• Eigenvalues and eigenstates
• Measurements of energy, angular momentum, and position in quantum systems
• Quantum probabilities
• Superposition of states
• Quantum calculations in multiple representations
• Time dependence
• Degeneracy

Estimated Time: 30-45 minutes

Students calculate probabilities for energy, angular momentum and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. The purpose of this activity is to help students build an understanding of when they can expect a quantity to be time dependent and to give them more practice moving between representations.

• Eigenstates & Eigenvalues
• Familiarity with the postulates of quantum mechanics, particularly those having to do with measurement
• The energy and angular momentum eigenstates and eigenvalues of a particle confined to a ring
• Calculating probabilities using Dirac “bra-ket” and wavefunction notation
• Familiarity with time evolution in quantum mechanics (see Analyzing the Probabilities of Time-evolved States for an example activity from the Spins course)

• Tabletop Whiteboard with markers
• A handout for each student

It is a good idea at the start of this activity to give students the initial state as a linear combination of energy eigenstates and ask them to write $\vert\psi(t)\rangle$ on small whiteboards. This provides a chance to remind students how to write a state as function of time before they begin to wrestle with the probabilities.

• Degeneracy: students may experience some difficulty due to the degeneracy of some states, in particular that you have to include all the states that share that eigenvalue..

$$P_{E={m^2\,\hbar^2\over 2I}}=\vert \langle m\vert \psi\rangle\vert^2+\vert \langle -m\vert \psi\rangle\vert^2$$

• Probability v. Probability Density:
  • Students struggle with the two different ways of finding probability: for discrete and continuous measurements.
  • Most recognize that they need to do an integral, but are not sure whether you square the norm before or after you do the integral.

• Notation: Depending on how much you have done beforehand, some students still struggle with writing the energy eigenstates for the ring in wave function notation: $$\vert m\rangle\doteq\frac{1}{\sqrt{2\pi}}e^{im\phi}$$

• Some students will recognize the exponential cross terms in the probability amplitude can be written as a sine. Point this out in the wrap up, but it is more important that they take the integral and see that the position probability changes with time.

• Some will state without showing that the energy and angular momentum probabilities do not change with time. Ask them to show this explicitly to make sure that everyone in the group understands why and because it makes the comparison to position probability much easier - you see the cross terms go away for energy and angular momentum.

This is a good activity to have a group present their results. This allows the whole class to see the worked out solution without redoing it for them, but still allows you to point out the important features of the problem.

• Remind them how to deal with degeneracy.
• Reiterate the two ways of finding a probability and how they are connected.
  • For discrete measurements:$$P_{a_n}=\vert\langle a_n \vert \psi\rangle \vert^2\doteq\left|\int_{-\infty}^{+\infty}\Phi_n^*(x)\psi(x)dx\right|$$
  • For continuous measurements:$$P_{a<x<b}=\sum_{x=a}^b\vert\langle x \vert \psi\rangle \vert^2 \doteq\int_a^b\vert\psi(x) \vert^2 dx$$
• Mention that the exponential cross terms can often be written as a sine or cosine (depending on the phase)
• In discussing under what circumstances measurement probabilities change with time, it is a good idea to connect it to earlier activities (from Spins and Waves courses) where they saw that probabilities were only time-dependent if the operator did not commute with the Hamiltonian.

• Quantum Ring Sequence: This is a part of a sequence of activities and homework problems that use a particle confined to a ring as a touchstone example.
• Students readily grasp the strategy of finding probability amplitudes “by inspection” when they are given an initial state written as a sum of eigenstates. We find that students then find it extremely difficult to find probability amplitudes of wavefunctions that are not written this way (i.e. using an integral to find the expansion coefficients of a function). This activity should be followed up with an activity (Superposition States for a Particle Confined to a Ring) and/or homework that allow students to practice this more general method.