Physics 561 Homework Five

Physics 561 Homework Five
1. Warm up
Consider a system of angular momentum j = 1 whose state is given by:
|Ψ〉 = α|1, 1〉 + β|1, 0〉 + γ|1, −1〉 (1)
In the basis |j, m〉 of simultaneous eigenstates of J 2 and J z . Calculate the mean value of the
angular momentum, 〈 ⃗ J〉. Also calculate the mean squared values of the three components
of the angular momentum, 〈J 2 x〉, 〈J 2 y 〉, 〈J 2 z 〉.
2. Momentary Commutation of Reason
Given ⃗ L = ⃗ R × ⃗ P the orbital angular momentum of a particle, find the commutators:
[L i , R j ], [L i , P j ], [L i , R 2 ], [L i , P 2 ], [L i , ⃗ R · ⃗P ]. Explain physically why the angular momentum
does not commute with the linear momentum in quantum mechanics, by appealing to
the relationship of angular momentum to rotation and of linear momentum to translation.
3. Quadrupole Hamiltonian
Exercise 6 in complement F V I . I attach it as a scan for those who don’t have the book.
4. Reciprocal Rotation
In the coordinate representation, the eigenfunctions of orbital angular momentum are the
spherical harmonics. How would one express the eigenfunctions of orbital angular momentum
in the momentum representation? (The answer is easier to derive than you might
think).
5. Transverse Variance
Assume that a particle in a spherically symmetric potential is known to be in an eigenstate
of L 2 and L z , with eigenvalues l(l + 1)¯h 2 and m¯h respectively. Prove that the expectation
values between |l, m〉 states of the transverse components of the angular momentum are:
〈L x 〉 = 〈L y 〉 = 0 (2)
and
〈L 2 x〉 = 〈L 2 y〉 = l(l + 1)¯h2 − m 2¯h 2
2
Interpret this result classically.
(3)
6. Hydrogen Preview
Consider the operators ⃗ X, ⃗ P = −i¯h ⃗ ∇ and ⃗ L = ⃗ X × ⃗ P in the coordinate representation.

( ( )
(a) Prove that P 2 = ⃗P · ˆX)
ˆX · P ⃗ − 1 ⃗ ( )
P ·
r
ˆX × L ⃗ where r ˆX = X. ⃗ Be careful of
operator ordering.
(b) Use this result to show that the momemtum squared, which is closely related to the
kinetic energy operator, can be written as a piece that depends only on r plus a piece
that depends on the magnitude of the angular momentum operator:
P 2 = −¯h 2 ( ∂
∂r 2 + 2 r
∂
∂r
)
+ L2
r 2 (4)
This expression can be used to simplify the Schrodinger equation for a particle in a
spherically symmetric potential, such as in the hydrogen atom.