Updated: 23 Feb 2026

These are the problems that we will solve during the problem class in week 16. You are not expected to attempt these questions in advance of the problem class. We will work through them in sequence, going through the solutions after everybody has time to attempt the questions.

I have included additional questions for you to attempt / discuss if you complete the questions before we go over the solution. If you also complete these, please work through Problem Sheet: Week 17

Problems¶

Problem 1

In this question we will study angular momentum in 3D quantum mechanics.

Write down the formula from classical mechanics for the angular momentum $\Lvec$ of a particle, in terms of its position $\rvec$ and momentum $\pvec$ .
Write out each component $L_x$ , $L_y$ and $L_z$ explicitly.
Use part 1 to write down the quantum mechanical angular momentum operator $\hat{\Lvec} = (\hat{L}_x, \hat{L}_y, \hat{L}_z)$ .
Confirm that each component of $\hat{\Lvec}$ is a Hermitian operator. What is the physical significance of this?
Hint: What is $(\hat{A}\hat{B})^\dagger$ in terms of $\hat{A}^\dagger$ and $\hat{B}^\dagger$ (from 1st year)?
Calculate $[\hat{L}_x, \hat{L}_y]$ .
What is the physical significance of your answer?
Use part 4 to make a guess (without calculation) for $[\hat{L}_y, \hat{L}_z]$ and $[\hat{L}_x, \hat{L}_z]$ .

Problem 2

Consider the 3D infinite box potential, of volume $L^3$ , with walls at $x = 0$ , $x = L$ , $y = 0$ , $y = L$ , $z = 0$ and $z = L$ .

Write down the energies levels $E_\mathbf{n}$ of the box.
What is the ground state energy of the box, and what is the associated (vector of) quantum numbers $\mathbf{n} = (n_x, n_y, n_z)$ ?
State what it means for an energy level to be degenerate.
What is the degeneracy of the next set of energy levels, and what are the associated quantum numbers $\mathbf{n}$ ?
Find all of the energy levels that have energy equal to $\frac{38 \hbar^2 \pi^2}{2ML^2}$ , and hence determine the degeneracy of these levels.
Consider the state $\ket{\psi_\mathrm{in}} = \frac{1}{\sqrt{3}}(\ket{E_{(2,1,1)}} + \ket{E_{(1,2,1)}} + \ket{E_{(1,1,2)}})$ . If this is the state of a particle at $t = 0$ , use the superposition principle to write down the state $\ket{\psi(t)}$ at time $t$ .
Is this state a stationary state?

Solutions¶

Solutions

Tuesday:

Thursday:

Problem Class Week 17 Problems

Problems¶

Solutions¶