08angular - P460 - angular momentum 1 Orbital Angular...

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Unformatted text preview: P460 - angular momentum 1 Orbital Angular Momentum In classical mechanics, conservation of angular momentum L is sometimes treated by an effective (repulsive) potential Soon we will solve the 3D Schr. Eqn. The R equation will have an angular momentum term which arises from the Theta equations separation constant eigenvalues and eigenfunctions for this can be found by solving the differential equation using series solutions but also can be solved algebraically. This starts by assuming L is conserved (true if V(r)) 2 2 2 mr L ] , [ = = L H dt L d r r P460 - angular momentum 2 Orbital Angular Momentum Look at the quantum mechanical angular momentum operator (classically this causes a rotation about a given axis) look at 3 components operators do not necessarily commute = r h r r r r i p p r L z 1 cos sin sin cos ) ( ) ( ) ( x y x y z z x z x y y z y z x y x i yp xp L x z i xp zp L z y i zp yp L = = = h h h z y x y z z x z x y z y x x y y x L i x y i z y x z x z z y L L L L L L h h h = = = = ) ( )] )( ( ) )( [( ] , [ 2 2 P460 - angular momentum 3 Side note Polar Coordinates Write down angular momentum components in polar coordinates (Supp 7-B on web,E&R App M) and with some trig manipulations but same equations will be seen when solving angular part of S.E. and so and know eigenvalues for L 2 and L z with spherical harmonics being eigenfunctions = + = + = h h h i L i L i L z y x ) sin cot cos ( ) cos cot (sin ] ) (sin [ 2 2 2 sin 1 sin 1 2 2 + = h L lm lm m lm lm l m lm z lm z Y l l Y Y L Y m L Y L l 2 sin sin 1 2 2 2 2 2 2 ) 1 ( ] ) (sin [ 2 2 h h h + = = = = P460 - angular momentum 4 Commutation Relationships Look at all commutation relationships since they do not commute only one component of L can be an eigenfunction (be diagonalized) at any given time different all same indices any tensor L i L L or L L L L L L L i L L L i L L L i L L ijk k ijk j i z z x x y y y x z x z y z y x , 1 ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ = = = = = = = = = = h h h h P460 - angular momentum 5 Commutation Relationships but there is another operator that can be simultaneously diagonalized (Casimir operator) y z y x y y z y z x y z y y x z x y x x z x z y x z x x y x z z y x z z z z y x L L L L L L L L L L L L L L L L L L L L L L L L L L L L g u L L L L L L L L L L L L L L L L ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( : sin ) ( ) ( ] , [ 2 2 2 2 2 2 2 2 2 2 2 + = + = + = + = = + + = = + + = P460 - angular momentum 6 Group Algebra...
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This note was uploaded on 10/02/2009 for the course PHYS 460 taught by Professor Johnson,c during the Spring '08 term at Northern Illinois University.

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08angular - P460 - angular momentum 1 Orbital Angular...

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