PHGN 520 Homework 09: Angular Momentum I
Due date: April 21, 2016
Problem 1 (45%) A free particle (V = 0) can be analyzed using either the planewave
basis {|ki} or the spherical-wave basis {|E, lmi}, obeying these normalization conditions:
hk0 |ki = δ 3 (k − k0 ),
hE 0 , l0 m0 |E, lmi = δ(E − E 0 )δl,l0 δm,m0 .
prove In real-space representation,
ψE,lm (r) = hr|E, lmi = cl jl (kr)Ylm (r̂),
where jl (x) is the spherical Bessel function, Ylm (r̂) the spherical harmonics, and r̂ the radial
(also called the directional) unit vector, r̂ = rr . Similarly, in the k-space,
ψE,lm (k) = hk|E, lmi = glE (k)Ylm (k̂).
(1) [25%] prove that
2 2
~
~k
glE (k) = √ δ
−E .
2µ
µk
Here µ is the mass of the particle, to distinguish from the magnetic quantum number m.
2 2
2 2
~ k
(Hint: Obviously glE (k) = Aδ 2µ − E , since energy E = ~2µk . Then use the normalization condition for {|E, lmi} and properties of Dirac-δ function to prove that |A|2 =
~2
.)
µk
(2) [20%] Then use the addition theorem (Eq. 5.17 in lecture notes),
l
4π X m 0 0 ∗ m
Pl (cos γ) =
[Y (θ , φ )] Yl (θ, φ),
2l + 1 m=−l l
where Pl (x) are Legendre polynomials and γ is the angle between two unit vectors described
by (θ0 , φ0 ) and (θ, φ), and the well-known expression (Eq. 5.18 in lecture notes)
ik·r
e
=
∞
X
(2l + 1)il jl (kr)Pl (cos θ),
l=0
where θ is the angle between k and r: k · r = kr cos θ, prove that
r
il 2µk
cl =
.
~
π
(Hint: consider hr|ki =
eik·r
.)
(2π)3/2
1
Problem 2 (35%) For a particle in a potential V (r), prove that
d
hLi = hNi,
dt
where L is the orbital angular momentum operator and N is the torque operator:
N = r × (−∇V ).
(Note: It is then easily to verify that for a central potential V (r) = V (r), the orbital angular momentum is conserved, just like in classical mechanics. This is a crucial conclusion in
quantum mechanics.)
Problem 3 (20%)
(a) [5%] prove that hLx i = hLy i = 0 in a state |lmi.
(b) [5%] prove that in these states
1
hL2x i = hL2y i = ~2 [l(l + 1) − m2 ].
2
(c) [10%] prove that ∆Lx · ∆Ly satisfy the uncertainty principle:
1
h(∆A)2 ih(∆B)2 i ≥ |h[A, B]i|2 .
4
Which states have the minimum uncertainty?
2