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Multiparticle and Multichannel Methods

Stephen L. Adler

Published Apr 27, 1995• 0 citations
Publisher

Abstract

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.

DOI
10.1093/oso/9780195066432.003.0009

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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.
Abstract We proceed in this chapter to discuss scattering theory and related aspects of the bound-state problem in quaternionic quantum mechanics. We begin with a detailed discussion of a simple one-dimensional example, the delta function potential model. From this model will emerge the result (Adler, 1988) that will be the principal theme of this and the succeeding two chapters: with the standard ray representative choice of Sec. 4.2, for short-range potentials the asymptotic scattering states and the S-matrix in quaternionic quantum mechanics are always complex ℂ(1, i).
Abstract Besides the bounded, periodic classical motion of particles in potentials as shown in Fig. 5.2 (for energy E1 ), there is also the possibility of unbound states which are not localized in space and not repetitive in time; these correspond to particles incident on the potential and which subsequently “bounce “ off (energy E2 in Fig. 5.2) or temporarily change their speeds as they go over the potential (energy E3 ). Classical mechanics, which solves for the exact trajectories in either case, makes little distinction between the two types of motion aside from some tech- nical details,
Abstract Just as in classical mechanics, the study of single particle systems in one dimen- sion provides invaluable experience in the use of quantum mechanical concepts and techniques. To extend these ideas to more realistic applications (to be able to enter the Quantum World of Part II) we also need to develop the formalisms necessary to handle multiparticle systems. In this chapter we will discuss separ- able systems (in Section 14.2), the important special case of two-body systems (Section 14.3), notation for spin-1/2 wavefunctions (Section 14.4), and lastly, in Section 14.5, the important constraints placed on quantum wavefunctions of multiparticle systems due to indistinguishability.
Most of our knowledge about the structure of matter and interaction between particles is derived from scattering experiments. From a theoretical point of view, scattering problems are concerned with the continuous (and positive) energy eigenvalues and unbound eigenfunctions of the Schro¨dinger equation. We have already encountered one-dimensional examples of scattering in Chapter 4, in the discussion of reflection and transmission of an incident particle with definite momentum from a potential step or barrier. In this chapter we consider scattering from a more formal point of view. We will confine our discussion to elastic scattering (scattering without loss or gain of energy by the projectile) from central potentials although many of the concepts and results are applicable to inelastic scattering.
The real Hilbert space formalism developed within the quaternionic quantum mechanics ([Formula: see text]) is fully applied to the simple model of the autonomous particle. This framework permits novel insights within the usual description of the complex autonomous particle, particularly concerning the energy of a non-stationary motion. Through the appraisal of the physical role played by a fully quaternionic scalar potential, an original self-interaction within the quaternionic autonomous particle has been determined as well. Scattering processes are considered to illustrate these novel features.
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.