385lec22 - PHYS 385 Lecture 22 Spherical harmonics II Lecture 22 Spherical harmonics II What's important associated Legendre functions quantized

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PHYS 385 Lecture 22 - Spherical harmonics - II 22 - 1 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. Lecture 22 - Spherical harmonics - II What's important : associated Legendre functions quantized angular momentum Text : Gasiorowicz, Chap. 11 In the previous lecture, we separated out the angular part of the Schrödinger equation and solved for the azimuthal part ( ) of the wavefunction as well as the special case m = 0 of the polar function ( ) in the equation: sin d d sin d d + l ( l + 1)sin 2 - m 2 [ ] = 0 (1) The solution set for m = 0 is the Legendre polynomials ( ) = P L (cos ), and we showed that l is required to have integer values. Associated Legendre functions When m 0, more work is required to solve the equation. First, perform the substitution x = cos( ) so that Eq. (1) becomes Eq. (8) in the previous lecture: d dx 1 - x 2 ( 29 dP dx + l ( l + 1) - m 2 (1 - x 2 ) P = 0. (2) Expanding the first term: d 2 P l m ( x ) dx 2 - 2 x dP l m ( x ) dx - x 2 d 2 P l m ( x ) dx 2 + l ( l + 1) - m 2 1 - x 2 P l m ( x ) = 0 (3) or (1 - x 2 ) d 2 P l m ( x ) dx 2 - 2 x dP l m ( x ) dx + l ( l + - m 2 1 - x 2 P l m ( x ) = (4) This equation looks like trouble because of the (1- x 2 ) in the denominator of the square brackets.
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This note was uploaded on 09/07/2009 for the course PHYS 385 taught by Professor Davidboal during the Spring '09 term at Simon Fraser.

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385lec22 - PHYS 385 Lecture 22 Spherical harmonics II Lecture 22 Spherical harmonics II What's important associated Legendre functions quantized

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