Introduction

Student Conversations

• Degeneracy: Students may experience some difficulty due to the degeneracy of some states, in particular that you are calculating probability of measuring a certain value, 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 for a continuous quantity, but are not sure when to square (before integration or after).

  \(\left|\int \phi_n^*(x)\Psi(x,t) dx\right|^2\) vs. \(\int\left|\Psi(t) \right|^2 dx\)

  In particular, many students will forget to do the squaring for the calculation on the left because \(\int \phi_n^*(x) \Psi(x,t) dx\) looks a lot like \(\int \Psi^*(x,t) \Psi(x,t) dx\).

• 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. In particular, they may forget to include the normalization constant. \[|m\rangle\doteq\frac{1}{\sqrt{2\pi}}e^{im\phi}\]

• When calculating the probability of finding the particle in a region, some students will recognize some of the exponential cross terms can be written as a sine function \(2i\sin \omega t = e^{i\omega t}-e^{-i\omega t}\). This is a nice detail to point out in the wrap up, but it is more important that the students take the integral over position 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 calculate this explicitly to make sure that everyone in the group understands why (because the time-dependent phases norm square to 1 and there are no cross terms remaining). Understanding this calculation makes the comparison to position probability much easier - you see the cross terms go away for energy and angular momentum.

Wrap-up

• Remind students how to deal with degeneracy.
• Reiterate the two ways of finding a probability and how they are connected.
  • For discrete measurements:\[P_{a_n}=\vert\langle a_n \vert \psi\rangle \vert^2\doteq\left|\int_{0}^{2\pi}\Phi_n^*(x)\psi(x)dx\right|\]
  • 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.

Extensions and Related Material

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 another activity and/or homework from the Quantum Ring Sequence that allows students to practice this more general method.