In this chapter we discuss the quantum mechanics of the scattering of particles by a fixed object. Scattering is a very important feature of many physical processes and often performs an important role in the process of performing and recording an experimental result. Thus when we perform an x-ray diffraction experiment we record the diffraction pattern created when x-rays are scattered by a crystal, while nearly all the information we possess about the energy levels and structure of nuclei and fundamental particles has been obtained from experiments in which beams of particles (e.g., protons or neutrons) are scattered from targets containing the nuclei or particles under investigation. Indeed, any visual observation we make simply by looking at something involves the detection by our eyes of light scattered from the object we are looking at. It is important therefore that we understand something of the quantum mechanics of the scattering processes, and this will be the aim of the present chapter.
In this chapter we discuss the quantum mechanics of the scattering of particles by a fixed object. Scattering is a very important feature of many physical processes and often performs an important role in the process of performing and recording an experimental result. Thus when we perform an x-ray diffraction experiment we record the diffraction pattern created when x-rays are scattered by a crystal, while nearly all the information we possess about the energy levels and structure of nuclei and fundamental particles has been obtained from experiments in which beams of particles (e.g., protons or neutrons) are scattered from targets containing the nuclei or particles under investigation. Indeed, any visual observation we make simply by looking at something involves the detection by our eyes of light scattered from the object we are looking at. It is important therefore that we understand something of the quantum mechanics of the scattering processes, and this will be the aim of the present chapter.
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.
The scattering process is equivalent to the collision process and relates the physics of gaseous states, solid state physics (electrons colliding with phonons in the crystal lattice), plasma physics and so on. When we cannot directly visualize particles, scattering can be a valuable probe for their study. An example is the hydrogen atom that scattering experiments confirm it to have a nucleus with an electron orbiting around it. Scattering experiments also confirm protons to have some constituents called quarks. So the scattering theory is pivotal in understanding of the quantum nature of particles. In principle, we can formulate all problems of quantum mechanics in the language of scattering theory.
This chapter begins with a brief outline of some of the key motivations for considering a quantum theory: the early attempts to determine the spectral distribution of energy density of black bodies; stability of atoms and molecules; specific heats of solids; interference and diffraction of light beams; polarization of photons. The experimental foundations of wave mechanics are then presented in detail, but in a logical order quite different from its historical development: photo-emission of electrons by metallic surfaces, X- and γ-ray scattering from gases, liquids and solids, interference experiments, atomic spectra and the Bohr hypotheses, the experiment of Franck and Hertz, the Bragg experiment, diffraction of electrons by a crystal of nickel (Davisson and Germer), and measurements of position and velocity of an electron. The need for a quantum theory In the second half of the nineteenth century it seemed that the laws of classical mechanics, developed by the genius of Newton, Lagrange, Hamilton, Jacobi and Poincaré, the Maxwell theory of electromagnetic phenomena and the laws of classical statistical mechanics could account for all known physical phenomena. Still, it became gradually clear, after several decades of experimental and theoretical work, that one has to formulate a new kind of mechanics, which reduces to classical mechanics in a suitable limit, and makes it possible to obtain a consistent description of phenomena that cannot be understood within the classical framework. It is now appropriate to present a brief outline of this new class of phenomena, the systematic investigation of which is the object of the following sections and of chapters 4 and 14.
Quantum mechanics is a theory to explain and predict the behavior of particles such as electrons, protons, neutrons, atomic nuclei, atoms, and molecules, as well as the photon–the particle associated with electromagnetic radiation or light. From quantum theory we obtain the laws of chemistry as well as explanations for the properties of materials, such as crystals, semiconductors, superconductors, and superfluids. Applications of quantum behavior give us transistors, computer chips, lasers, and masers. The relatively new field of molecular biology, which leads to our better understanding of biological structures and life processes, derives from quantum considerations. Thus, quantum behavior encompasses a large fraction of modern science and technology. Quantum theory was developed during the first half of the twentieth century through the efforts of many scientists. In 1926, E. Schrödinger interjected wave mechanics into the array of ideas, equations, explanations, and theories that were prevalent at the time to explain the growing accumulation of observations of quantum phenomena. His theory introduced the wave function and the differential wave equation that it obeys. Schrödinger's wave mechanics is now the backbone of our current conceptional understanding and our mathematical procedures for the study of quantum phenomena. Our presentation of the basic principles of quantum mechanics is contained in the first three chapters. Chapter 1 begins with a treatment of plane waves and wave packets, which serves as background material for the subsequent discussion of the wave function for a free particle.
Quantum scattering, a fundamental phenomenon in quantum mechanics, is very important in understanding the interactions between particles at the microscopic level.1,2 Quantum scattering plays a pivotal role in various fields of physics, chemistry, and even beyond, influencing areas such as materials science, quantum computing, and nuclear physics among others.3,4 Theoretical concepts such as the Schrödinger equation, scattering amplitude, and scattering cross-section are highlighted in this, along with their significance in describing particle interactions with potential energy fields.5,6
Abstract The new wave mechanics has been eminently successful in correlating and accounting for the large amount of experimental data on the periodic properties of the atom. The applications of the new theory to aperiodic phenomena, though not nearly so numerous, have been attended with no less success; indeed, it is in these experiments on free electrons, in which diffraction patterns similar to those produced by beams of X-rays and light are obtained, that the wave nature of electrons is so clearly and objectively demonstrated. Such diffraction effects have been obtained by a number of investigators by scattering beams of homogeneous electrons in crystals, thin films and by a ruled grating. Similar effects are obtained when electrons are scattered by complex molecules, owing to the symmetrically situated nuclei in the molecular structure, and when α-particles are scattered by helium owing to interference between the waves of the scattered incident particles and recoiling helium nuclei.