Homework 8
(due Wednesday, Oct. 31)
1. Commutators:
(a) Prove the identity:
[AB, C] = A[B, C] + [A, C]B.
(1)
(b) What is the similar identity for [A,BC]?
(c) Compute the cummutator [Lx Ly , Lz Ly ].
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
q
J− |j, mi = h̄ j(j + 1) − m(m − 1)|j, m − 1i
J± = Jx ± iJy
Jz |j, mi = h̄m|j, mi.
(2)
(3)
(4)
(5)
(a) For j = 2 compute the three matrices, hj, m′ |Jα |j, mi for α = x, y, z. These are
5×5 matrices with the top left position being m′ = m = 2 and the bottom right
position being m′ = m = −2:
(b) 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.
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
s
(−1)l (2l + 1)!
Yll (θ, φ) = l
(sin θ)l eilφ
2 l!
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 = 1 spherical harmonics:
hl = 1, m|l = 1, mi =
Z
π
sin(θ)dθ
0
Z
2π
0
dφ|Y1m (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.