• Eigenvalues and eigenstates
• Measurements of energy and angular momentum in quantum systems
• Quantum probabilities
• Superposition of states
• Expectation values
• Time dependence
• Degeneracy

Estimated Time: 20 minutes

Students calculate the expectation value of energy and angular momentum 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.

• 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 expectation values using Dirac “bra-ket” notation
• Familiarity with time evolution in quantum mechanics

• Tabletop Whiteboard with markers
• A handout for each student

• 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).

• 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.

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

• Notation: as with earlier activities, students are usually comfortable at this point with doing expectation values in bra-ket notation, but fewer are comfortable with using wavefunction notation. For groups that finish early, ask them to use the other method to compare.

This activity provides an opportunity to contrast two methods of finding expectation values.

1. Carry out the explicit and messy differentiation and integration on the given state.
2. Recast the initial state as a linear combination of eigenstates and carry out the much simpler calculations on these eigenstates.

Generally, students in the class will be mixed in the approach they choose. By emphasizing this when you wrapup this activity, students have the opportunity to sort out for themselves the benefits of each method. One of the thrusts of the first activity is to get students to make this comparison explicitly.

• 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.