Wave Functions for Nuclear Shell Theory by a Variation Method

A system of $A$ nucleons with two-nucleon potentials $V(i, k)$ acting between each pair of them is taken as a model for the nucleus. A variation method for obtaining the best approximation to the wave function of this system as a linear combination of a given set of functions, ${\psi }_1, \cdots , {\psi }_q$, is presented. In view of the evidence for shell effects, each ${\psi }_{\alpha }$ is assumed to describe $A$ identical particles moving in a central field $V^{\prime }$. In general, the amplitudes of only a small number of ${\psi }_{\alpha }$ are large. For a central Gaussian interaction $V(i, k)$ and a harmonic oscillator potential $V^{\prime }$, this method can account for the binding energy of one neutron in ${\mathrm{O}}^{17}$, but not for the total binding energy of ${\mathrm{O}}^{16}$. The model is compared with a spectroscopic one for a double closed shell core plus outer nucleons.