PHYS2012 Quantum Physics Assignment

Due date: 10th September 2024 11.59pm

Question 1

A beam of spin-1/2 particles are prepared in the following quantum state:

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:

Answer the following questions.

1. Show that the above state is normalised. Prove that the state

is also normalised.

2. Using Born’s Rule, calculate the probability of getting outcomes +ℏ/2 and −ℏ/2 when measuring in the z-direction (Sz) 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 Sn (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.

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, over many repetitions of the experiment with the same input state. 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⟩ = cos2/θ| + ⟩ + e iϕsin2/θ|−⟩ 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. It may be helpful to start by considering what the measurements of Sx tell you about where the state |ψ⟩ lives on the Bloch sphere, specifically, is there anything you can deduce about the angles θ or ϕ? After considering Sx, then consider Sz measurements.

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 B. The initial spin state of the electron is |ψ(t = 0)⟩ = |+⟩x. Your goal is to make the spin precess to the state |+⟩y by applying two uniform. magnetic fields, one in the y-direction followed by one in the x-direction. Answer the following questions.

(a) Consider the following experiment:

• First, you apply a magnetic field B = Byyˆ in the y-direction for a time ty, and then turn it off.

• Immediately after that, you apply a magnetic field B = Bxx in the x-direction for a time tx, and then turn it off.

What times ty and tx should you choose to ensure that the final state is |+⟩y?

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 By.

(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 B = Bxˆx + Byyˆ, where Bx = By. 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 |+⟩y?

Write your answer in terms of the charge of the electron e, the mass of the electron me, and the magnetic field strengths Bx = By.

(c) For Bx = By 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-2/1 state about the direction of a uniform. magnetic field derived in lectures.