes mathematical (and intuitive) sense only under very special circumstances, namely in those where a system actually behaves like particles do: - As quasiparticles, they are useful in the domain, where the geometric optics perspective applies. - Before and after a scattering event in a dilute gas or a particle collider ring, particles can be considered as being essentially free. During the very short interaction time itself, the scattering process cannot be described by a particle picture, but needs a hydromechanical description or a kinetic description via Kadanoff-Baym equations. Already at the very beginning of modern quantum mechanics, when Born found his probability interpretation, the scattering process of single particles in an external classical field was understood (see Section 14.1 of the Appendix) to be a wave process relating initially and finally stationary particles, freely moving in their center of mass frame. It is a historical accident that one continues to use the name particle in the many microscopic situations, where it is grossly inappropriate to think of it with the classical meaning of a tiny bullet moving through space. When we restrict the use of the particle concept to where it is appropriate, or when we do not think of particles as “objects”, in both cases all mystery is gone, and the foundations become fully rational. The inappropriate focus on the particle aspect of quantum mechanics created the appearance of mystery; common sense is restored by focussing instead on the field aspect. To substantiate the above summary view, we give in Section 12.1 an analysis of the photoelectric effect and in Section 12.2 a review of the origin of particle tracks. This leads in Section 12.3 to a discussion about the reality of particles. In the remainder of the chapter, we discuss the relation between fields and particles in more detail. Section 12.4 poses the problem how the nonrelativistic multiparticle description arises from relativistic quantum field theory. Section 12.5 considers the case of free quantum fields, and Section 12.6 the case of interacting quantum fields. Section 12.7, the final section, discusses the semiclassical view of quasiparticles in the approximation familiar from geometric optics.
This chapter analyses the problem of the asymptotic convergence of scattering states for a collision of two structureless particles. For scattering theory, convergence of state vectors and operators is required to be strong, as a more stringent condition than weak convergence. In non-relativistic collisions, the velocity of the scattering particles is negligible with respect to the speed of light c. The accomplished analysis is of direct use for studying the asymptotic convergence of scattering states. An adequate physical interpretation of the obtained results is in order. The asymptotically free scattering states, which are introduced in an unambiguous manner, will become of decisive importance for a proper physical interpretation of the obtained results. The obtained strong limit of the quantum-mechanical free wave packet U 0 (t)χ represents a physically intuitive criterion for the asymptotic freedom, which encompasses the essential requirements imposed on the outgoing and incoming wave packets.