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277 Appendix Spherical Harmonic Functions and Wigner 3-j Symbols We briefly introduce the definition of the spherical harmonic functions. First we define Θ lm m l l m l l m i l l m l m P m i l l m l m P m θ θ θ () =− ( ) + ( ) − ( ) + ( ) ( ) ≥ = + ( ) − ( ) + ( ) ( ) < 1 2 1 2 0 2 1 2 0 ! ! cos ! ! cos for for (A.1) where P l m are associated Legendre polynomials and the phase factor i l is after Landau and Lifshitz [1989]. This definition is different from the usual one; however, this choice is the most natural from the viewpoint of the theory of the addition of angular momenta. Define Φ m im e φ π φ () = 1 2 (A.2) The normalized spherical harmonic function is given by Y lm lm m θφ θ φ , ( ) = () () Θ Φ (A.3) where we note that Y Y lm l m l m , * , , , θφ θφ ( ) =− ( ) ( ) − − 1 (A.4) The normalized spherical harmonic functions with different l or m are orthonormal: Y Y d lm l m ll mm 1 1 2 2 1 2 1 2 * , , , θφ θφ θφ δ δ ( ) ( ) ( ) = ∫ Ω (A.5) where d dd Ωθφ θθφ , sin ( ) = . The addition theorem holds for Legendre polynomials:

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  • 277

    Appendix

    Spherical Harmonic Functions andWigner 3-j Symbols

    We briefly introduce the definition of the spherical harmonic functions. First wedefine

    lmm l

    lm

    ll

    m

    il l m

    l mP m

    il l m

    l mP m

    ( ) = ( ) +( ) ( )+( ) ( )

    = +( ) ( )+( ) ( )


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