PHYS2012 Quantum Physics Assignment
Due date: 14th September 2025 11.59pm
Question 1
A beam of spin-1/2 particles are prepared in the following quantum state:
|ψ⟩=|+⟩y+√3e−iπ/4|−⟩y.(1)
Answer the following questions.
1. Normalise this quantum state vector.
2. What are the possible results of a measurement of the spin component Sy, and with what
probabilities do they occur?
3. Calculate the expectation value ⟨Sy⟩and the uncertainty ∆Syfor this state. How does this
quantity relate to your answer to Part 2 of this question? H int: Make sure you write the
vector in the correct basis!
4. What are the possible results of a measurement of the spin component Sz, and with what
probabilities do they occur?
5. Calculate the expectation value ⟨Sz⟩for this state.
Question 2
Consider a beam of spin-1/2 particles prepared in the quantum state:
|ψ⟩1=−i√3
2|+⟩+1
2|−⟩.(2)
Answer the following questions.
1. Show that the above state is normalised. Prove that the state
|ψ⟩2=−i√3e−
iπ
3
2|+⟩+e−
iπ
3
2|−⟩.(3)
is also normalised.
2. Using Born’s Rule, calculate the probability of measuring the spin in the z-direction (ie.
measuring Sz) and getting outcomes +ℏ/2 and −ℏ/2 for both |ψ⟩1and |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 n and the associated spin operator Sn (written as a matrix) for
which the state |ψ⟩1is an eigenvector with eigenvalue +ℏ/2.
Hint: Start by comparing the state |ψ⟩1with the state |+⟩nin your formula sheet.
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