Homework 8: Due November 5
1. This problem is designed to fill in some gaps in our derivation of Schwinger’s approach to angular
momentum. Specifically, it concerns implementing finite rotations on the creation and annihilation
operators.
a. Using the definitions of the operators J
ˆ
r
in terms of the creation and annihilation
operators for +/- given in Eqs. 3.8.13 to show that ⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−=
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
−
+
+
+
−
+
+
a
a
J
a
ak
kˆ
ˆ
2
ˆ
,
ˆ
ˆ
σ
where k
σ
is a Pauli matrix.
b. From the result in a. show that
() ()
(
)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅+=
⎟
⎠
⎞
⎜
⎝
⎛⋅−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎠
⎞
⎜
⎝
⎛⋅+
−
+
+
+
−
+
+
a
a
niJni
a
a
Jni kˆ
ˆ
ˆ
2/sin(12/cos
ˆ
ˆ
exp
ˆ
ˆ
ˆ
ˆ
exp
σθθθθ
rr
where 1is the identity matrix. Hint: consider infinitesimal results and integrate
2. Using the fact that 0
)!()!(
)()(
,mjmj
aa
mj mjmj
−+
=
−+
−
++
+ derive eq. 3.5.41 starting from Eqs. 3.8.13 and using
only the commutation relations for the creation and annihilation operators for the harmonic oscillator.
3. Conventionally one quantizes the angular momentum in terms of eignenstates in the z-direction. In
principle, one could take any direction. The purpose of this problem is to relate an eigenstate of Jn ˆ
ˆ
r
⋅(where n
ˆ
is an arbitrary unit vector) in terms of eigenstates in the z-direction. In principle this can be implemented
straightforwardly using the Wigner –D matrices. In this problem, however, I want you to use Schwinger’s
approach directly.
a. As a first example, I want you to consider a state with fixed j (i.e. an eigenstates of 2
ˆ
J and
of Jn ˆ
ˆr
⋅with eigenvalue of j (i.e. the maximum possible value) and express it as sum of states
with good mj,. That is, if we define
ψ
such that
ψψ
)1(
ˆ2+= jjJ and
ψψ
jJn =⋅ ˆ
ˆ
r
and express
ψ
as ∑
=mmjmc
ψ
, I want you to use Schwinger’s approach to find the
coefficients m
c.
b. As a second example, I want you to consider a state with fixed j (i.e. an eigenstates of 2
ˆ
J
and of Jn ˆ
ˆr
⋅with eigenvalue of j-1 (i.e. one les than the the maximum possible value) and express
it as sum of states with good mj,. That is, if we define Φ such that Φ+=Φ )1(
ˆ2jjJ and
Φ−=Φ⋅ )1(
ˆ
ˆjJn r and express Φ as
∑
=Φ mmjmd , I want you to use Schwinger’s
approach to find the coefficients m
d.
Sakurai Chapter 3: 13, 15, 16
