This chapter reviews quantum mechanics required to pursue the quantum charged particle beam optics, the central theme of this book. All physical phenomena are quantum mechanical at the fundamental level. Classical mechanics observed in the macroscopic world is an approximation. That is why it raises a curiosity when classical mechanics is found to be very successful at a microscopic level in the design and operation of beam devices like an electron microscope or a particle accelerator. This chapter gives an introductory review of nonrelativistic and relativistic quantum mechanics just at the level necessary for presenting the formalism of quantum charged particle beam optics. All the basic equations of quantum mechanics, Schrödinger’s time-evolution equation, the Klein-Gordon equation, and the Dirac equation, are given in detail. The Feshbach-Villars formalism of the Klein-Gordon equation and the Foldy-Wouthuysen transformation of the Dirac equation, forming the basis for the development of quantum charged particle beam optics, are described.
This chapter reviews quantum mechanics required to pursue the quantum charged particle beam optics, the central theme of this book. All physical phenomena are quantum mechanical at the fundamental level. Classical mechanics observed in the macroscopic world is an approximation. That is why it raises a curiosity when classical mechanics is found to be very successful at a microscopic level in the design and operation of beam devices like an electron microscope or a particle accelerator. This chapter gives an introductory review of nonrelativistic and relativistic quantum mechanics just at the level necessary for presenting the formalism of quantum charged particle beam optics. All the basic equations of quantum mechanics, Schrödinger’s time-evolution equation, the Klein-Gordon equation, and the Dirac equation, are given in detail. The Feshbach-Villars formalism of the Klein-Gordon equation and the Foldy-Wouthuysen transformation of the Dirac equation, forming the basis for the development of quantum charged particle beam optics, are described.
The purpose of this chapter is to introduce the Dirac equation, which will provide us with a basis for developing the relativistic quantum mechanics of electronic systems. Thus far we have reviewed some basic features of the classical relativistic theory, which is the foundation of relativistic quantum theory. As in the nonrelativistic case, quantum mechanical equations may be obtained from the classical relativistic particle equations by use of the correspondence principle, where we replace classical variables by operators. Of particular interest are the substitutions In terms of the momentum four-vector introduced earlier, this yields In going from a classical relativistic description to relativistic quantum mechanics, we require that the equations obtained are invariant under Lorentz transformations. Other basic requirements, such as gauge invariance, must also apply to the equations of relativistic quantum mechanics. We start this chapter by reexamining the quantization of the nonrelativistic Hamiltonian and draw out some features that will be useful in the quantization of the relativistic Hamiltonian. We then turn to the Dirac equation and sketch its derivation. We discuss some properties of the equation and its solutions, and show how going to the nonrelativistic limit reduces it to a Schrödinger-type equation containing spin.
The chapter covers some of the essential aspects of relativistic quantum mechanics. The Klein-Gordon equation and its solution in coulombic fields, how to interpret probability density and probability current derived from the Klein-Gordon equation, derivation of the Dirac equation and its covariant form, probability density and probability current for Dirac equation, positive and negative energy eigenvalues in the case of a free particle, constants of motion for relativistic treatment, presence of spin magnetic moment and spin-orbit interaction energy in the Dirac equation, and the solution of the Dirac equation for central potentials and for the hydrogen atom are presented with elegant mathematical treatment and physical interpretations. A modern and reader friendly approach is adopted for derivation of the Dirac equation, constants of motion, and for the solution of the Dirac equation for central potentials. Solved examples and unsolved exercises relating to fundamental concepts of relativistic quantum mechanics are added towards the end of chapter.
Abstract Quantum mechanics was created mainly to account for microscopic phenomena that occur on a scale of order IO−10 m. It has turned out to be an astonishingly successful theory which applies on a scale down to at least 10−18 m. Quantum theory has become established as part of the essential core of physics. The purpose of this introductory chapter is twofold. Firstly, in Sections 1.1–1.7 we review the formalism of elementary nonrelativistic quantum mechanics. In this review we attempt to give enough of the terminology and basics of the subject so that a reader who is not already familiar with quantum mechanics will be able to follow the applications considered later in the book. For the reader who finds our discussion too brief, the literature on quantum mechanics is vast and we recommend in particular Refs 1–5. The second purpose of this chapter is to present the basic tools of operator methods as they are used in this book: thus in Sections 1.8–1.11 we discuss commutation relations, shift operators, and the virial, hypervirial, and Hellmann Feynman theorems.
