classes_winter09_113AID28_P_23_113A_W09 - L 2 operator is...

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Preview #23 (Chem113A W’09, due Fri. 2/27/09 at 12:05pm in class) Assigned reading: Engel 7.4—7.5 Attendance record : I am [ ] present at or [ ] absent from this class meeting (see date and time above). Name: ________________________ Forming study groups is permissible, but you must construct your solutions independently. By writing down my name, I confirm that I strictly obey the academic ethic code when doing this preview and my statement on attendance (above) is correct. Please write down the names of everyone who you worked with on this preview in the space above. [1] Spherical harmonics are common eigenfunctions of L 2 and L z operators. [1](a) If two QM operators commute, they can share common eigenfunctions.
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Unformatted text preview: L 2 operator is defined as: L 2 = L x 2 + L y 2 + L z 2 . Please use Eq. (7.31) and commutator formula (ref. lecture notes or formula sheets) to prove that L 2 and L z commute, i.e., [ L 2 , L z ] = 0. [1](b) According to the “common eigenket theorem”, therefore L 2 and L z can share common eigenfunctions, which are spherical harmonics Y l,m . Please use the l =1 m =+1 spherical harmonics Y 1,+1 as an example to prove by algebra that it is indeed an eigenfunction L z . Please calculate the corresponding eigenvalue. [Hint: you can find L z in Eq. (7.30)]...
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This note was uploaded on 03/29/2009 for the course CHEM 113A taught by Professor Lin during the Winter '07 term at UCLA.

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