Lecture20 - CHEM 356 Lecture 20 Fall 2009 1 Orbital Angular...

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CHEM 356, Lecture 20, Fall 2009 1 Orbital Angular Momentum Operators: Simultaneous Eigenfunctions. We saw that the differential equation governing the spherical harmonic functions 𝑌 ( 𝜃, 𝜙 ) was separable, with 𝑌 ( 𝜃, 𝜙 ) = Θ( 𝜃 )Φ( 𝜙 ), giving sin 𝜃 Θ( 𝜃 ) 𝑑 𝑑𝜃 ( sin 𝜃 𝑑 Θ 𝑑𝜃 ) + 𝛽 sin 2 𝜃 + 1 Φ( 𝜙 ) 𝑑 2 Φ 𝑑𝜙 2 , with 𝛽 2 𝐼𝐸 2 for the rigid rotor model. Because we already know the eigenfunctions Φ( 𝜙 ) = 1 2 𝜋 e 𝑖𝑚𝜙 , 𝑚 = 0 , ± 1 , ± 2 , ⋅ ⋅ ⋅ , we obtain the equation sin 𝜃 Θ( 𝜃 ) 𝑑 𝑑𝜃 ( sin 𝜃 𝑑 Θ 𝑑𝜃 ) + 𝛽 sin 2 𝜃 𝑚 2 = 0 , or, equivalently, sin 𝜃 𝑑 𝑑𝜃 ( sin 𝜃 𝑑 𝑑𝜃 ) Θ( 𝜃 ) + [ 𝛽 sin 2 𝜃 𝑚 2 ]Θ( 𝜃 ) = 0 , for Θ( 𝜃 ). If we now change variables from 𝜃 to 𝑥 using 𝑥 = cos 𝜃 , with 𝑑 𝑑𝜃 = 𝑑𝑥 𝑑𝜃 𝑑 𝑑𝑥 = 𝑑 cos 𝜃 𝑑𝜃 𝑑 𝑑𝑥 = sin 𝜃 𝑑 𝑑𝑥 , we get an equivalent equation in terms of 𝑥 , namely, sin 2 𝜃 𝑑 𝑑𝑥 [ sin 2 𝜃 𝑑 𝑑𝑥 ] Θ( 𝑥 ) + [ 𝛽 sin 2 𝜃 𝑚 2 ]Θ( 𝑥 ) = 0 . We may rewrite sin 2 𝜃 in terms of 𝑥 as 1 𝑥 2 , so that the equation for Θ( 𝑥 ) can be written entirely in terms of 𝑥 and derivatives with respect to 𝑥 as (1 𝑥 2 ) 𝑑 𝑑𝑥 [ (1 𝑥 2 ) 𝑑 𝑑𝑥 ] Θ( 𝑥 ) + [ 𝛽 (1 𝑥 2 ) 𝑚 2 ]Θ( 𝑥 ) = 0 . An equivalent version of this equation, and the one that is most commonly given, is 𝑑 𝑑𝑥 [ (1 𝑥 2 ) 𝑑 𝑑𝑥 ] Θ( 𝑥 ) + [ 𝛽 𝑚 2 1 𝑥 2 ] Θ( 𝑥 ) = 0 .
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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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