Analytic Relativistic Three-Fermion Harmonic-Oscillator Wave Functions

Abstract

The hyperspherical method is applied to the three-body Dirac equation. Three spin −1/2 particles of equal masses are considered. A solution for the three-fermion wave function in a given configuration consists of an eight-component radial wave function in the two-component Dirac notation. A central diagonal quadratic harmonic oscillator two-body potential energy is added to the relativistic mass and kinetic-energy operators. Harmonic-oscillator-type Gaussian solutions of the three-body Dirac equation, analytic in the energy and mass, are found for the various natural-parity configurations likely to be important in the three-body descriptions of the nucleon, if one chooses an appropriate two-body potential. The configurations considered are the (1+/2)3, the (1−/2)21+/2 positive-parity configurations, and the (1+/2)21−/2 as well as (1−/2)3 configurations of negative parity. Analytic solutions are obtained for a relativistic parameter defined as S = (E− 3M) / (E+ 3M) ranging from zero to one. This is from the completely non-relativistic to the extreme relativistic case. E is the total energy of the system and M is the rest mass of each of the fermions. Other more realistic potentials will couple these configurations together into a coupled set of equations. These analytic Gaussian-type solutions are convenient for studying the effect of using other potentials on a bound wave function that includes the relativistic mass and kinetic energy. For certain hyper-harmonic potentials these configurations will decouple from each other. This relativistic solution also allows one to determine the importance of relativistic effects by comparing results for a given potential in a non-relativistic Schrodinger equation to results using the same potential in this three-body Dirac equation approach. The analytic solutions found here remain tractable even in the limit of mass M tending to zero.