|+⟩ − i
2
|−⟩. (2)
Answer the following questions.
1. Show that the above state is normalised. Prove that the state
|ψ⟩2 = e iπ4
√
3
2
|+⟩ − e iπ4 i
2
|−⟩. (3)
is also normalised.
2. Using Born’s Rule, calculate the probability of measuring Sz and getting outcomes +ℏ/2
and −ℏ/2 for both |ψ⟩1 and |ψ⟩2.
3. Comment on your results in Part 2, in particular, on the impact of the overall phase on
measurement outcomes.
4. Identify a spatial vector n⃗ and the associated spin operator Sn⃗ (written as a matrix) for
which the state |ψ⟩1 is an eigenvector with eigenvalue +ℏ/2.
Hint: Start by comparing the state |ψ⟩1 with the state |+⟩n in your formula sheet.
1
5. What is the other eigenvector of this same spin operator, with eigenvalue −ℏ/2?
6. Calculate the inner product between these two eigenvectors.
Question 3
Consider the following set of Stern-Gerlach experiments. In this experiment the spins ejected from
the source are not random, but are in a specific quantum state |ψ⟩. Your job is to determine the
state |ψ⟩ from the outcomes of measuring the spin component Sx, Sy, or Sz on many copies of |ψ⟩.
The measurement outcomes are shown below. The statistics for Sy are intentionally left blank.
Answer the following questions.
1. Based on the measurement data above, determine the state vector |ψ⟩ that describes the
spin-1/2 particles exiting the source.
Hint: This question involves using the Born rule in reverse. Consider which general state
|ψ⟩ = cos θ2 |+⟩+ eiϕsin θ2 |−⟩ on the Bloch sphere, or alternatively which general state |ψ⟩ =
a |+⟩+ b |−⟩, is consistent with the measured probabilities. If using the Bloch sphere, think
about what the measured probabilities indicate about θ and ϕ of a general state |ψ⟩ on the
Bloch sphere.
2. Based on the state |ψ⟩ that you have inferred, what are the possible results of a measurement
of the spin component Sy, and with what the probabilities do they occur? Are they consistent
with the measured data?
Hint: Remember that there may be some statistical fluctuations in the data due to the
sample size.
Question 4
An electron is placed in a controllable magnetic field B⃗. The initial spin state of the electron
is |ψ(t = 0)⟩ = |+⟩. Your goal is to make the spin precess to the state |+⟩x by applying
uniform magnetic fields. Answer the following questions.
(a) Consider the following experiment:
2
• First, you apply a magnetic field B⃗ = Bxxˆ in the x-direction for a time tx, and
then turn it off.
• Immediately after that, you apply a magnetic field B⃗ = Bz zˆ in the z-direction for
a time tz, and then turn it off.
What times tx and tz should you choose to ensure that the final state is |+⟩x?
Write your answer in terms of the charge of the electron e, the mass of the electron me,
and the magnetic field strengths Bx and Bz.
(b) Now consider a different experiment:
• Instead of applying the magnetic fields sequentially, you apply them simultaneously
with equal strengths. The resultant magnetic field is B⃗ = Bxxˆ+Bz zˆ, where Bx =
Bz. You apply this field for a time t, and then turn it off.
What time t should you choose to ensure that the final state is |+⟩x?
Write your answer in terms of the charge of the electron e, the mass of the electron me,
and the magnetic field strengths Bx = Bz.
(c) For Bx = Bz which of these experiments is quicker to perform?
Hint: Use the Bloch sphere picture to reason about this question. You may invoke the general
solution for the precession of a spin- 12 state about the direction of a uniform magnetic field
derived in lectures.