Infinite square well. Finding the eigenvalues and eigenstates of an inifinite square well potential is one of the very first problems every student of quantum mechanics encounters. Using what you learnt last term in the Oscillations, waves and optics course, can you qualitatively discuss what the eigenstates look like when written as momentum wave functions?
Associated Jupyter notebook.

What happens after a measurement? Consider two physical observables, the Hamiltonian Ĥ and another observable Â. Describe the differences in the time evolution of the quantum state obtained after measuring the physical property associated with either observable.

Compatible observables and angular momentum. The square of the angular momentum operator J² commutes with the angular momentum components J₁ and J₃. This means that J² and J₁ share a common set of eigenstates, and that J² and J₃ also share a common set of eigenstates. However, J₁ and J₃ do not commute, so they do not share a common set of eigenstates. How is this possible?

Identical particles. If you want to expand on some of the topics covered during the course, and prepare for the third year, a good option is to follow the videos on identical particles. Whether you do Physics, Chemistry, or Materials in Part II, the quantum mechanics of systems with many identical electrons will take centre stage, so learning how to describe systems of identical particles will get you ready for next year.