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The Open Nuclear & Particle Physics Journal

A Theorem Relative to the Centrifugal Force Exerted Over a Relativistic Quantum Particle

M. Grado-Caffaro

Published May 17, 2013• 0 citations
Publisher

Abstract

We prove that a free relativistic rotor (or rotator) Klein-Gordon field is massless.To this end, the magnitude of the centrifugal force and its associated quantum operator for a free particle with a circular motion are determined in Special Relativity.A partial differential equation including the centrifugal force operator is derived.Finally, from the conjunction of the above equation and the Klein-Gordon equation, the desired result is reached.

DOI
10.2174/1874415x01306010001

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We prove that a free relativistic rotor (or rotator) Klein-Gordon field is massless.To this end, the magnitude of the centrifugal force and its associated quantum operator for a free particle with a circular motion are determined in Special Relativity.A partial differential equation including the centrifugal force operator is derived.Finally, from the conjunction of the above equation and the Klein-Gordon equation, the desired result is reached.
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.
In this paper, we show that the Klein-Gordon quantum-mechanical operator, operating on the function ${e}^{\frac{\mathrm{iS}}{\ensuremath{\hbar}}}$, where SSS is the relativistic action of a free particle or of a particle in a field defined by a four vector, is completely equivalent to the relativistic Hamilton-Jacobi equation, provided one takes into account the vanishing of the divergence of the energy-momentum four vector. From this we see that classical relativistic mechanics can be formulated in terms of operators which are identical with those used in quantum mechanics.
We underscore the need of relativistic quantum mechanics in dealing with particles. Various notations used in relativity are defined and explained. We derive the Klein–Gordon equation for free particle as well as for particle in electromagnetic field. We obtain a solution and interpret it. We derive the Dirac equation for free particle as well as for particle in electromagnetic field. Plane wave solution to the Dirac equation is obtained. We show how spin of electron, idea of antiparticle , phenomena of pair production and pair annihilation follow from the Dirac equation.
A system of two equations is found that has solutions which coincide with the solutions of the Klein‐Gordon equation in the rest frame. This system includes the Schrödinger equation for a free neutral spinless particle. Using the Schrödinger equation as an additional condition for solving the Klein‐Gordon equation in the rest frame leads to two Helmholtz equations. Helmholtz equations can be solved by specifying a particle model and boundary conditions. One of the Helmholtz equations leads to discreteness of the rest masses of relativistic particles.
We present Heisenberg's equation of motion for the radial variable of a free non-relativistic particle in D dimensions. The resulting radial force consists of three contributions: (i) the quantum fictitious force which is either attractive or repulsive depending on the number of dimensions, (ii) a singular quantum force located at the origin, and (iii) the centrifugal force associated with non-vanishing angular momentum. Moreover, we use Heisenberg's uncertainty relation to introduce a lower bound for the kinetic energy of an ensemble of neutral particles. This bound is quadratic in the number of atoms and can be traced back to the repulsive quantum fictitious potential. All three forces arise for a free particle: “Force without force”.