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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 These satisfy the orthogonality relation ( ) ( ) 2 2 2 . im in im in mn e e d e e d * = = ` (13.1) The corresponding orthonormal functions are ( ) 1 . 2 im m e = (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  + =  + (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 * =  (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 * = (13.5) The complete solution of Laplaces equation in spherical polar coordinates is 2 ( ) 1 , . l l m lm lm l l l m l B A r Y r + = = = + (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  + 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 + + + =  + + =  + =  (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 and ( ) , , . r Let be the angle between these two vectors. The addition theorem relates the Legendre polynomial of degree l in cos to spherical harmonics of the angles ( ) , and ( ) , : ( ) ( ) ( ) 4 cos , , ....
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This note was uploaded on 12/02/2011 for the course PHYS 809 taught by Professor Macdonald during the Fall '11 term at University of Delaware.
 Fall '11
 MacDonald

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