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.
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.
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.
This chapter presents a descriptive introduction to the field of collision phenomena from the theoretical point of view, with the primary purpose of enumerating the fundamental themes of research as well as the relevant basic physical aspects. In the nonstationary treatment, collision is understood as a physical phenomenon which essentially develops in several time stages. Collisions between elementary particles are beyond the primary scope of the present work, because they would require relativistic quantum mechanics, considering, e.g., a target as an elementary particle, built from mesons, quarks, etc. Asymptotic convergence of scattering states is the most essential characteristic of the collision problem. This sole characteristic makes the collision phenomenon substantially different from the problem of finding bound states of the examined physical system for certain given interactions among the constituents.
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.
Some results of the the dependent formalism in scattering theory are presented. The asymptotic condition with an observable algebra A acting on a small Hilbert space, h, is imposed. H is given by H = H/sub 0/ + V, where H/sub 0/ is the free Hamiltonian on h and V the potential present. The demonstrations given are based on the convergence properties of the spectral measurements associated with the vectors, exp(-iHt) psi, psi being the scattering state. It is shown thnt the OMEGA /sup plus or minus / operators plus or minus infinity), where g belongs to a group of dense h vectors and where T/sub t/ is a family of limited operators in A. A necessary and sufficient condition is given for OMEGA /sup plus or minus / to exist and to permit a choice of T/sub t/ unitary operators to be made. (auth)
Abstract In this chapter we explain how particle scattering can be described by local observables. First we introduce the concept of the S-matrix, which is an idea common to classical and quantum mechanics. We show the construction of incoming and outgoing states which have a certain asymptotic behaviour at time −∞ and +∞, and we also show that the constructed states have an acceptable physical asymptotic behaviour. Then we derive the representation of the S-matrix given by these scattering states and discuss its properties.