On the geometric foundations of nuclear shell structure

Abstract

Cartan's geometric theory of partial differential equations is applied to a system of Schrödinger equations. It is shown that the choice of a Riemann manifold which is a torus is equivalent to using a many-body neutron and proton potential commonly used in nuclear theory. The theory is applied to spinless, ground-state systems using the Dirichlet principle to minimise the energy, to obtain the neutron-proton ratios, Coulomb and binding energies of nuclei. A shell structure naturally manifests itself from the choice of the manifold.