On the Iteration Method and Its Application to the Oxygen Problem

Past calculations for nuclei heavier than He lead to approximate binding energies considerably less than the experimental values. There are two possible explanations, still in need of clarification: (1) the Hamiltonian is wrong, (2) the convergence of the variation or perturbation methods used is too slow to yield sufficiently accurate results with a reasonable amount of work. A variational method which is essentially equivalent to a perturbation calculation including first-, second-, and third-order contributions to the energy yields strong evidence for believing that in the lowest state of the selected Hamiltonian the oxygen nucleus has a finite radius and the binding energy is greater than that of four $\alpha$-particles. The Hamiltonian is assumed in the form of "Wigner's first approximation." It is the simplest form which leads to saturation. Methods of extrapolation permit an estimate of its true binding energy, and an approximate value of about 230 $m{c}^2$ is obtained. Slowness of convergence is also demonstrated. This paper is concerned with an analysis of the evidence which leads to the above conclusions. The variational method used is the "iteration method" described in Appendix I. It presents many interesting features, notably it permits an estimate of the lowest eigenvalue of the Hamiltonian. The formalism necessary for the calculation of the matrix elements is derived in Appendices II-IV.