First-order differential equations for single-particle quantum mechanical systems.

Coupled first-order differential forms of a single-particle Schrödinger equation are presented. These equations are convenient to solve efficiently using the widely available ordinary differential equation solvers. This is particularly true because the solutions to the differential equation are two sets of complementary functions that share simple and consistent mathematical relationships at the boundary and across the domain for a given potential. The differential equations are derived from an integral equation obtained using the Green's function for the kinetic operator, making them universally applicable to various systems. These equations are applied to the Yukawa potential -e^{-αr}/r to calculate the critical screening parameter α=1.19061242106061770534277710636105 using a standard quadruple precision calculation, which is the most accurate compared to similar calculations in the past that confirm the first 30 significant figures. Also reported is the interesting coincident point with the eigenvalue, α=-E=0.274376862689408994894705268554458.