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Unformatted text preview: CHEM 356, Lecture 20, Fall 2009 1 Orbital Angular Momentum Operators: Simultaneous Eigenfunctions. We saw that the differential equation governing the spherical harmonic functions ( , ) was separable, with ( , ) = Θ( )Φ( ), giving sin Θ( ) ( sin Θ ) + sin 2 + 1 Φ( ) 2 Φ 2 , with ≡ 2 ℏ 2 for the rigid rotor model. Because we already know the eigenfunctions Φ( ) = 1 √ 2 e , = 0 , ± 1 , ± 2 , ⋅⋅⋅ , we obtain the equation sin Θ( ) ( sin Θ ) + sin 2 − 2 = 0 , or, equivalently, sin ( sin ) Θ( ) + [ sin 2 − 2 ]Θ( ) = 0 , for Θ( ). If we now change variables from to using = cos , with = = cos = − sin , we get an equivalent equation in terms of , namely, − sin 2 [ − sin 2 ] Θ( ) + [ sin 2 − 2 ]Θ( ) = 0 . We may rewrite sin 2 in terms of as 1 − 2 , so that the equation for Θ( ) can be written entirely in terms of and derivatives with respect to as (1 − 2 ) [ (1 − 2 ) ] Θ( ) + [ (1 − 2 ) − 2 ]Θ( ) = 0 . An equivalent version of this equation, and the one that is most commonly given, is [ (1 − 2 ) ] Θ( ) + [ − 2 1 − 2 ] Θ( ) = 0 . CHEM 356, Lecture 20, Fall 2009 2 This equation is known as the associated Legendre equation, and is dealt with in detail in mathematical physics courses: it is one of the classic differential equations, and is of the Frobenius type, so that it is trickier to solve than was the Hermite ODE that we’ve dealt with previously....
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This note was uploaded on 02/28/2011 for the course CHEM 356 taught by Professor Prof.iaskjd during the Fall '09 term at Waterloo.
 Fall '09
 Prof.Iaskjd
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