The asymptotic condition: A qualitative look Suppose that we have prepared an initial quantum state |ψ at t = 0. We may view |ψ as a square integrable wavepacket ψ(0, x) = x|ψ .Assume that this wavepacket does not populate any bound states of the studied problem.According to the central assumption of the scattering theory, our wavepacket behaves for t = 0 as follows: As t increases from 0 to +∞, the wavepacket ψ(t, x) = x|ψ(t) tends to leave the interaction region (= region of x ∈ (a, b) with nonzero potential) and becomes thus freely evolving at very large positive times.[ Note that some portion of ψ(0, x) would generally escape to the right (x → +∞) and some portion to the left (x → -∞).] Similarly, as t decreases from 0 to -∞, the wavepacket ψ(t, x) = x|ψ(t) tends to leave the interaction region and becomes thus freely evolving at very large negative times.What we just described is the so called Asymptotic Condition (AC).It has crucial importance for all our subsequent elaborations.
The asymptotic condition: A qualitative look Suppose that we have prepared an initial quantum state |ψ at t = 0. We may view |ψ as a square integrable wavepacket ψ(0, x) = x|ψ .Assume that this wavepacket does not populate any bound states of the studied problem.According to the central assumption of the scattering theory, our wavepacket behaves for t = 0 as follows: As t increases from 0 to +∞, the wavepacket ψ(t, x) = x|ψ(t) tends to leave the interaction region (= region of x ∈ (a, b) with nonzero potential) and becomes thus freely evolving at very large positive times.[ Note that some portion of ψ(0, x) would generally escape to the right (x → +∞) and some portion to the left (x → -∞).] Similarly, as t decreases from 0 to -∞, the wavepacket ψ(t, x) = x|ψ(t) tends to leave the interaction region and becomes thus freely evolving at very large negative times.What we just described is the so called Asymptotic Condition (AC).It has crucial importance for all our subsequent elaborations.
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.
The purpose of these lectures is to give an accessible and self contained introduction to quantum scattering theory in one dimension. Part A defines the theoretical playground, and develops basic concepts of scattering theory in the time domain (Asymptotic Condition, in- and out- states, scattering operator S^). The aim of Part B is then to build up, in a step-by-step fashion, the time independent scattering theory in energy domain. This amounts to introduce the Lippmann-Schwinger equation for the stationary scattering states (denoted as ∣ψE(±1)±⟩), to discuss fundamental properties of ∣ψE(±1)±⟩, and subsequently to construct S^ and T^ operators in terms of ∣ψE(±1)±⟩. Physical contents of the S^ and T^ operators is then illuminated by deriving explicit formulas for the probability of transmission/reflection of our quantum particle through/from the interaction region of the potential. An illustrative numerical example is given, which also highlights an existence of scattering resonances. Finally, Part C elaborates the nonhermitian scattering theory (Siegert pseudostate formalism), which offers an extremely powerful tool suitable for clear cut understanding of the resonance phenomena.
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.
Basic notions and theWigner–Weisskopf theory Although the notions of bound states, scattering and quantum transitions are well defined in the quantum theory, the description of an unstable system, involved in the process of decay, has remained an outstanding issue for many years. The problem is fundamental, since it concerns the nature of irreversible processes, one of the most important issues in statistical mechanics and the theme that is central to this book. The theory of decay is intimately connected with scattering theory and necessarily contains mathematical ideas and methods. We shall try to explain these points carefully as we get to them. We treat elsewhere in the book the ideas of Boltzmann, Van Hove and Prigogine on irreversible phenomena. The tools that are developed there are basically approximate, although very useful. One can argue that the basic rigorous characteristic of an irreversible process is that, as represented in terms of the evolution of a state in the Hilbert space of the quantum theory, it must be a semigroup. This type of evolution, resulting in an operation Z ( t ) on a state ψ , should satisfy the property Z ( t 2 ) Z ( t 1 ) = Z ( t 1 + t 2 ). (17.1) The argument is as follows. If the system evolves in time t 1 and is stopped, then evolves further at time t 2 , since the process has no memory, the total evolution should be as if the system evolved from the initial state to a state at t 1 + t 2 independently of the fact that it was done in two stages (Piron, 1976). Since the process is irreversible, the operator Z ( t ) may have no inverse. Such an evolution is called a semigroup . As we shall see, it is not possible to obtain such an evolution law in the framework of the standard quantum theory (Horwitz et al. , 1971), but recently much work has been done, and methods have been developed, based on ideas of Sz.-Nagy and Foias (1976), such as the theory of Lax and Phillips (1967) and its extension to the quantum theory (Strauss et al. , 2000) in which semigroup evolution can be achieved.
This chapter examines the most critical properties, which make the collision phenomenon substantially different from the corresponding problem of bound states in quantum mechanics. In order to elaborate the treatments further in the course of the presented analysis, it was necessary to employ the quantum-mechanical entities, such as the equations of time evolution, as well as the state vectors and the Hilbert space of eigenstates. A given picture of quantum mechanics can be more practical than formalism for the examined physical phenomenon. Hence, the choice of the optimal picture of quantum mechanics becomes a matter of considerable practical importance. The initial state Ψ0i is experimentally prepared, i.e. characterized by means of the impulse, polarization, spin, isospin and other internal quantum numbers of the constituent parts of the scattering system.
We give a full and self contained account of the basic results in N-body scattering theory which emerged over the last ten years: The existence and completeness of scattering states for potentials decreasing like r-μ, [Formula: see text]. Our approach is a synthesis of earlier work and of new ideas. Global conditions on the potentials are imposed only to define the dynamics. Asymptotic completeness is derived from the fact that the mean square diameter of the system diverges like t2 as t → ±∞ for any orbit ψt which is separated in energy from thresholds and eigenvalues (a generalized version of Mourre's theorem involving only the tails of the potentials at large distances). We introduce new propagation observables which considerably simplify the phase–space analysis. As a topic of general interest we describe a method of commutator expansions.