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Unformatted text preview: 1 PHYS809 Class 13 Notes Spherical harmonics The separated equation for the azimuthal angle has solutions . im e φ ± These satisfy the orthogonality relation ( ) ( ) 2 2 2 . im in im in mn e e d e e d π π φ φ φ φ φ φ πδ *- ⋅ = ⋅ = ∫ ∫ ` (13.1) The corresponding orthonormal functions are ( ) 1 . 2 im m e φ φ π Φ = (13.2) The spherical harmonic functions are produced by combining this function with a normalized associated Legendre function ( ) ( ) ( ) ( ) ( ) ! 2 1 , 1 cos . 4 ! m m m im l l l m l Y P e l m φ θ φ θ π- + = - + (13.3) The Condon – Shortley phase factor ( ) 1 m- is introduced for convenience in the quantum theory of angular momentum. Be aware that in some books this phase factor is placed in the definition of ( ) m l P x and in others in the spherical harmonic. The equation ( ) ( ) ( ) ( ) ( ) ! 1 , ! m m m l l l m P x P x l m-- = - + can be used to show that ( ) ( ) ( ) , 1 , . m m m l l Y Y θ φ θ φ *- = - (13.4) By construction, the spherical harmonics are orthonormal over a spherical surface: ( ) ( ) 1 2 1 2 1 2 1 2 , , sin . m m l l l l m m sphere Y Y d d θ φ θ φ θ θ φ δ δ * = ∫ (13.5) The complete solution of Laplace’s equation in spherical polar coordinates is 2 ( ) 1 , . l l m lm lm l l l m l B A r Y r ψ θ φ ∞ + = =- = + ∑ ∑ (13.6) Since the electrostatic potential is a real function of position but the spherical harmonics are complex functions of the angular variables, we see that the pair of terms involving the coefficients A lm and A l,-m must combine in such a way as to make the solution real. Parity of the spherical harmonics Consider the transformation . → - x x In spherical polar coordinates this transformation is ( ) ( ) , , , , . r r θ φ π θ φ π →- + Since ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ! 2 1 , 1 cos 4 ! ! 2 1 1 1 cos 1 4 ! 1 , . m m m im im l l m l m m m im l l m l l m l Y P e e l m l m l P e l m Y φ π φ π θ φ π θ π θ π θ φ +- +- + = -- +- + = --- + = - (13.7) Hence the spherical harmonics have parity ( ) 1 . l- Addition theorem for spherical harmonics Consider two vectors x and ′ x with spherical polar coordinates ( ) , , r θ φ and ( ) , , . r θ φ ′ ′ ′ Let γ be the angle between these two vectors. The addition theorem relates the Legendre polynomial of degree l in cos γ to spherical harmonics of the angles ( ) , θ φ and ( ) , : θ φ ′ ′ ( ) ( ) ( ) 4 cos , , ....
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- Fall '11
- Spherical Harmonics, Alm, Associated Legendre function