Normal States of Nuclear Three- and Four-Body Systems

Abstract

A variational method employing orthogonal (Hermite) functions in linear combination is used for calculating the binding energies of ${\mathrm{H}}^{3}$ and ${\mathrm{He}}^{4}$, with the following choice of nuclear constants: $A=35.60 \mathrm{Mev}; a=2.25\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13} \mathrm{cm}; g=0.20,$ and an error function potential. Two types of coordinates are used for ${\mathrm{H}}^{3}$; (1) normal, (2) individual particle coordinates. Their advantages and disadvantages are discussed. With the use of a certain limited set of functions of normal coordinates the energy has been depressed from -6.21 Mev in zeroth approximation to -7.21 Mev, while a suitable set of functions involving only individual particle coordinates reduced it from -6.16 Mev to -6.84 Mev. The second Schr\"odinger perturbation is less effective than the latter scheme by 0.22 Mev. Functions of different symmetry, called into play by the Heisenberg operators, are found to contribute 0.07 Mev on the basis of a modified variation method. ${\mathrm{He}}^{4}$ has been treated only with the use of individual particle coordinates (Hartree method). A similar group of functions lowers the energy from -24.81 Mev in zeroth approximation to -25.85 Mev, which is better than the effect of Schr\"odinger's perturbation theory by 0.25 Mev. General estimates of convergence limits are given.