Relativistic Dynamics of a Quantum System

Gintautas P Kamuntavičius,G Kamuntavičius

doi:10.4236/jamp.2017.57121

Abstract

In this paper, we examine quantum systems with relativistic dynamics. We show that for a successful description of these systems, the application of Galilei invariant nonrelativistic Hamiltonian is necessary. To modify this Hamiltonian to relativistic dynamics, we require precise relativistic kinetic energy operators instead of nonrelativistic ones for every internal (Jacobi) coordinate. Finally, we introduce and investigate the Schr&oumldinger equation with relativistic dynamics for two-particle systems with harmonic oscillator and Coulomb potentials.

Highlights

The definition of relativistic quantum mechanical momentum operator, introduced in [1], allows us to modify the Schrödinger equation for free particle, taking into account basic ideas of special relativity theory (SRT)
The introduced method of relativistic corrections of bound state energies of twoparticle systems with harmonic oscillator potential presents the eigenvalues as functions of one dimensionless parameter β < 2 3
Our investigation has shown that successful relativistic dynamics application needs Galilei invariant formalism for the description of a quantum system

Summary

Introduction

The definition of relativistic quantum mechanical momentum operator, introduced in [1], allows us to modify the Schrödinger equation for free particle, taking into account basic ideas of special relativity theory (SRT). We have to carefully investigate the main conclusions of SRT and quantum mechanics to find the possibility of consistence of both theories. - the general relation between radius vector r and time t of the event as seen in the reference frame S and the radius vector r′ and time t′ of the same event in the reference frame S′ , which is moving with uniform velocity relative to S, is defined by Lorentz transformation, which implies invariance of the space-time interval, defined as the scalar product of these four-vectors: s2= (ct )2 − r 2= (ct′)2 − (r′)2 ;. The main problem is—how and to what extent can these statements be applied in the theory of quantum systems?

Quantum System’s Schrödinger Equation

Internal Hamiltonian of a Two-Particle System

The Harmonic Oscillator Potential

The Coulomb Potential

Results

Conclusions