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Problems 155
B
m 5 3
m 5 2
m 5 1
m 5 0
m 5 21
m 5 22
m 5 23
Figure 5.6 Possible orientations in a magnetic ®eld B of the orbital angular momentum
vector L for the case l 3
Problems
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^
5.1 Show that each of the operators L x , L y , L z is hermitian.
5.2 Evaluate the following commutators:
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^
^
^
(a) [L x , x] (b) [L x , ^ p x ] (c) [L x , y] (d) [L x , ^ p y ]
^
5.3 Using the commutation relation (5.10b), ®nd the expectation value of L x for a
system in state jlmi.
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^
5.4 Apply the uncertainty principle to the operators L x and L y to obtain an expres-
^
^
sion for ÄL x ÄL y. Evaluate the expression for a system in state jlmi.
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^
^ 2
5.5 Show that the operator J commutes with J x and with J y .
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5.6 Show that J and J ÿ as de®ned by equations (5.18) are adjoints of each other.
5.7 Prove the relationships (5.19a)±(5.19g).
5.8 Show that the choice for c ÿ in equation (5.24) is consistent with c in equation
(5.22).
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5.9 Using the raising and lowering operators J and J ÿ, show that
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^
hjmjJ x jjmihjmjJ y jjmi 0
5.10 Show that
^ 2
^ 2
2
1
hjmjJ jjmihjmjJ jjmi [j(j 1) ÿ m ]" 2
x y 2
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^
^
^
5.11 Show that jj, mi are eigenfunctions of [J x , J ] and of [J y , J ]. Find the
eigenvalues of each of these commutators.