PHGN 520 Homework 11: Angular Momentum III
Due date: Thursday 5 PM, May 05, 2016
Problem 1 (30%)
(1) [12%] By explicitly writing down the 3 × 3 matrix for Jy with j = 1, show that for j = 1
only,
2
Jy
Jy
Jy
=1−i
sin β −
(1 − cos β).
exp −iβ
~
~
~
(2) [18%] Using the previous result to prove that

1
(1 + cos β) − √12 sin β
2

d(j) (β) =  √1 sin β
cos β
2

1
(1 − cos β) √12 sin β
2
1
(1
2
− cos β)



sin β  .

1
(1
+
cos
β)
2
− √12
Problem 2 (30%) Adding angular momenta j1 = j2 = 1, one can form j = 0, 1, 2 states.
Using either the ladder operator method or the recursion relation, express all nine {|jmi}
eigenkets in terms of {|j1 , j2 ; m1 , m2 i}.
Notes: Since here j1 = j2 = 1, one can use +, 0, − to denote m1,2 = 1, 0, −1, respectively;
therefore, |1, 1; +1, −1i can be written simply as |+, −i, and so forth. You can express a
combined eigenket as, for instance, |1, 1i =
√1 |+, 0i
2
−
√1 |0, +i.
2
Problem 3 (40%) A system of three (non-identical) spin-half particles, whose spin operators are S1 , S2 and S3 , is governed by the Hamiltonian
H = AS1 · S2 /~2 + B(S1 + S2 ) · S3 /~2 ,
with A and B two constants. Find its energy levels and their degeneracies.
Hints: It is a rather simple problem using the technique of addition of angular momentum;
don’t waste your time by solving this problem in brute force.
1