The early twentieth century saw two major revolutions in the way physicists understand the world. One was quantum mechanics itself and the other was the theory of relativity. Important results also emerged when these two ideas were brought together: some of these have been referred to in earlier chapters-in particular the fact that fundamental particles such as electrons have intrinsic angular momentum (spin) was stated to be a relativistic effect. This chapter explores the relationship between relativity and quantum mechanics more deeply A full understanding of relativistic quantum mechanics is well outside the scope of this book, but many of the important results can be understood at this level and these will be discussed in this chapter. After a short summary of the main results of special relativity, we show how combining this with the time-dependent Schro¨dinger equation leads to a new wave equation known as the Dirac equation. We show how the Dirac equation requires particles such as electrons to have intrinsic angular momentum (spin) and we explore some of its other consequences. The chapter concludes with an outline of some more advanced ideas known as quantum field theory.
The early twentieth century saw two major revolutions in the way physicists understand the world. One was quantum mechanics itself and the other was the theory of relativity. Important results also emerged when these two ideas were brought together: some of these have been referred to in earlier chapters-in particular the fact that fundamental particles such as electrons have intrinsic angular momentum (spin) was stated to be a relativistic effect. This chapter explores the relationship between relativity and quantum mechanics more deeply A full understanding of relativistic quantum mechanics is well outside the scope of this book, but many of the important results can be understood at this level and these will be discussed in this chapter. After a short summary of the main results of special relativity, we show how combining this with the time-dependent Schro¨dinger equation leads to a new wave equation known as the Dirac equation. We show how the Dirac equation requires particles such as electrons to have intrinsic angular momentum (spin) and we explore some of its other consequences. The chapter concludes with an outline of some more advanced ideas known as quantum field theory.
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.
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.
In this paper, the deformed Special Relativity, which leads to an essentially new theoretical context of quantum mechanics, is presented. The formulation of the theory arises from a straightforward analogy with the Special Relativity, but its foundations are laid through the hypothesis on breakdown of the velocity-momentum parallelism which affects onto the Einstein equivalence principle between mass and energy of a relativistic particle. Furthermore, the derivation is based on the technique of an eikonal equation whose well-confirmed physical role lays the foundations of both optics and quantum mechanics. As a result, we receive the angular deformation of Special Relativity which clearly depicts the new deformation-based theoretical foundations of physics, and, moreover, offers both constructive and consistent phenomenological discussion of the theoretical issues such like imaginary mass and formal superluminal motion predicted in Special Relativity for this case. In the context of the relativistic theory, presence of deformation does not break the Poincaré invariance, in particular the Lorentz symmetry, and provides essential modifications of both bosons described through the Klein-Gordon equation and fermions satisfying the Dirac equation. On the other hand, on the level of discussion of quantum theory, there arises the concept of emergent deformed space-time, wherein the presence of angular deformation elucidates a certain new insight into the nature of spin, as well as both the Heisenberg uncertainty principle and the Schrödinger wave equation.
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.
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.
This chapter starts with the historical development of quantum mechanics and explanations of key concepts behind fundamental discoveries such as Planck's hypothesis, the photoelectric effect, Bohr's atomic model, Compton scattering, the de Broglie hypothesis, the Pauli exclusion principle, the Schrödinger equation, Heisenberg quantum mechanics and the uncertainty principle, the Born interpretation of wave function, the Davisson and Germer experiment, and Dirac relativistic quantum mechanics, which made important contributions to the foundation of modern quantum mechanics. Concepts like the particle properties of radiation, the wave properties of matter, quantized energy and momentum, and the idea that one can no longer know exactly where a single particle like an electron is at any instant of time are brought out clearly through this chapter. A list of important quantum mechanical parameters along with solved examples and unsolved exercises on the key concepts are included towards the end.