This chapter considers the basic problem of the scattering of a beam of particles by an immovable scattering centre which exerts a force on the moving particles depending only on the separation r. The concept of scattering cross-section is very useful in dealing with the interaction of a beam of particles with a scattering centre. The chapter discusses the scattering problem by solving the time-independent Schrodinger equation. Conservation of energy implies that, after the scattering is complete, the freely-moving deflected particles will have the same energy and de Broglie wavelength as the incident particles. In order to simplify the geometry, consider a parallel flux of particles moving parallel to a line of incidence through the immovable scattering centre. Conservation of energy implies that, after the scattering is complete, the freely-moving deflected particles will have the same energy and de Broglie wavelength as the incident particles.
This chapter considers the basic problem of the scattering of a beam of particles by an immovable scattering centre which exerts a force on the moving particles depending only on the separation r. The concept of scattering cross-section is very useful in dealing with the interaction of a beam of particles with a scattering centre. The chapter discusses the scattering problem by solving the time-independent Schrodinger equation. Conservation of energy implies that, after the scattering is complete, the freely-moving deflected particles will have the same energy and de Broglie wavelength as the incident particles. In order to simplify the geometry, consider a parallel flux of particles moving parallel to a line of incidence through the immovable scattering centre. Conservation of energy implies that, after the scattering is complete, the freely-moving deflected particles will have the same energy and de Broglie wavelength as the incident particles.
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.
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.
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.
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.
In this chapter, the Schrödinger equation is “derived” for particles that can be described by de Broglie waves. The Schrödinger equation is very different from the corresponding equation of motion in classical mechanics. In order to illustrate the fundamental differences between the two theories, one of the simplest problems of particle dynamics is solved in both Newtonian and quantum mechanics. This simple example also helps to show that quantum mechanics is the fundamental theory and classical mechanics is an approximation, a remarkably good approximation, when considering macroscopic objects. The solution of the Schrödinger equation is presented for a particle inside a box and the quantization condition is derived. The amazing possibility of quantum tunneling when a particle is incident on a barrier of height larger than the energy of the incident particle is also discussed. Finally the three-dimensional Schrödinger equation is solved for the hydrogen atom.