Question 1
PHYS2012 Quantum Physics Assignment
Due date: 12th September 2023 11.59pm
A beam of spin-1/2 particles are prepared in the following quantum state:
|ψ⟩ = |+⟩x + √2eiπ/4|−⟩x. (1)
Answer the following questions.
1. Normalise this quantum state vector.
2. What are the possible results of a measurement of the spin component Sx, and with what probabilities do they occur?
3. Calculate the expectation value ⟨Sx⟩ and the uncertainty ∆Sx for this state. How does this quantity relate to your answer to Part 2 of this question? Hint: 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= 23|+⟩−2i|−⟩. (2)
Answer the following questions.
1. Show that the above state is normalised. Prove that the state
3|+⟩−eiπ i|−⟩. (3) 22
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 ⃗n and the associated spin operator S⃗n (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.
2. Using Born’s Rule, calculate the probability of measuring Sz and getting outcomes +ħ/2
is also normalised.
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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⟩ = cosθ2 |+⟩+eiφsinθ2 |−⟩ 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:
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• 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 = Bzzˆ 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 + Bzzˆ, 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-21 state about the direction of a uniform magnetic field derived in lectures.
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