Homework 10
(due Wednesday, Dec. 5)
This is the last homework assignment! There is one problem on spin and one problem
on addition of angular momentum.
1. Spin:
The Hamiltonian of an electron in a magnetic field of magnitude B and direction n̂ is
H = µB B n̂ · ~σ .
(1)
In spherical coordinates the unit vectors n̂ is
n̂ = sin(θ) cos(φ)x̂ + sin(θ) sin(φ)ŷ + cos(θ)ẑ.
(2)
(a) Verify that the general spin eigenstates given in Eq. 4.155 in the book are indeed
eigenstates of the Hamiltonian in Eq. (1) with the unit vector given in Eq. (2).
(Just multiply the vector times the Hamiltonian.)
(b) Next suppose that the magnetic field is B x̂. Let the spin initially be pointing in
the +ẑ direction,
!
1
|ψ(t = 0)i =
.
(3)
0
Determine |ψ(t)i.
(c) We will now interpret this |ψ(t)i in two ways. First, compute the expectation
value of Sx , Sy , and Sz for |ψ(t)i in part (b). Sketch how the expectation is
rotating in three dimensions.
(d) Next using the exact eigenstates (Eq. 4.155) determine the θ and φ time dependence for |ψ(t)i in part (b). Compare the θ and φ time dependence to the
expectation value time dependence of part (c).
(e) Repeat (b)-(d) for a magnetic field of B ŷ with the spin sill initially pointing in
the +ẑ direction.
2. Addition of angular momentum:
In this problem we will compute the Clebsch-Gordon coefficients for adding spin 1 and
spin 21 angular momentum. The results will be compared to those in Table 4.8 of the
book.
(a) List the (six) possible pairs of m1 and m2 for j1 = 1 and j2 =
m = m1 + m2 values for these two states.
1
.
2
Give the
(b) The state with m1 = 1 and m2 = 12 has total angular momentum 23 . To avoid
confusion use the notation of the book and call this state |1, 1i| 12 , 12 i. Apply the
lowering operator to this state to get the the three other states with j = 3/2.
Again following the notation of the book denote these states at | 23 , mi for m =
1
, − 12 , − 32 .
2
(c) Find a state which is orthogonal to | 32 , 21 i, but still has an m value of 12 . Normalize
it and make the coefficient of |1, 1i| 12 , − 21 i positive. Verify that this state has total
angular momentum 12 by acting with S 2 on it.
(d) Apply the lowering operator, S− , to the state of (c) to obtain the j = 21 , m = − 21
state.
(e) Compare your results in (a)-(d) to those in Table 4.8.