Gebremedhin Daniel, Weatherford Charles
Physics Department, Florida A&M University, Tallahassee, Florida 32307, USA.
Phys Rev E. 2023 Oct;108(4-2):045301. doi: 10.1103/PhysRevE.108.045301.
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.
给出了单粒子薛定谔方程的耦合一阶微分形式。这些方程便于使用广泛可用的常微分方程求解器进行高效求解。尤其如此，因为微分方程的解是两组互补函数，对于给定势，它们在边界处和整个定义域内具有简单且一致的数学关系。这些微分方程是从使用动能算符的格林函数得到的积分方程推导而来的，这使得它们普遍适用于各种系统。将这些方程应用于汤川势-e^{-αr}/r，使用标准四倍精度计算得出临界屏蔽参数α = 1.19061242106061770534277710636105，与过去类似计算相比，这是最精确的，过去的计算确认了前30位有效数字。还报告了与特征值的有趣重合点，α = -E = 0.274376862689408994894705268554458。
