The Padua Algorithm for the Computation of Deuteron wave functions, Binding Energy and Other Properties

The deuteron is a bound state of a neutron and proton in $^{3}$S$_{1}$ and $^{3}$D$_{1}$ states. The wavefunctions that represent the $^{3}$S$_{1}$(u) {\&} $^{3}$D$_{1}$(w) are obtained by solving a Rarita-Schwinger (RS) equation. In this presentation, an algorithm pioneered by a Padua group [1] for solving the RS equation is described. The Padua algorithm yields accurate results with fewer mesh points (N=300 or more) compared to standard methods [2]. The algorithm relies on the transformation x= tan$^{-1}$(r). This truncates the infinite integration domain 0$<$r$<\infty $ to 0$<$x$<$ $\pi $/2. This enables a grid that is uniform in x to explore the inner most region of r with a finer mesh than the asymptotic region. After second-order central difference approximations are used to replace the derivatives in the modified RS equation, it reduces to a standard eigenvaue problem of the form AY = E$^{2}$Y, where A is a 2Nx2N matrix, N the maximum number of steps, E the binding energy and Y is u(n) (1$<$n$<$N) and w(n) (N+1$<$n$<$2N). n is an index for individual steps. A FOTRAN program with a subroutine from IMSL is used to solve the RS equation for the Reid Soft Core Potential to illustrate the algorithm. [1] T.A. Minnelli, A.Pacolini, and C. Villi, Nuovo Cimento, 101, (1991) p.1626 [2] W.H.Press et al. Numerical Recipes, 1986, Ch. 16