Neutron-Proton Scattering with Spin-Orbit Coupling. II. Variational Formulation and Effective Range Theory

The variational methods of Schwinger et al. are used to give variationally correct expressions for the three independent real parameters in the scattering matrix (${\delta }_{J\alpha }$, ${\delta }_{J\beta }$, and ${\epsilon }_J$) for the "mixed" states, such as ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}+{}_{}{}^3{}_{}{}^{}D_1^{}{}_{}{}^{}$, of the neutron-proton system with spin-orbit coupling. The relation between these variational principles and the effective range theory is discussed briefly. The effective range expansions are then derived nonvariationally. The two leading terms are given for $k^5{cot\delta }_{1\beta }$ and for $k^{-2\mathrm{}}{tan\epsilon }_1$. The three leading terms are given for $k{cot\delta }_{1\alpha }$.