Introduction to Quantum Mechanics · Problem 4.63

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · spherical_harmonics_normalization

Problem

Work out the normalization factor for the spherical harmonics, as follows. From Section 4.1.2 we know that Yℓm=KℓmeimϕPℓm(cos⁡θ);Y_\ell^m = K_\ell^m e^{im\phi} P_\ell^m(\cos\theta); the problem is to determine the factor KℓmK_\ell^m (which I quoted, but did not derive, in Equation 4.32). Use Equations 4.120, 4.121, and 4.130 to obtain a recursion relation giving Kℓm+1K_\ell^{m+1} in terms of KℓmK_\ell^m. Solve it by induction on mm to get KℓmK_\ell^m up to an overall constant, C(ℓ)≡Kℓ0C(\ell) \equiv K_\ell^0. Finally, use the result of Problem 4.25 to fix the constant. You may find the following formula for the derivative of an associated Legendre function useful: (1−x2)dPℓmdx=−1−x2 Pℓm+1−mxPℓm.(4.226)(1-x^2)\frac{dP_\ell^m}{dx} = -\sqrt{1-x^2}\, P_\ell^{m+1} - m x P_\ell^m. \qquad (4.226)

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