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.
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.
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.
Development of the mathematical concept of scattering theory proceeded quite slowly because of difficulties in the mathematical problems themselves, as well as the lack of collaboration among mathematicians and physicists. The scattering experiment plays one of the leading roles in the part of physics devoted to measurement, irrespective of whether one is concerned with particle collisions or with scattering of photons on various substances. The first direct confirmation of the existence of the stationary states of atoms came in 1914 through Franck-Hertz’s experiment on inelastic scattering of electrons on the mercury target vapour. Collision problems include at least 80 percentage of the entire research themes and activities in the domain of atomic physics theory. Active and passive diagnostic techniques with particle beams are mainly based upon the problem of scattering in which electron transfer takes place.
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.
The theory of scattering consists in studying the collision also called dispersion, between two or more particles. This chapter studies the simplest case of the collision between two particles without spin. Historically, the discovery of the quantum world and the particles was made by studying collision processes. Many physics experiments are based on collisions, or more precisely of scattering. In 2012 particle physicists announced the long awaited discovery of the Higgs Boson in the collision processes at the Large Hadron Collider (LHC). An incoming spherical wave does not correspond to an usual experimental situation. In an experimental situation, the incoming wave is more often modeled by an incident plane wave.
At the atomic scale, the distinction between particles and waves breaks down. The realization, early in the twentieth century, that light consists of discrete packages, or “photons,” prompted the converse suggestion that particles could behave like waves. Confirmation of this idea came in 1927, from an experiment involving the scattering of electrons from a crystal. Today, “wave-particle duality” is central to quantum mechanics, and the wave properties of subatomic particles are widely used in microscopy and crystallography. Electron diffraction techniques are widely used to characterize atomic arrangements on solid surfaces and inside thin films, and to investigate the structures of molecules in gases. The dual nature of waves and particles is part of the “spookiness” of quantum mechanics, which cannot be explained in any classical way. It used to be that one could only imagine an experiment that would simultaneously demonstrate the particle and wave nature of electrons.
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned with the way in which electrons are arranged around the nucleus and the processes by which these arrangements change. This comprises ions, neutral atoms and, unless otherwise stated, it can be assumed that the term atom includes ions. The term atomic physics can be associated with nuclear power and nuclear weapons, due to the synonymous use of atomic and nuclear in Standard English. Physicists distinguish between atomic physics -which deals with the atom as a system consisting of a nucleus and electrons- and nuclear physics, which studies nuclear reactions and special properties of atomic nuclei. As with many scientific fields, strict delineation can be highly contrived and atomic physics is often considered in the wider context of atomic, molecular, and optical physics. Physics research groups are usually so classified. Atomic physics primarily considers atoms in isolation. Atomic models will consist of a single nucleus that may be surrounded by one or more bound electrons. It is not concerned with the formation of molecules (although much of the physics is identical), nor does it examine atoms in a solid state as condensed matter. It is concerned with processes such as ionization and excitation by photons or collisions with atomic particles. While modelling atoms in isolation may not seem realistic, if one considers atoms in a gas or plasma then the time-scales for atom-atom interactions are huge in comparison to the atomic processes that are generally considered. This means that the individual atoms can be treated as if each were in isolation, as the vast majority of the time they are. By this consideration atomic physics provides the underlying theory in plasma physics and atmospheric physics, even though both deal with very large numbers of atoms. Electrons form notional shells around the nucleus. These are normally in a ground state but can be excited by the absorption of energy from light (photons), magnetic fields, or interaction with a colliding particle (typically ions or other electrons). Electrons that populate a shell are said to be in a bound state. The energy necessary to remove an electron from its shell (taking it to infinity) is called the binding energy. Any quantity of energy absorbed by the electron in excess of this amount is converted to kinetic energy according to the conservation of energy. The atom is said to have undergone the process of ionization. If the electron absorbs a quantity of energy less than the binding energy, it will be transferred to an excited state. After a certain time, the electron in an excited state will "jump" (undergo a transition) to a lower state. In a neutral atom, the system will emit a photon of the difference in energy, since energy is conserved. If an inner electron has absorbed more than the binding energy (so that the atom ionizes), then a more outer electron may undergo a transition to fill the inner orbital. In this case, a visible photon or a characteristic x-ray is emitted, or a phenomenon known as the Auger effect may take place, where the released energy is transferred to another bound electron, causing it to go into the continuum. The Auger effect allows one to multiply ionize an atom with a single photon. There are rather strict selection rules as to the electronic configurations that can be reached by excitation by light — however there are no such rules for excitation by collision processes. One of the earliest steps towards atomic physics was the recognition that matter was composed of atoms. It forms a part of the texts written in 6th century BC to 2nd century BC such as those of Democritus or Vaisheshika Sutra written by Kanad. This theory was later developed in the modern sense of the basic unit of a chemical element by the British chemist and physicist John Dalton in the 18th century. At this stage, it wasn't clear what atoms were although they could be described and classified by their properties (in bulk). The invention of the periodic system of elements by Mendeleev was another great step forward. The true beginning of atomic physics is marked by the discovery of spectral lines and attempts to describe the phenomenon, most notably by Joseph von Fraunhofer. The study of these lines led to the Bohr atom model and to the birth of quantum mechanics. In seeking to explain atomic spectra an entirely new mathematical model of matter was revealed. As far as atoms and their electron shells were concerned, not only did this yield a better overall description, i.e. the atomic orbital model, but it also provided a new theoretical basis for chemistry (quantum chemistry) and spectroscopy. Since the Second World War, both theoretical and experimental fields have advanced at a rapid pace. This can be attributed to progress in computing technology, which has allowed larger and more sophisticated models of atomic structure and associated collision processes. Similar technological advances in accelerators, detectors, magnetic field generation and lasers have greatly assisted experimental work.