The new wave equations of relativistic and non-relativistic quantum mechanics

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.

In this paper we introduce models of short wave-long wave interactions in the relativistic setting. In this context the nonlinear Schr\"odinger equation is no longer adequate for describing short waves and is replaced by a nonlinear Dirac equation. Two specific examples are considered: the case where the long waves are governed by a scalar conservation law; and the case where the long waves are governed by the augmented Born-Infeld equations in electromagnetism.

Examination of the Einstein energy-momentum relationship suggests that simple unbound forms of matter exist in a four-dimensional Euclidean space. Position, momentum, velocity, and other vector quantities can be expressed as Euclidean four-vectors, with the magnitude of the velocity vector having a constant value, the speed of light. We see that charge may be simply a manifestation of momentum in the new fourth direction, which implies that charge conservation is a form of mo...

A derivation is presented of the quantummechanical wave equations based upon the Equity Principle of Einstein's General Relativity Theory. This is believed to be more generic than the common derivations based upon Einstein's energy relationship for moving particles. It is shown that Schrodinger's Equation, if properly formulated, is relativisticly covariant. This makes the critisized Klein-Gordon Equation for spinless massparticles obsolete. Therefore Dirac's Equation is pres...

In this work consideration is given to massless and massive gauge-invariant spin 0 and spin 1 fields (particles) within the scope of a theory of the generalized relativistic wave equations with an extended set of the Lorentz group representations. The results obtained may be useful as regards the application of a relativistic wave-equation theory in modern field models.

The properties of relativistic particles in the quasiclassical region are investigated. The relativistic semiclassical wave equation appropriate in the quasiclassical region is derived. It is shown that the leading-order WKB quantization rule is the appropriate method to solve the equation obtained.

Ehrenfest theorem is proven in relativistic quantum theory of charged particles, moving under the influence of an external electromagnetic field. In order to extend the classic Ehrenfest result to the relativistic domain we bypassed the problems with the relativistic position operator by deriving directly Newton's second law. Our approach is characterized by its universality. The detailed form of the wave equation is not needed. All that is required is the existence of the co...

The Schr\"odinger equation is universally accepted due to its excellent predictions aligning with observed results within its defined conditions. Nevertheless, it does not seem to possess the simplicity of fundamental laws, such as Newton's laws of motion. Various insightful attempts have been made to elucidate the rationale behind the Schr\"odinger equation. This paper seeks to review existing explanations and propose some prospectives on the derivation of the Schr\"odinger ...

In this chapter we examine the quantised electromagnetic (EM) field in the context of a Schr\"odinger equation for single photons. For clarity we consider only a one-dimensional system. As a universal tool for calculating the time-evolution of quantum states, a Schr\"odinger equation must exist that describes the propagation of single photons. Being inherently relativistic, however, critical aspects of both special relativity and quantum mechanics must be combined when quanti...

This article devoted to relativistic dynamics of a charged massive particle in an electroscalar field. It represents a continuation of paper [1] where the authors constructed a non-relativistic theory which describes transverse electromagnetic waves along with longitudinal electroscalar ones, responsible for the wave transport of the Coulomb field. A new type of relativistic force exerted by electroscalar field on an electrically charged particle and the relativistic law of s...