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arXiv:0708.0052

An Introduction to Relativistic Quantum Mechanics. I. From Relativity to Dirac Equation

Maria Cristina De Sanctis

Published Aug 1, 2007• 1 citation
arXivHTMLPDF

Abstract

By using the general concepts of special relativity and the requirements of quantum mechanics, Dirac equation is derived and studied. Only elementary knowledge of spin and rotations in quantum mechanics and standard handlings of linear algebra are employed for the development of the present work.
DOI
10.48550/arxiv.0708.0052

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Chapter 19 introduces relativistic quantum mechanics, with the aim of providing the student with a basic knowledge of the subject and a link to more advanced courses in relativistic quantum theory. It begins by showing that when the Schrödinger equation is modified to make it consistent with special relativity, the result is the Klein–Gordon equation, which gives better qualitative agreement with experiment. Introducing the electron spin into the Schrödinger equation leads to the Pauli equation, which gives rise to important spin effects. By considering both special relativity and spin, one obtains the Dirac and the van der Waerden equations. The main properties and implications of Dirac’s equation are discussed, including the electron’s zitterbewegung. The solution of Dirac’s equation for the free particle is presented in detail, the hydrogen-like atom is shown to be exactly solvable, and the particle in an external electromagnetic field is included as a complementary material.
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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.
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.
Abstract Conventional methods for teaching spin at a university level generally follow four main approaches. The first introduces spin into the Schrödinger equation artificially, leaving its physical meaning unclear. The second connects spin to the representation of the rotation group, which clarifies its nature but requires knowledge of group theory. The third approach derives from field theory, separating angular momentum into orbital and intrinsic parts, the latter corresponding to spin; however, this cannot be done within quantum mechanics alone. The fourth derives electron spin from Dirac’s 1928 relativistic equation, which, though correct, has led to the misconception that spin is a purely relativistic effect. A detailed analysis of Dirac’s 1928 papers shows that the concept of spin arises from the need to construct an equation linear in both energy and momentum to preserve Born’s probabilistic interpretation. This linearization of the Klein–Gordon equation reveals that spin is conceptually independent of relativity. Historical and pedagogical perspectives can together clarify conceptual difficulties and promote a deeper understanding of fundamental ideas in physics. Drawing on selected passages from Dirac’s original work, the present discussion offers broader conceptual and cultural considerations that introduce the notion of spin with limited mathematical requirements and can be applied independently of the didactic approach adopted.
We develop a new concept of quantum mechanics which is based on a generalized space-time and on an action vector space similar to it. Both spaces are provided by algebraic properties. This allows to calculate the Dirac matrixes and to derive quantum mechanics equations from structure equations of the specified algebras. A new interpretation of the wave function is given as differential of the action vector. A generalization of the Dirac equation for 8-component wave function is derived. It is interpreted as the equation for two leptons of the same generation. A procedure of the approximate description of free leptons is formulated. The generalized equation of quantum mechanics is reduced to the Dirac, Pauli and Schrödinger equations by the sequential use of this procedure. We explain the existence of three lepton generations.