Variational Calculation of Electron Scattering by a Static Potential

The Schwinger variational method, for the approximate determination of scattering amplitude, is tested for accuracy in the case of the elastic scattering of electrons from the static potential $V\left(\mathrm{r}\right)=-\frac{e^2{V}_0{e}^{-\lambda r}}{r}$, by using eight different forms of trial wave functions. The results are compared by checking the closeness of fit of the associated scattering amplitude with an exact solution to the problem. In the course of the calculation a number of expressions, of use in more complicated problems, were obtained and are here recorded. The parameter values used in the test were $a_0\lambda =\frac{8}{3}$, $k^2={\left(0.72\mathrm{}\lambda \right)}^2$, $V_0=7.8\mathrm{}$, where $a_0$ is the first Bohr orbit radius for hydrogen.