Two-Nucleon Hard-Core Interaction in a Common Harmonic-Oscillator Potential

The problem of two nucleons with no interaction except a repulsive hard core, assuming that both move in a common harmonic-oscillator potential, is considered. Analytic forms are obtained for the wave functions and the energy eigenvalues which are found to depend on the radius of the hard core. The method is applied to the calculation of the spin-orbit splitting in ${\mathrm{O}}^{17}$, treating the two-nucleon spin-orbit interaction as a perturbation and using the hard-core wave functions as the unperturbed wave functions. A value of 5.95 MeV is obtained for the spin-orbit separation for the $J=\frac{5}{2}$ and $J=\frac{3}{2}$ levels of ${\mathrm{O}}^{17}$.