A previously defined analytic technique of constructing matrix elements from the Bernstein-polynomials (B-poly) has been applied to Schr¨odinger equation. This method after solving generalized eigenvalue problem yields very accurate eigenenergies and eigenvectors. The numerical eigenvectors and eigenvalues obtained from this process agree well with exact results of the hydrogen-like systems. Furthermore, accuracy of the numerical spectrum of hydrogen equation depends on the number of B-polys being used to construct the analytical matrix elements. Validity of eigenvalues and quality of the constructed wavefunctions is verified by evaluating the Thomas-Reiche-Kuhn (TRK) sum rules. Excellent numerical agreement is seen with exact results of hydrogen atom.
Bhatti, Muhammad I. "Analytic Matrix Elements of the Schrödinger Equation." Adv. Stud. Theor. Phys 7.11 (2013). http://dx.doi.org/10.12988/astp.2013.13002
Adv. Stud. Theor. Phys