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lecture11_umn - Lecture 11 Angular Momentum Eigenvalues...

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Lecture 11 Angular Momentum Eigenvalues of the Angular Momentum Operators October 5 2009
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Angular Momentum Angular momentum is a vector quantity , defined as the cross product of the position vector and the momentum vector . (11-1) where x , y , and z are the components of the position vector, and p x , p y , and p z are the components of the momentum vector. A 3 x 3 determinant may be evaluated by Cramer's rule. L = i j k x y z p x p y p z
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The Three Components of Angular Momentum (11-2) The components of L , L x , L y , and L z , are the terms in parentheses preceding the corresponding unit vectors. In the absence of a torque on a system , angular momentum is a conserved quantity , just as linear momentum is conserved in the absence of a force on a system. i j k x y z p x p y p z = yp z " zp y ( ) i + zp x " xp z ( ) j + xp y " yp x ( ) k http://en.wikipedia.org/wiki/File:Torque_animation.gif
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Magnitude of Angular Momentum The magnitude of the angular momentum is | L | 2 (or L 2 , since it is just a number). From eq. 11-2: (11-3) In a quantum mechanical system , the momentum components themselves are the operators (11-4) where q is either x , y , or z . Let us now consider the commutation properties for any two components of the angular momentum . L 2 = L x 2 + L y 2 + L z 2 p q = " i h # q
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Commutator of L x and L y For an arbitrary function f (11-5) T he commutator is not zero, but instead involves the operator L z . L x , L y [ ] f = " i h ( ) y # z " z y $ % ( ) " i h ( ) z f x " x f z $ % ( ) " " i h ( ) z x " x z $ % ( ) " i h ( ) y f z " z f y $ % ( ) * + , , , , , - . / / / / / = " h 2 y f x + yz 2 f x z " xy 2 f z 2 " z 2 2 f x y + xz 2 f y z " yz 2 f x z + z 2 2 f x y + xy 2 f z 2 " x f y " xz 2 f y z $ % ( ) ) ) ) = " h 2 y x " x y $ % ( ) f = i h L z f
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Momentum By symmetry of the operators: (11-6) So, from the uncertainty principle , we see that we can never know more than one component of the angular momentum to perfect accuracy . In three dimensions: we can know the angular momentum only to within a circle defining the base of a cone having height equal to the one component we can know. L
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This note was uploaded on 12/14/2010 for the course CHEM 3502 taught by Professor Staff during the Fall '08 term at Minnesota.

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lecture11_umn - Lecture 11 Angular Momentum Eigenvalues...

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