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Problem_Set_4

Problem_Set_4 - Chemistry 120A Spring Semester 2009 Problem...

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Chemistry 120A Spring Semester, 2009. Problem Set 4 Due Wed. March 4, 2009. 1. Working with angular momentum operators. (a) Given the definition of the angular momentum operators in Cartesian coordinates, explicitly show that ˆ L x , ˆ L y ! " # $ = i ! L z , ˆ L x , ˆ L z ! " # $ = % i ! L y , and ˆ L y , ˆ L z ! " # $ = i ! L x . These 3 relations can be compactly summarized as ˆ L i , ˆ L j ! " # $ = i ! % ijk L k where the permutation tensor ˆ L i , ˆ L j ! " # $ = i ! ijk L k is 1 for a cyclic permutation (xyz, yzx, zxy), -1 for a permutation that needs one swap of indices to be cyclic, and 0 otherwise. (b) Beginning from the basic definition of spherical polar coordinates, derive the explicit expression for L in spherical polar coordinates. (c) Beginning from an explicit eigenfunction of angular momentum, ( ) ! " # i m l 2 exp sin 2 , 2 2 2 15 4 1 = = = , explicitly show that the action of L correctly generates the next lower m value (you can do it again if you want!). (d)
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Problem_Set_4 - Chemistry 120A Spring Semester 2009 Problem...

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