In order to understand the properties of nanoparticles, it is very much required to have knowledge of quantum mechanics. This chapter introduces the basic concept of quantum mechanics. This chapter starts with quantum theory for radiation. It has been observed that the phenomena related to intense radiation like interference, diffraction, etc., could be well described with the help of Maxwell’s wave theory. But in the case of weakly intensed light, wave theory cannot be used to predict the related experimental phenomena. In this regard, quantum theory has been developed considering the particle nature of radiation. This evaluation of quantum theory has been briefly discussed in this chapter. In the world of matter, Newton’s law is successfully used to predict the position, velocity, acceleration, etc., of any macroscopic particle. But this couldn’t be used for particles of tiny dimensions like electrons, neutrons, etc. The difference is that for macroscopic particles we can measure their position and velocity with some accuracy, but for tiny particles, called quantum particles, it is difficult to determine their position and velocity at any given point in time. Wave theory has been presented by Schrödinger to predict features of quantum particles. We have given a few examples where Schrödinger’s wave theory can successfully explain quantum particles’ behavior under static as well as dynamic conditions. But in a real system, we cannot use this theory. We have discussed a few approximation methods to be used with Schrödinger’s wave theory in the case of a real system.
The aim of relativistic quantum mechanics is to describe the finer details of the structure of atoms and molecules, where relativistic effects become nonnegligible. It is a sort of intermediate realm, between the familiar nonrelativistic quantum mechanics and fully relativistic quantum field theory, and thus it lacks the simplicity and elegance of both. Yet it is a necessary tool, mostly for quantum chemists. Pilkuhn's book offers to this audience an up-to-date survey of these methods, which is quite welcome since most previous textbooks are at least ten years old. The point of view of the author is to start immediately in the relativistic domain, following the lead of Maxwell's equations rather than classical mechanics, and thus to treat the nonrelativistic version as an approximation. Thus Chapter 1 takes off from Maxwell's equations (in the noncovariant Coulomb gauge) and gradually derives the basic aspects of Quantum Mechanics in a rather pedestrian way (states and observables, Hilbert space, operators, quantum measurement, scattering,. Chapter 2 starts with the Lorentz transformations, then continues with the Pauli spin equation and the Dirac equation and some of their applications (notably the hydrogen atom). Chapter 3 is entitled `Quantum fields and particles', but falls short of treating quantum field theory properly: only creation/annihilation operators are considered, for a particle in a box. The emphasis is on two-electron states (the Pauli principle, the Foldy--Wouthuysen elimination of small components of Dirac spinors, Breit projection operators. Chapter 4 is devoted to scattering theory and the description of relativistic bound states. Chapter 5, finally, covers hyperfine interactions and radiative corrections. As we said above, relativistic quantum mechanics is by nature limited in scope and rather inelegant and Pilkuhn's book is no exception. The notation is often heavy (mostly noncovariant) and the mathematical level rather low. The central topic is the description of atoms and molecules, including relativistic effects. The author fulfils this program in a reasonable way and offers a valuable tool to the targeted audience. I am not overly enthusiastic about the end result, but I might be prejudiced. Clearly, going further would require the full power of quantum field theory, but this is clearly beyond the scope of the book.
Abstract This chapter is seeking relativistic wave equations, which are invariant under Lorentz transformations, in an attempt to obtain a quantum mechanical description of relativistic particles. First, the chapter starts with the Klein–Gordon equation and then it discusses the Dirac equation. It also takes the non-relativistic limit of the Dirac equation to derive the Schrödinger equation with two additional terms, the Zeeman term and the spin–orbit coupling term. These two terms emerge naturally from the Dirac equation, and thus the spin, as an internal quantum number which behaves like angular momentum, is clearly identified. Finally, the existence of antimatter is clearly supported by the nature of the solutions to the Dirac equation.