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PHYS2012 Quantum Physics Assignment

September 8, 2022

Question 1

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

|ψ⟩ = 5|+⟩x + eiπ/3|−⟩x . (1)

Answer the following questions.

1. Normalise this state vector.

2. What are the possible results of a measurement of the spin component Sz , and with what probabilities do they occur?

3. Calculate the expectation value ⟨Sz ⟩ for this state. How does this quantity relate to your answer to part (b)?

4. What are the possible results of a measurement of the spin component Sx , and with what probabilities do they occur?

5. Calculate the expectation value ⟨Sx ⟩ and the uncertainty ∆Sx for this state.

Question 2

Consider a beam of spin-1/2 particles prepared in the quantum state:

1 i^3

|ψ⟩ = 2 |+⟩ + 2 |−⟩ . (2)

Answer the following questions.

1. Identify a spatial vector ⃗n and the associated spin operator S⃗n (written as a matrix) for which the state |ψ⟩ is an eigenvector with eigenvalue +ℏ/2.

Hint: Start by comparing the state |ψ⟩ with the state |+⟩n in your formula sheet.

2. What is the other eigenvector of this same spin operator, with eigenvalue −ℏ/2?

3. 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⟩ . In order to determine this state, you measure the spin components Sx , Sy , and Sz . The results are shown below. The measurement statistics for Sy have been left blank intentionally.

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: Think about what the measurement probabilities tell you about the components of the quantum state |ψ⟩ . The Bloch sphere may be helpful if you like to think visually.

2. Based on the state |ψ⟩ that you have obtained, 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 measurement data? Remember that there may be some statistical fluctuations due to the sample size.

Question 4

An electron is placed in a controllable magnetic field . The initial spin state of the electron is |ψ(t = 0)⟩ = |+⟩ . You would like to cause the spin to precess to the state |+⟩北 by applying uniform magnetic fields. Answer the following questions.

1. Consider the following experiment. First, you apply a magnetic field = B北 in the x- direction for a time t北 , and then turn it off. After that, you apply a magnetic field = Bz in the z-direction for a time tz , and then turn it off. What times t北 and tz should you choose to ensure that the final state is |+⟩北 ? Give your answer in terms of the charge of the electron

e, the mass of the electron me , and the magnetic field strengths B北 and Bz . Hint: Remember that quantum states are equivalent up to an overall phase.

2. Now consider a different experiment. Instead of applying the magnetic fields sequentially, you apply them simultaneously. The resultant magnetic field is = Bz + B北 , where B北 = Bz . You apply these fields for some time t, and then turn both fields off. What time t should you choose to ensure that the final state is |+⟩北 ? Give your answer in terms of the charge of the electron e, the mass of the electron me , and the magnetic field strengths B北 and Bz .