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08angular - Orbital Angular Momentum In classical mechanics...

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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 equation’s 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 0 ] , [ 0 = = L H dt L d r r
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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 φ 1 0 0 0 cos sin 0 sin cos φ φ φ φ ) ( ) ( ) ( 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
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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 φ φ θ φ θ φ θ φ φ θ φ = + = + = 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 φ θ θ θ θ θ + = 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 + = = = Φ Θ = θ θ θ θ θ
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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 0 ] , [ 0 ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ ± = = = = = = = = = = ε ε h h h h
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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 0 ) ( ) ( ] , [ 2 2 2 2 2 2 2 2 2 2 2 + = + = + = + = = + + = = + + =
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P460 - angular momentum 6 Group Algebra The commutation relations, and the recognition that there are two operators that can both be diagonalized, allows the eigenvalues of angular momentum to be determined algebraically similar to what was done for harmonic oscillator an example of a group theory application. Also
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