Principle of Minimal Energy in Relativistic Schroedinger Theory

TL;DR

This paper develops a variational principle for relativistic quantum systems that incorporates particle interactions dynamically, enabling approximate solutions for bound states like positronium with improved physical accuracy.

Contribution

It introduces a new variational principle for relativistic bound states that includes dynamical interactions, extending the Ritz principle to relativistic quantum mechanics.

Findings

Successfully reproduces positronium groundstate energy

Provides a framework for variational solutions in relativistic quantum systems

Incorporates dynamical particle interactions into energy minimization

Abstract

The Hamilton-Lagrange action principle for Relativistic Schr\"odinger Theory (RST) is converted to a variational principle (with constraints) for the stationary bound states. The groundstate energy is the minimally possible value of the corresponding energy functional and the relativistic energy eigenvalue equations do appear as the corresponding variational equations. The matter part of these eigenvalue equations is a relativistic generalization of the well-known Ritz principle in non-relativistic quantum mechanics which however disregards the dynamical character of the particle interactions. If the latter are included in the proposed principle of minimal energy for the bound states, one obtains a closed dynamical system for both matter and gauge fields. The new variational principle enables the development of variational techniques for solving approximately the energy eigenvalue equations. As a demonstration, the positronium groundstate is treated in great detail. Here a simple exponential trial function is sufficient in order to reproduce the (exact) result of conventional quantum mechanics where the relativistic and spin effects are neglected.