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Analytic matrix elements of the Schr dinger equation

Keywords: Hydrogen-like systems , Schr dinger equation , Quantum System , Bernstein-Polynomials , Energy Spectrum

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Abstract:

A previously defined analytic technique of constructing matrix elementsfrom the Bernstein-polynomials (B-poly) has been applied toSchr¨odinger equation. This method after solving generalized eigenvalueproblem yields very accurate eigenenergies and eigenvectors. The numericaleigenvectors and eigenvalues obtained from this process agreewell with exact results of the hydrogen-like systems. Furthermore, accuracyof the numerical spectrum of hydrogen equation depends on thenumber of B-polys being used to construct the analytical matrix elements.Validity of eigenvalues and quality of the constructed wavefunctionsis verified by evaluating the Thomas-Reiche-Kuhn (TRK) sumrules. Excellent numerical agreement is seen with exact results of hydrogenatom.

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