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Angular States in Quantum Mechanics

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3. Let denote the eigenstates of L2 and Lx; i.e.

L2 = l(l+1)h-bar2 and Lx = m*h-bar*

a. Explain briefly why you can always express any given as a superposition of spherical harmonics Yl'm' with l'=l.
b. In particular, for each m = +1,0,-1, find the constants a,b,c such that


is a normalized eigenstate of Lx, and verify that , , are orthogonal. This problem should be solved algebraically, using Lx = ½ (L++L-), and the orthogonality of the spherical harmonics.

c. Suppose a system is in the state . If Lz is measured what possible values could be found, and with what probabilities? Calculate the uncertainty z in this state. You should be able to find these quantities without doing any explicit angular integrations.


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