Lect27_OrbAngMom - Lecture 27: Orbital Angular Momentum...

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Lecture 27: Orbital Angular Momentum Phy851 Fall 2009
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The General Theory of Angular Momentum Starting point: – Assume you have three operators that satisfy the commutation relations: – Let: Conclusions: – Simultaneous eigenstates of J 2 and J z exist – They must satisfy: – Where the quantum numbers take on the values: y x z y x iJ J J J J J J ± = + + = ± 2 2 2 2 y x z x z y z y x J i J J J i J J J i J J h h h = = = ] , [ ] , [ ] , [ m j m m j J m j j j m j J z , , , ) 1 ( , 2 2 h h = + = 1 , ) 1 ( ) 1 ( , ± ± + = ± m j m m j j m j J h j j j j j m j , 1 , , 2 , 1 , , 3 , 2 5 , 2 , 2 3 , 1 , 2 1 , 0 + + = = K K
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Orbital Angular Momentum For orbital angular momentum we have: So that: In coordinate representation we have: P R L r r v × = x y z YP XP L = v L →− i h r r × r ( ) φ θ = × sin 1 1 0 0 det r r r r e e e r r r r r r r + = e e r r sin 1 + = e e i L r r h r sin 1
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The z-component of L L z is defined by: L e L z z r r =
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Lect27_OrbAngMom - Lecture 27: Orbital Angular Momentum...

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