PHGN 520 Homework 10: Angular Momentum II
Due date: April 28, 2016
Problem 1 (40%) Show that e
− iπL x
/
~ | l, m i = | l, − m i , where L x is the orbital angular momentum along the x direction.
( Hint : you need to rigorously prove that the rotated state on the left is an eigenstate of L
2 and L z mathematically, and its physical meaning is rather obvious.)
Problem 2 (30%)
(1) [12%] Prove that exp ip
0
( r · σ )
= cos p
0 r
+ i r · σ r sin p
0 r
,
~ ~ ~ where σ are the Pauli Matrices, r is the position vector (not an operator), and p
0 is a constant momentum. ( Note : Here only the math matters, i.e., r and p
0 can be an arbitrary vector and scalar, respectively, provided that p
0 r/
~ is dimensionless.)
(2) [10%] If a Hamiltonian is given by H = ε σ · ˆ , where ε is a constant and
ˆ = ( n x
, n y
, n z
) = (sin θ cos φ, sin θ sin φ, cos θ ) .
Determine the time evolution operator U ( t ) as a 2 × 2 matrix.
  a
(3) [8%] Then if a state at t = 0 is given by | ψ (0) i =
 
, obtain | ψ ( t ) i .
b
Problem 3 (30% + 40% bonus)
(1) [30%] Evaluate j
X m 0 = − j m
0
| d
( j ) m
0 m
( β ) | 2 for any allowed value of j (integer or half-integer). ( Hint : Consider the rotated operator
J z
, and check your answer for j = 1 / 2 and j = 1.)
(2) [40%] (Bonus Problem) Prove that, for any allowed value of j , j
X m
0
= − j m
0 2 | d
( j ) m 0 m
( β ) | 2
=
1
2 j ( j + 1) sin
2
β +
1
2 m
2
(3 cos
2
β − 1) .
( Hint : You might solve this problem in many ways, and one approach is to examine the rotational properties of J
2 z
.)
1