A new variable step method for the numerical integration of the one- dimensional Schrödinger equation
A new variable step method is presented for the numerical integration of the one-dimensional Schrödinger equation of the form \[ y''(r)=(U(r)- E)y(r)=f(r)y(r) \] with the boundary condition \(y(0)=0\), \(\lim_{r\to \infty}y(r)=0\) and the continuity condition for y(r) and y'(r) at any arbitrary point \(r_ 0\), \(0<r_ 0<+\infty\), where U(r) is a given potential function, r is the internuclear distance. The authors were inspired by the work of \textit{A. D. Raptis} and \textit{J. R. Cash} [Comput. Phys. Commun. 36, 113-119 (1985; Zbl 0578.65086)]. This variable step method makes use of the ``canonical functions'' approach. At each interval, an estimate of the step-size h is determined in order to obtain the desired local accuracy. For the practical case of a given potential function with polynomial function with polynomial interpolations, the step-size h is directly given by a simple formula. The method is applied to the Morse potential function and numerical examples are presented.
scientific article; zbMATH DE number 4068200