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Elementary relativistic quantum mechanics

Efstratios Manousakis

Published Nov 1, 2015• 0 citations
Publisher

Abstract

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.

DOI
10.1093/acprof:oso/9780198749349.003.0031

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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.
This chapter introduces the concepts of the relativistic theory to basic problems of quantum physics, in particular, the extension of the Schrodinger equation to a relativistic form. Starting from the fundamental laws of relativity, the chapter evaluates the Klein–Gordon equation and investigates if the equation can be considered a generalization of the non-relativistic Schrodinger equation to the case of the relativistic energy. The chapter shows that the wave function, which is a solution to the Klein–Gordon equation, cannot be connected with the probability wave function. For this reason, the Dirac equation is used to solve the problems faced by the Klein–Gordon equation. The Dirac equation also includes the spin. The advantage of the Dirac equation over the Schrodinger and Klein–Gordon equations is that it naturally includes the spin. In the Schrodinger and Klein–Gordon equations, the spin is not present and has to be added to the wave function manually whenever the wave function determines a system with the spin.
It is not possible to use the Schrödinger equation to describe the motion of a particle with relativistic effects since it is not invariant relative to the Lorentz transformation. However, the attempt to generalize quantum mechanics in the relativistic domain on the basis of the one particle theory is somehow difficult. So we find the appropriate equations on the basis of the relativistic invariance. To obtain the non-relativistic equations, we take the limit as the speed of light tends to infinity. This obviously recovers the Schrödinger equation. Under this heading, we extend the non-relativistic Schrödinger equation in the description of particles with speeds comparable with that of light. This extension should be consistent with the Lorentz transformation [ 12 ]. The characteristic feature of the relativistic wave equations is the spin which provides a useful gauge in the applicability of particular equations that describe particular kinds of particles. We have two types of relativistic equations, i.e., the equation with spin-zero due to Schrödinger and spin-half due to Dirac that describes the electron.
The Klein-Gordon equation describes the wave-like behavior of spinless particles since it is Lorentz invariant. While it seemed initially ripe for explaining the electronic structure of the hydrogen atom, the lack of a unconditional positive probability density really limited its applications. Yet, it is intimately connected with fermions. Any solution to the Dirac equation is automatically a solution to the Klein-Gordon equation. What is even more surprising, the Klein-Gordon equation for a free particle turns into the Schrödinger equation in the non-relativistic limit. In this work we show that these problems disappear when we use the relativistic mass instead of the rest mass. While the Klein-Gordon equation losses its Lorentz invariance because of this transformation, it gains most of the features present the Schrödinger equation, including the unconditional positivity of probability density, while keeping most of its relativistic characteristics intact, including the matter-wave dispersion relation. What is even more surprising, the non-relativistic, quasi-static limit of the Klein-Gordon equation with relativistic mass is simply the Schrödinger equation under all possible conditions. So, it can be argued that this Klein-Gordon equation is a sort of relativistic Schrödinger equation.
A relativistic wave equation describes the motion of a particle that has speed approaching that of light. A characteristic feature of relativistic wave equation is that the spin of the particle is built into the theory from the start and cannot be added afterwards as Pauli added electron spin to Schrödinger non-relativistic equation. In the next section is described a spin zero particle known as Klein Gordon equation. The subsequent section will deal with Dirac equation describing spin 1 / 2 particles. We shall continue to use three dimensional vector notations rather than the the four dimensional notation of special relativity theory. However, the invariance of an equation under Lorentz transformation can usually be inferred from its symmetry between the space coordinate and time.
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.