ANALYTIC RELATIVISTIC 3-FERMION HARMONIC-OSCILLATOR WAVE-FUNCTIONS

The hyperspherical method is applied to the three-body Dirac equation. Three spin-1/2 particles of equal masses are considered. A solution f or the three-fermion wave function in a given configuration consists o f an eight-component radial wave function in the two-component Dirac n otation. A central diagonal quadratic harmonic oscillator two-body pot ential energy is added to the relativistic mass and kinetic-energy ope rators. Harmonic-oscillator-type Gaussian solutions of the three-body Dirac equation, analytic in the energy and mass, are found for the var ious natural-parity configurations likely to be important in the three -body descriptions of the nucleon, if one chooses an appropriate two-b ody potential. The configurations considered are the (1/2(+))(3), the (1/2(-))(2)1/2(+) positive-parity configurations, and the (1/2(+))(2)1 /2(-) as well as (1/2(-))(3) configurations of negative parity. Analyt ic solutions are obtained for a relativistic parameter defined as S = (E - 3M)/(E + 3M) ranging from zero to one. This is from the completel y non-relativistic to the extreme relativistic case. E is the total en ergy of the system and M is the rest mass of each of the fermions. Oth er more realistic potentials will couple these configurations together into a coupled set of equations. These analytic Gaussian-type solutio ns 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 wi ll decouple from each other. This relativistic solution also allows on e to determine the importance of relativistic effects by comparing res ults 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 t he limit of mass M tending to zero.