Mass-Energy Relation in Quantum Theory

Abstract

It does not seem to be generally recognized that the mass-energy relation is not contained implicitly in the quantum theory, but is something superposed upon it, and without any clear guarantee as to its consistency with that theory. Even classical relativistic particle dynamics does not provide an explanation of defect.The first step in the paper is to produce a theory of rest-mass defects founded upon a concept in which a nucleon consists of a number of subparticles, or as they will be called, which specks obey conservation of relativistic energy and momentum in a suitably chosen frame of reference, and with rest masses which are regarded as absolute constants. The theory leads to a situation in which the nucleons act as individual entities conserving momentum as a whole but with rest masses which can vary with the mutual proximities of the nucleons one to another. However, in place of a corresponding conservation of kinetic energy, we have conservation of $\ensuremath{\Sigma}({\ensuremath{\tau}}_{n}+{m}_{0n}{c}^{2})$, where ${\ensuremath{\tau}}_{n}$ is the ordinary relativistic kinetic energy, so that ${\ensuremath{\tau}}_{n}={m}_{0n}{c}^{2}[{(1\ensuremath{-}{{\ensuremath{\beta}}_{n}}^{2})}^{\ensuremath{-}\frac{1}{2}}\ensuremath{-}1]$.So far there is no mention of potential energy. However, at this stage we postulate that the ${m}_{0n}$ vary with the relative coordinates of the nucleons so that ${m}_{0n}{c}^{2}$ really becomes the representative of what is ordinarily regarded as a potential energy $V$. Indeed, ${m}_{0n}{c}^{2}$ becomes $V+\mathrm{const}$. We thus arrive at ${\ensuremath{\tau}}_{n}+V=\mathrm{const}$, but with the ${m}_{0n}$ occurring in ${\ensuremath{\tau}}_{n}$ being functions of relative coordinates.Normally, the next procedure would be to express ${\ensuremath{\tau}}_{n}$ in terms of the relativistic momentum, and then replace the momentum by the usual operator to form an equation for the $\ensuremath{\psi}$ function. A transformation to center-of-gravity coordinates and relative coordinates might be expected to lead to an expression for $\ensuremath{\psi}$ of the form $\ensuremath{\psi}=\ensuremath{\varphi}\ensuremath{\chi}$, where $\ensuremath{\varphi}$ is a function only of the relative coordinates and $\ensuremath{\chi}$ is a function of the center-of-gravity coordinates. It might then be expected that the mass-energy relation could reveal itself in the fact that the equation for $\ensuremath{\chi}$ represented that of a single entity moving with constant velocity and with a rest mass $\ensuremath{\Sigma}{m}_{n\ensuremath{\infty}}\ensuremath{-}\frac{{\ensuremath{\epsilon}}_{B}}{{c}^{2}}$, where the ${m}_{n\ensuremath{\infty}}$ are the rest masses of the nucleons when at infinity, and ${\ensuremath{\epsilon}}_{B}$ is the binding energy which, in turn, would be yielded by the $\ensuremath{\varphi}$ equation, as the negative of the lowest eigenvalue for the energy in that equation. Realization of this form for the rest mass of the nucleus is the aim of the paper.The $\ensuremath{\psi}$ equation formed from the classical relativistic Hamiltonian in the above manner is of nonlinear form, and it resists an immediate accommodation of itself to the above procedure. However, by use, for the nucleus, of a pseudomomentum slightly different from the true relativistic momentum as the quantity which is to be replaced by the operator, and by a slight change in the manner of making the transformation to the operator form, the desired results are achieved. Both modifications of the normal procedure revert to that procedure for the case of small velocities.