By shifting the emphasis from the concept of trajectory to the concept of probability density it is possible to incorporate the uncertainty principle into classical mechanics. This amendment in the nonrelativistic classical theory is sufficient to derive the Schrödinger equation for a general potential. In order to show that the approach has general validity it is necessary to show that it can be generalized to the classical relativistic dynamics. In this paper it is shown how this generalization is achieved for a free particle, and as a result the Dirac instead of the Klein–Gordon equation is obtained. It is shown that the spin and the magnetic moment of charged particles are classical in character because their correct values are calculated as the averages over the classical (relativistic) phase space density, subject to the constraint imposed by the uncertainty principle. Since the Dirac equation has direct connection to the classical (relativistic) dynamics the problem of the positive and negative energy states is discussed.
By shifting the emphasis from the concept of trajectory to the concept of probability density it is possible to incorporate the uncertainty principle into classical mechanics. This amendment in the nonrelativistic classical theory is sufficient to derive the Schrödinger equation for a general potential. In order to show that the approach has general validity it is necessary to show that it can be generalized to the classical relativistic dynamics. In this paper it is shown how this generalization is achieved for a free particle, and as a result the Dirac instead of the Klein–Gordon equation is obtained. It is shown that the spin and the magnetic moment of charged particles are classical in character because their correct values are calculated as the averages over the classical (relativistic) phase space density, subject to the constraint imposed by the uncertainty principle. Since the Dirac equation has direct connection to the classical (relativistic) dynamics the problem of the positive and negative energy states is discussed.
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.
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.
Despite the success of the Dirac equation, the incorporation of the special theory of relativity into quantum mechanics predicts some paradoxes, like that the spin prediction from Dirac equation can be only identified with non- relativistic approximations (Pauli and Foldy-Wouthysen), as well as the spin predication is a relativistic quantum phenomena because the spin prediction is a necessary requirement of the relativistic quantum mechanics only. In this paper we show that the derivation of the spin and its magnetic moment can be done with a pure classical treatment. Since we start from the classical physical laws and the classical relativity principle to get the linear Schrödinger equation as a result the derivation of the spin and its magnetic moment can be done with a pure classical treatment. This approach result a Schrödinger equation, in which we show that the spin of the electron is a non relativistic quantum phenomenon too.
Covariant classical particle dynamics is described, and the associated covariant relativistic particle quantum mechanics is derived. The invariant symmetric bracket is defined on the space of quantum amplitudes, and its relation to a generalized Hamiltonian dynamics and to a covariant Schrödinger type equation is shown. Examples for relativistic potential problems are solved. Mathematically and physically acceptable probability densities for the Klein-Gordon equation and for the Dirac equation are derived, and a new continuity equation for each case is given. The quantum distribution for mass is discussed, and unambiguous representations of four-velocity and four-acceleration operators are given.
We derive the equations of relativistic quantum mechanics from a modified version of classical electrodynamics, where probability is replaced by potential. As a result, a particle is not a localized entity, in the classical sense, but has a localized energy extremum. The particle/wave aspect of matter is inherent in the particle/wave equation describing elementary particles. Furthermore, the Heisenberg uncertainty and Planck-Einstein-de Broglie relations, and the Klein-Gordon, Dirac and Proca equations follow naturally from the particle/wave equation. In addition, we incorporate a new and more physical interpretation of spin angular momentum.
In this chapter, we investigate the Dirac equation, which is named after P. A. M. Dirac, who is one of the fathers of quantum field theory. The Dirac equation is a relativistic quantum mechanical wave equation for spin-1/2 particles (e.g. electrons), which was derived by Dirac in 1928. The difficulties in finding a consistent single-particle theory from the Klein–Gordon equation led Dirac to search for an equation that had a positive-definite conserved probability density and was first order both in time and space. One can show that these two conditions imply that a matrix equation is required. The reason why the Klein–Gordon equation did not yield a positive-definite probability density is connected with the second-order time derivative in this equation, which arises because the Klein–Gordon equation is related to the relativistic energy–momentum relation E 2 = m 2 + p 2 via the correspondence principle that includes a term E 2 . Thus, a ‘better’ Lorentz covariant wave equation with a positive-definite probability density should have a first-order time derivative only. However, the equivalence of time and space coordinates in Minkowski space requires that such an equation also have only first-order space derivatives.