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lecture_24_ang_mom2

lecture_24_ang_mom2 - Chem 120A Spring 2007 READING Angular...

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Chem 120A Angular Momentum, continued 03/16/07 Spring 2007 Lecture 24 READING: Ratner and Schatz: Chapter 4 Atkins and Friedman: Chapter 4.1-4.7 Angular momentum From classical physics we know that the angular momentum is vector L = vector r × vector p = vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle i j k x y z p x p y p z vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle . (1) In order to construct quantum analog of classical angular momentum, we replace classical variable x , p x , y , p y , and z , p z by the corresponding quantum operators, which in position representation have the form ˆ x = x · , ˆ p x = i ¯ h x , etc. Thus the angular momentum operator in quantum mechanics is ˆ L = ( ˆ y ˆ p z ˆ z ˆ p y ) i +( ˆ z ˆ p x ˆ x ˆ p z ) j +( ˆ x ˆ p y ˆ y ˆ p x ) k . (2) There are also operators for each component of the angular momentum, as defined by Eq. 2, ˆ L x = ˆ y ˆ p z ˆ z ˆ p y ˆ L y = ˆ z ˆ p x ˆ x ˆ p z ˆ L z = ˆ x ˆ p y ˆ y ˆ p x , (3) and ˆ L 2 = ˆ L 2 x + ˆ L 2 y + ˆ L 2 z . (4) Recall the commutation relations for position and momentum, [ ˆ r i , ˆ p j ] = i ¯ h δ i j , i , j = x , y , z . (5) These commutations can be used to derive the commutation relations of angular momentum. The result is the following relations: bracketleftbig ˆ L x , ˆ L y bracketrightbig = i ¯ h ˆ L z bracketleftbig ˆ L y , ˆ L z bracketrightbig = i ¯ h ˆ L x bracketleftbig ˆ L z , ˆ L x bracketrightbig = i ¯ h ˆ L y and bracketleftbig ˆ L 2 , ˆ L i bracketrightbig = 0 , i = x , y , z . (6) Chem 120A, Spring 2007, Lecture 24 1
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These commutation relations tell us that if we measure the angular momentum with respect to a particular direction, then measure the angular momentum in another direction, the results are not the same as they would be if we perform the reverse process. This is a consequence of the noncommutivity of rotations.
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