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

Estimated Time: 30-90 minutes

Students make energy and angular momentum calculations for a particle confined to a ring in a particular initial state that is a linear combination of energy eigenstates. These calculations are done in Dirac “bra-ket” notation, matrix notation and in wavefunction notation. One of the main purposes of this activity is to help students see the parallel between similar calculations in these three representations and to connect those calculations explicitly to the postulates of quantum mechanics.

• 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”, matrix and wavefunction notation

• Tabletop Whiteboard with markers
• A handout for each student.

This activity flows naturally from a lecture in which the eigenstates for energy and angular momentum on a ring are found. Many of the calculations done here are similar to calculations they have done before, but this activity emphasizes the different representations we use for quantum calculations and highlights when each representation is most useful.

The first activity begins with a reminder to the students that that an arbitrary state $|\Phi\rangle$ can be written in the $L_z$ eigenbasis as

$$ \eqalign{\left| \Phi\right\rangle &\doteq \pmatrix{\vdots \cr \langle 2|\Phi\rangle \cr \langle 1|\Phi\rangle \cr \langle 0|\Phi\rangle \cr \langle -1|\Phi\rangle \cr \langle -2|\Phi\rangle \cr \vdots} = \pmatrix{\vdots \cr a_{2} \cr a_{1} \cr a_{0} \cr a_{-1} \cr a_{-2} \cr \vdots}} $$

Including this in the introduction to this activity should help students avoid confusion about the ordering of the elements in the column vectors used in this activity.

• Notation:
  • For the bra-ket notation, at this point, many students can find probabilities “by inspection.” It is a good idea to encourage them to write out their process at least once for each notation in order to be able to better compare them.
  • The three initial states on the handout are identical, but it may take most of the activity for students to realize this, especially for the wavefunction notation.
    • In particular, students struggle with the fact that the coefficient and the normalization constant are combined in the wave function notation (e.g. “Where does this $\pi$ come from?”))
    • If they do recognize that they are the same state, encourage them to do at least one calculation in each notation in order to show that they get the same answer.
• Zero: students may be confused in the case where they measure zero probability or when the observed value/eigenvalue for a quantity is zero.
• Degeneracy: students may experience some difficulty due to the degeneracy of some states,
  • you have to include all states that have the same eigenvalue
  • it is the sum of the square of the norm of the coefficients and not the square of the sum of coefficients.

$$P_{E={4\hbar^2\over 2I}}=\vert \langle 2\vert \psi\rangle\vert^2+\vert \langle -2\vert \psi\rangle\vert^2\neq \vert \langle 2\vert \psi\rangle+\langle -2\vert \psi\rangle\vert^2$$

• Students commonly attempt to determine the values resulting from a quantum experiment by allowing the operator corresponding to the observable of interest to act on the initial state. Students who do this should be encouraged to consider the nature of this transformation (it's a vector, not a scalar) and to recognize that the transformation does not necessarily yield an eigenvector (the state of the system should be an eigenstate after the measurement).

There are several main ideas to bring up in the wrap-up discussion:

• Notation: the fact that all three states are the same state.
  • This also serves as a great introduction to a mini-lecture reminding students that you can write any state $|\psi\rangle$ as a linear combination of eigenstates $|m\rangle$.
• This is a good opportunity to emphasize to students that there are only a limited number of different types of quantum calculations they can do once they know the eigenstates and eigenvalues for a system. In particular this activity encourages them to tie these measurements directly to the postulates of quantum mechanics. In general this should be relatively straight forward, however, students may struggle to relate the quantum postulates and calculations they know in bra-ket and matrix notation to the calculations they do in the position basis. This activity may lead to a mini-lecture/review of how wavefunctions can be written as states in the continuous, position basis using bra-ket notation. Quantum Text 5.3

$$|\psi\rangle = \sum_m c_m |m\rangle$$

• Degeneracy: reiterate that the probability of a degenerate eigenvalue is the sum of the square of the norm of the coefficients.

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

• This activity can be used to help review and solidify students understanding of the formalism used to make quantum calculations. This is especially important if you expect to proceed from this activity to calculations of the rigid rotor and hydrogen atom which use the same formalism with substantially more challenging eigenfunctions.

• 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.