General spherical harmonic bra-ket overlap integrals of trigonometric functions
Giuseppe Lingetti, Paolo Pani
TL;DR
This paper derives closed-form formulas for spherical harmonic overlap integrals of trigonometric functions using Clebsch-Gordan coefficients, aiding complex wave equation problems in physics.
Contribution
It provides new analytical expressions for overlap integrals in terms of double sums of Clebsch-Gordan coefficients, applicable to various physical theories.
Findings
Closed-form formulas for overlap integrals derived
Formulas expressed as double sums of Clebsch-Gordan coefficients
Applicable to non-separable wave equations in physics
Abstract
Closed formulas in terms of double sums of Clebsch-Gordan coefficients are computed for the evaluation of bra-ket spherical harmonic overlap integrals of a wide class of trigonometric functions. These analytical expressions can find useful application in problems involving non-separable wave equations, e.g. general-relativistic perturbation theory, electromagnetism, quantum theory, etc, wherein the overlap integrals arise from the coupling among different angular modes.
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