Homework 8
(due Friday, Nov. 1)
This is the homework assignment on angular momentum. You will get experience working
with angular momentum commutators, evaluating angular momentum expectation values,
and computing properties of the spherical harmonics.
1. Commutators: In the following you may wish to use the identity
[AB, C] = ABC − ACB + ACB − CAB = A[B, C] + [A, C]B.
(1)
(a) Show that the commutator [L2 , Lz ] is equal to zero.
(b) Compute the commutator [L2x Ly , Lx ].
(c) Compute the commutator [L3y , Lx ].
(d) Compute the commutator [L2z Ly , Lx ].
(e) Add the results from (b)-(d) to compute [L2 Ly , Lx ]. Does the result make sense?
2. Matrix elements: The angular momentum operators acting on the angular momentum
eigenstates, |j, mi, may be determined by
q
J+ |j, mi = h̄ j(j + 1) − m(m + 1)|j, m + 1i
(2)
q
J− |j, mi = h̄ j(j + 1) − m(m − 1)|j, m − 1i
J± = Jx ± iJy
Jz |j, mi = h̄m|j, mi.
(3)
(4)
(5)
(a) For j = 1 compute the three matrices, hj, m′ |Jα |j, mi for α = x, y, z. These are
3×3 matrices with the top left position being m′ = m = 1 and the bottom right
position being m′ = m = −1:


h1, 1|Jα|1, 1i
h1, 1|Jα|1, 0i
h1, 1|Jα|1, −1i

h1, 0|Jα|1, 0i
h1, 0|Jα|1, −1i 
Jα =  h1, 0|Jα|1, 1i
.
h1, −1|Jα|1, 1i h1, −1|Jα|1, 0i h1, −1|Jα |1, −1i
(6)
(b) Check that these matrices obey the angular momentum commutation relations:
[Jx , Jy ] = ih̄Jz , [Jy , Jz ] = ih̄Jx , and [Jz , Jx ] = ih̄Jy .
(c) For j = 3/2 compute the three matrices, hj, m′ |Jα |j, mi for α = x, y, z. (These
are 4×4 matrices.)
(d) Check that these matrices obey the angular momentum commutation relations:
[Jx , Jy ] = ih̄Jz , [Jy , Jz ] = ih̄Jx , and [Jz , Jx ] = ih̄Jy . You may use Matlab to do
this (print out your work).
3. Spherical harmonics:
(a) Using the lowering operator
!
∂
∂
,
− + i cot(θ)
∂θ
∂φ
−iφ
L− = h̄e
Eq. (3), and the spherical harmonic for m = l
(−1)l
Yl (θ, φ) = l
2 l!
l
s
(2l + 1)!
(sin θ)l eilφ
4π
determine all the spherical harmonics, Ylm (θ, φ), for l = 0, 1, and 2. You can
check your results with those in the book.
(b) Verify the normalization of the l = 2 spherical harmonics:
hl = 2, m|l = 2, mi =
Z
π
sin(θ)dθ
0
Z
2π
0
dφ|Y2m (θ, φ)|2 = 1.
Hint: do a change of variables to u = cos(θ).
(c) Verify the orthogonality of the l = 1 spherical harmonics:
′
hl = 1, m |l = 1, mi =
for m 6= m′ .
Z
π
sin(θ)dθ
0
Z
2π
0
′
dφY1m ∗ (θ, φ)Y1m (θ, φ) = 0.