The Shell Model – Simplicity from Complexity

Abstract

In order to apply the shell model to most nuclei, an effective ("residual") interaction between nucleons should be added to the single nucleon Hamiltonian. It cannot be the bare interaction which would introduce strong short range correlations into the shell model wave functions of nucleons moving independently in a potential well. For many years, many authors have been developing methods for calculating from the bare interaction, an effective interaction which may be used in the shell model. Only recently, the results seem to agree with experiment. During these years, successful shell model calculations have been carried out by using effective interactions determined consistently from measured energies of nuclei. Some old and some new results exhibit the success of this approach. In principle, shell model wave functions may be determined by a renormalized Hamiltonian which contains the effective interactions rather than the bare ones. Shell model wave functions may be used to calculate other observables if matrix elements are taken of renormalized operators. Such a program could succeed irrespective of the relation between shell model wave functions and the real ones. This would certainly be the case for ab initio calculations where a central potential well is not assumed and derivation of nuclear structure is attempted by using many body theory of nucleons interacting by the bare interaction. Still, there are observables which have been successfully calculated by shell model wave functions from real, un-renormalized operators. Several examples are briefly described below. This fact may be taken as some evidence that the "wounds" inflicted on shell model wave functions by the short range correlations occupy only a small volume and that there is a considerable overlap between shell model wave functions and the real ones. The calculated overlap between shell model and real wave functions may be obtained from derivations of the effective interaction. It is more difficult to see how the shell model may emerge as a good approximation from ab initio calculations. The question is whether the simplicity of the shell model as a good approximation will emerge from those complex calculations.