Principles of Quantum Mechanics · Problem 12.6.10

The Hydrogen Atom — R. Shankar

Physics — Quantum Mechanics · plane-wave-expansion

Problem

*(Optional).* Verify Eq. (12.6.41) given that (1)

\displaystyle\int_{-1}^{1} P_l(\cos\theta) P_{l'}(\cos\theta)\, d(\cos\theta) = [2/(2l+1)]\delta_{ll'}

(2)

P_l(x) = \dfrac{1}{2^l l!} \dfrac{d^l(x^2-1)^l}{dx^l}

(3)

\displaystyle\int_0^1 (1-x^2)^m\, dx = \dfrac{(2m)!!}{(2m+1)!!}

Hint: Consider the limit

kr \to 0

after projecting out

C_l