On the Nuclear Two-, Three- and Four-Body Problems

The simplest nuclear Hamiltonian satisfying all present requirements includes a Majorana-Heisenberg interaction $\mathrm{}\left\{\right(1-g)P+gPQ\}V\left(r\right)\mathrm{}$ between unlike particles and an attractive singlet interaction between like particles which is equal to that for unlike particles. The experimental mass defects of ${\mathrm{H}}^2$ and ${\mathrm{H}}^3$ together with the cross section $\sigma$ for slow neutron-proton scattering will determine the range $b$ and depth $B$ of the triplet well and the proportion $g$ of Heisenberg force (we use throughout the potential $B{e}^{-2\mathrm{}\frac{r}{b}}$). An exact analytic expression relating $\sigma$, $b$, $B$ and $g$ is derived for this potential and $g$ is found to be very insensitive to $\sigma$. An exact solution of ${\mathrm{H}}^2$ gives the relation between $B$ and $b$. The final relation which fixes the parameters is furnished by a Ritz-Hylleraas variational treatment of ${\mathrm{H}}^3$ with the above Hamiltonian and the wave function: $\psi =2^{-\frac{1}{2}}{\alpha }_1({\alpha }_2{\beta }_3-{\alpha }_3{\beta }_2){\phi }_1+6^{-\frac{1}{2}}\left({\alpha }_1\right({\alpha }_2{\beta }_3+{\alpha }_3{\beta }_2)-2\mathrm{}{\beta }_1{\alpha }_2{\alpha }_3){\phi }_2$ where ${\phi }_1$ and ${\phi }_2$ each represents an exponential times a power series in the interparticle distances of proper symmetry ${\phi }_2$ is brought in by the Heisenberg term; the Breit-Feenberg operator is used for the small triplet like-particle interaction). The convergence of energies obtained from successive improvements in $\psi$ is rapid and the eigenvalue may be closely estimated. After a relativistic correction is made we obtain: $b=1.73\mathrm{}\times {10}^{-13\mathrm{}}$ cm; $B=242\mathrm{} m{c}^2$ and