PHYS 402 Homework---Due March 4
1. A Stern-Gerlach apparatus is set up to measure the component of spin of an electron along the direction
in the x-z plane which makes an angle of θ relative to the z-axis. That is it is in the
direction xzn ˆ
)sin(
ˆ
)cos(
ˆ
θθ
+= A measurement is made and the electron us found to be spin up in this
direction
a. Compute the eigenvector of this state as given in the standard basis (oriented along z)
b. Suppose following this measurement a second Stern-Gerlach apparatus oriented in the same
direction measure the component of spin in the n
ˆ direction. What is the probability it will be
spin up?
c. Suppose instead that following the initial measurement a second Stern-Gerlach apparatus
oriented along the x
ˆ direction measures the spin component. What is the probability it will be
spin up?
d. Suppose instead that following the initial measurement a second Stern-Gerlach apparatus
oriented along the y
ˆ direction measures the spin component. What is the probability it will be
spin up?
e. Suppose instead that following the initial measurement a second Stern-Gerlach apparatus
oriented along the
z
ˆ
direction measures the spin component. What is the probability it will be
spin up?
2. Suppose two spin ½ particles combine to make a system and the state of the system (as expressed in a
basis of total spin and total third component of spin is known to be 1,10,10,0 2
1
2
1
2
1++=
ψ
a. What is the probability that second particle is up?
b. What is the probability that the first particle is down and the second particle is up?
c. What is the probability that both particles are up?
d. What is the probability that both particles are down?
3. Show that the most general rotationally invariant (ie. scalar) Hamiltonian for two spin ½ particles can be
represented as 21 ˆˆ
1
ˆ
ˆssbaH
r
r
⋅+= where a and b are constants and
1
ˆ
is the unit matrix. Hint: how many
states are there? What degeneracy do you expect?