Introduction.We give three applications of probability to scattering theory.The first question we consider is that of the "scattering length."We obtain a probabilistic expression for scattering length and use this expression to prove that in the case of an impenetrable obstacle ("hard" potential) the scattering length T is the same as the electrostatic capacity.Next, we briefly touch upon a problem which is inherently probabilistic-namely, scattering in a medium with random boundary.We find that if a large number of small hard spheres ("dust particles") are distributed at random in the medium, the effect is to shift all the eigenvalues by a constant.Our final topic is the so-called "inverse problem" of scattering theory: given the scattering operator, to compute the potential.For certain spherically symmetric potentials, this problem was solved in a famous paper of GelTand and Levitan.Here we show that their result can be included in the circle of potential-theoretic problems which can be solved by means of Brownian motion and Wiener integrals.We quickly sketch some of the physical background.We are concerned with the Schrodinger equation of non-relativistic quantum mechanics.In its stationary or time-independent form, this equation isThe "wave function" I/J depends on both the position vector fiEfi 3 and the wave number k.The real parameter k 2 is proportional to the energy.The function q(R) is called the "potential."It represents the interaction of the quantum-mechanical "particle" with some force field.The "scattering problem" is, roughly speaking, to determine the effect of this interaction asymptotically for large R.More precisely, we seek a solution of (1) which at |fl| = » is of the asymptotic form e iV2kz + fk(e)e iV ^k> ^l\\R\\ where e is the unit vector in the direction of R. The first term, e iV2kz , is an incident wave in the direction of the zaxis; the second term /fc(e)e iV ^I^H/||R||, is the "scattered wave."
Introduction.We give three applications of probability to scattering theory.The first question we consider is that of the "scattering length."We obtain a probabilistic expression for scattering length and use this expression to prove that in the case of an impenetrable obstacle ("hard" potential) the scattering length T is the same as the electrostatic capacity.Next, we briefly touch upon a problem which is inherently probabilistic-namely, scattering in a medium with random boundary.We find that if a large number of small hard spheres ("dust particles") are distributed at random in the medium, the effect is to shift all the eigenvalues by a constant.Our final topic is the so-called "inverse problem" of scattering theory: given the scattering operator, to compute the potential.For certain spherically symmetric potentials, this problem was solved in a famous paper of GelTand and Levitan.Here we show that their result can be included in the circle of potential-theoretic problems which can be solved by means of Brownian motion and Wiener integrals.We quickly sketch some of the physical background.We are concerned with the Schrodinger equation of non-relativistic quantum mechanics.In its stationary or time-independent form, this equation isThe "wave function" I/J depends on both the position vector fiEfi 3 and the wave number k.The real parameter k 2 is proportional to the energy.The function q(R) is called the "potential."It represents the interaction of the quantum-mechanical "particle" with some force field.The "scattering problem" is, roughly speaking, to determine the effect of this interaction asymptotically for large R.More precisely, we seek a solution of (1) which at |fl| = » is of the asymptotic form e iV2kz + fk(e)e iV ^k> ^l\R\ where e is the unit vector in the direction of R. The first term, e iV2kz , is an incident wave in the direction of the zaxis; the second term /fc(e)e iV ^I^H/||R||, is the "scattered wave."
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 aim of the lecture is to briefly describe the mathematical background of scattering theory for two- and three-particle quantum systems. We discuss basic objects of the theory: wave and scattering operators and the corresponding scattering matrix and illustrate them on the example of the Schrödinger equation. Our goal is to present time-dependent and stationary approaches and to describe the underlying mathematical methods. We also give a sketch of scattering theory for three interacting quantum particles including a difficult problem of the asymptotic completeness of scattering channels. Along with traditional results, we discuss new scattering channels arising for long-range pair interactions.
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 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.
Abstract A familiar classroom heuristic for hard-sphere scattering is that the low-energy cross section ‘equals the surface area’ because a long-wavelength particle can ‘feel’ the sphere from all directions. More precisely, this heuristic means that, when the wavelength is large compared with the size of the scatterer, the wave is sensitive to the global boundary condition imposed by the whole hard obstacle; in the zero-energy limit this global response is represented by the monopole term of an exterior Laplace problem. While useful as intuition, the heuristic prompts a natural question for students: is there a general connection between surface area and the threshold cross section of a non-spherical hard target? This pedagogical note, aimed at advanced undergraduates and beginning graduate students, shows how to answer that question systematically. For a perfectly hard obstacle the zero-energy Schrödinger equation reduces to an exterior Laplace problem with Dirichlet boundary conditions. Comparing its far-field monopole term with the corresponding electrostatic problem yields a direct mapping between the scattering length and the electrostatic capacitance of the same shape. The mapping provides a simple route to threshold cross sections for bodies of arbitrary shape, without relying on separability. Using published capacitances for several examples, we compare the threshold cross section with surface area and show that the equality for a sphere is not a general surface-area law.
We seek stationary state solutions to the nonlinear Schrodinger equation (NLS) for piecewise-constant potentials V(x). In this case, we are able to find complete stationary solutions, i.e. exact solutions which are separable in time and space, in terms of Jacobi elliptic functions. These special functions can be defined as an inverse integral and are an interpolation between the circular and hyperbolic trigonometric functions. As an important mathematical application of our exact solutions, we will seek a nonlinear generalization of the transfer matrix. When this matrix, which is commonly used in electromagnetic transmission problems, is applied to the coefficient vector characterizing the solution in the region to the left of the first boundary, it yields the coefficient vector for any region to the right. To solve this general problem, we begin with the nontrivial case of two boundaries, which corresponds to a potential step or barrier. This has a direct physical application to an atom laser incident on such a potential, as occurs in Bose-Einstein condensates on a chip. Our work provides the first step towards formulating a completely general nonlinear scattering theory.