Franz Schwabl Quantum Mechanics 4th Edit · Problem 5.2

The Central Potential I

Physics — Quantum Mechanics · spherical-harmonics-completeness

Problem

Show that the spherical harmonics satisfy the completeness relation (5.31). Use the addition theorem and the generating function for the Legendre polynomials 11−2ucos⁡ϑ+u2=∑l=0∞ulPl(cos⁡ϑ)  ,\frac{1}{\sqrt{1 - 2u\cos\vartheta + u^2}} = \sum_{l=0}^{\infty} u^l P_l(\cos\vartheta) \;, and prove that your result has the defining properties of the δ(2)\delta^{(2)}-function.

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