385lec21

# 385lec21 - PHYS 385 Lecture 21 - Spherical harmonics - I...

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PHYS 385 Lecture 21 - Spherical harmonics - I 21 - 1 ©2003 by David Boal, Simon Fraser University. All rights reserved; further copying or resale is strictly prohibited. Lecture 21 - Spherical harmonics - I What's important : Legendre polynomials spherical harmonics Text : Gasiorowicz, Chap. 11 Separation of variables In the previous lecture, we obtained an expression for the angular part of the wavefunction for central potenials which reads: - 1 sin sin + 1 sin 2 2 2 Y = Y l ( l + 1) (1) where the function Y is a function of the angles and in spherical polar coordinates. Recognizing that the and operations can be isolated by multiplying this expression by sin 2 , Eq. (1) can be packaged as sin sin Y + sin 2 l ( l + Y + 2 2 Y = 0 To effect the separation of variables, Y ( , ) is expressed as a product state: Y ( , ) = ( ) • ( ). (2) permitting the last equation to be written as 1 sin d d sin d d + sin 2 l ( l + = - 1 d 2 d 2 = m 2 , (3) after the usual division by and a small amount of algebraic manipulation. The separation constant m 2 on the rhs has been chosen with some foresight, which will be

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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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385lec21 - PHYS 385 Lecture 21 - Spherical harmonics - I...

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