Revision Problems - Quantum Mechanics

This is a short set of revision problems for you to attempt. Most of these questions should be straightforward, potentially after a little revision, especially those covering first year topics. If any of the questions prove challenging, then it is an indication that you may need to spend some more time revising these topics. We will make use of all of these topics extensively in this unit, so please do spend time making sure you are confident on these topics.

1Complex numbers¶

2Probabilities¶

3Quantum Physics¶

Exercise 5

Consider two quantum states $\ket{\psi_1} = \frac{1}{\sqrt{2}}(\ket{1} + \ket{2})$ and $\ket{\psi_2(\theta)} = \cos \theta \ket{1} + \sin \theta \ket{2}$ , where $0 \leq \theta < 2\pi$ , and $\ket{1}$ and $\ket{2}$ form an orthonormal basis.

Confirm that both $\ket{\psi_1}$ and $\ket{\psi_2(\theta)}$ are normalised quantum states.
Calculate the scalar product $\inner{\psi_1}{\psi_2(\theta)}$ . Hence find a value of $\theta$ such that $\ket{\psi_2(\theta)}$ is orthogonal to $\ket{\psi_1}$ . Call this value $\theta^*$ .
Show that the operator $\hat{A} = \ket{\psi_1}\bra{\psi_1} - \ket{\psi_2(\theta^*)}\bra{\psi_2(\theta^*)}$ is Hermitian. What are its eigenvalues and eigenvectors?
Consider measuring $\hat{A}$ on $\ket{\psi_2(\pi/2)}$ . What are the possible results of the measurement, what are their corresponding probabilities, and what is the state after measurement in each case?