Classical variational derivation and physical interpretation of Dirac's equation

Boris Podolsky

Eddington, in his book Relativity Theory of Protons and Electrons, makes statements to the effect that the invariance of Dirac's equation is an elementary consequence of the tensor form which it acquires in the new wave tensor calculus, in which it is merely an identity derivable on epistemological principles and not on physical hypothesis. It is shown in this article that the invariance of Dirac's equation referred to by Eddington is a purely formal one, while the invariance which is of particular interest in quantum mechanics depends upon the physical interpretation of the wave function. The generalization of physical interpretation suggested by Eddington's work, however, has the desirable quality of leaving Dirac's equation invariant in the physical sense. It is further shown that in the derivation of Dirac's equation Eddington makes use of the usual physical assumptions, but in somewhat disguised form.

Arjun Sharma

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.

成介 大高, Sigeyuki Otaka

Vasil G. Yordanov

In this paper, we present a stochastic approach to relativistic quantum mechanics. We formulate the three fundamental principles of this theory and derive the Dirac equations based on them. This approach enables us to bring more insight into the nature of Dirac's spinors. Furthermore, we provide a physical interpretation of the stochastic optimal control theory of quantum mechanics.