The intervals of oscillations in the solutions of the radial Schrödinger differential equation

Advances in Difference Equations

James Graham-Eagle¹  Dimitris M Christodoulou¹  Qutaibeh D Katatbeh² 

[1] Department of Mathematical Sciences, University of Massachusetts Lowell, Lowell, USA;Department of Mathematics and Statistics, Jordan University of Science and Technology, Irbid, Jordan

关键词: oscillations;    second-order linear differential equations;    analytical theory;    transformations;    Schrödinger differential equation;    Kummer’s differential equation;    Whittaker differential equation;    34A25;    34A30;    81Q05;   

DOI  :  10.1186/s13662-016-0777-7

学科分类：数学（综合）

来源: SpringerOpen

【 摘 要 】

We have previously formulated a program for deducing the intervals of oscillations in the solutions of ordinary second-order linear homogeneous differential equations. In this work, we demonstrate how the oscillation-detection program can be carried out for the radial Schrödinger equation when a Coulomb potential is used to describe the hydrogen atom. The method predicts that the oscillation intervals are finite in radius and their sizes are determined uniquely by the two quantum numbers n and ℓ. Numerical integrations using physical boundary conditions at the origin confirm this oscillatory behavior of the radial Coulomb wavefunctions. Two related differential equations due to Kummer and Whittaker and other attractive electrostatic potentials are also discussed in the same context.

【 授权许可】

CC BY