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Chapter4_7 - 2 2 = m l l d d d d The solution is(cos m l lm...

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PHY4604 R. D. Field Department of Physics Chapter4_7.doc University of Florida Associated Legendre Function The Angular Equation The angular equation is ) , ( ) 1 ( ) , ( ) ( 2 2 φ θ φ θ Y l l Y L op h + = or 0 ) , ( ) 1 ( ) , ( sin 1 ) , ( sin sin 1 2 2 2 = + + + φ θ φ φ θ θ θ φ θ θ θ θ Y l l Y Y , where I set 2 ) 1 ( h + = l l C . The Y( θ , φ ) are eigenvalues of the L 2 operator with eigenvalue 2 ) 1 ( h + l l . We look for solutions of the form ) ( ) ( ) , ( φ θ φ θ Φ Θ = Y . The equation becomes 0 1 sin ) 1 ( sin sin 1 2 2 2 = Φ Φ + + + Θ Θ φ θ θ θ θ θ d d l l d d d d and hence if we call the separation constant m 2 then 2 2 sin ) 1 ( sin sin 1 m l l d d d d = + + Θ Θ θ θ θ θ θ and 2 2 2 1 m d d = Φ Φ φ The Φ Equation: The solution of the Φ equation is easy with Φ = Φ 2 2 2 m d d φ with solution φ φ φ im im Be Ae + + = Φ ) ( . We will write this as φ φ im m e + = Φ ) ( and allow m to be positive and negative (and we will save the normalization until later). If we require that ) ( φ Φ be single-valued ( i.e. ) ( ) 2 ( φ π φ Φ = + Φ ) then m = 0, ±1, ±2, …. Note that m m m op z m i L Φ = Φ = Φ h h φ ) ( , which means that Φ m is an eigenfunction of the L z operator with eigenvalue
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Unformatted text preview: 2 2 = + + m l l d d d d . The solution is ) (cos ) ( m l lm AP = , with ) ( ) 1 ( ) ( | | 2 |/ | 2 x P dx d x x P l m m m l = and l l l l x dx d l x P ) 1 ( ! 2 1 ) ( 2 = with l = 0, 1, 2, and m = -l , -l +1, , + l . Spherical Harmonics: The normalized wave functions ) (cos |)! | ( 4 |)! | )( 1 2 ( ) ( ) ( ) , ( m l im m lm lm P e m l m l l Y + + + = = , where = (-1) m for m 0 and = 1 for m < 0. The Y lm form an orthonormal set, [ ] ' ' 2 0 0 ' ' sin ) , ( ) , ( mm ll lm m l d d Y Y = . Legendre Polynomials...
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