PHY4605–Introduction to Quantum Mechanics II
Spring 2005
Problem Set 1
Jan. 5, 2005
Due: Jan. 14, 2005
Reading: PH Notes, Griffiths Sec. 4.4.2
1. Inhomogeneous field
Consider a neutral spin-1/2 particle moving in a magnetic field
B = −αxî + (B0 + αz)k̂
(1)
(a) Verify that this field satisfies ∇ · B = 0.
(b) If at t = 0 the particle is in the state
Ã
|χi =
χ+
χ−
!
.
(2)
Find the general form of |χ(t)i at time t in terms of E0 = µ0 B0 and α.
[Hint: You might need the identity eiθn̂·~σ = cos θ + in̂ · ~σ sin θ]
(c) Take the initial state to be χ+ = χ− = 1, and assume B0 À αz, αx, i.e.
the gradients are small. Show that the spin up and spin down parts of the
wave function are actually moving in opposite directions by applying the
momentum operator p̂ to |χi, and specify the directions. Does the field
gradient in the x̂ direction have an important effect?
2. Stern-Gerlach effect.
(a) A beam of neutral spin-1/2 particles is created in initial state
s
|χi =
1
17
Ã
4
i
!
,
(3)
travelling in the x̂ direction so as to pass through an SG setup aligned with
B, ∇By k ŷ (an SG setup is defined to be a single region of inhomogeneous
field which splits the beam once). Of the two beams which emerge, the
one with spins anti-k to ŷ is then passed through a second SG setup with
B, ∇Bz k ẑ. Negelect all other components of the field and its gradients.
What fraction of the incoming spins end up in the outgoing beam with
spin up (k ẑ)?
(b) A beam of neutral spin S = 1 particles is prepared such that all particles
are in one of the eigenstates of Sz , and sent into a single SG setup with
B, ∇By k ŷ. Into how many beams is the original beam split, and what are
the relative intensities of the spots when imaged on a screen? Give results
for all possible values of Sz . [Hint: you will want to find the eigenvectors of
1
the matrices Sy and Sz . To save algebra, you may want to find them with
Maple using with(linalg):[eigenvects[A]]; See help (eigenvect) to
interpret output.
3. Oscillating B field. Griffiths Problem 4.33.
2