Angular States

(See attached file for full problem description)

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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

=aY11+bY10+cY1,-1

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.