Adapted from Tutorials in Physics: Quantum Mechanics, University of Washington. Information about this original tutorial and its documented impact on student understanding can be found in the following article:

Student ability to distinguish between superposition states and mixed states in quantum mechanics, Gina Passante, Paul J. Emigh, and Peter S. Shaffer, Phys. Rev. ST Phys. Educ. Res. 11, 020135 (2015).

Use interference to distinguish possible experimental results for superpositions and mixed states.

Estimated Time: 30 minutes

Wave functions for the infinite square well potential. Superposition principle for waves (pointwise addition). Relationship between wave function and probability density.

• Tabletop Whiteboard with markers
• A handout for each student

Valuable to start off with a SWBQ or Clicker Question focusing on whether or not a superposition and a mixed state are distinguishable or indistinguishable experimentally.

Then, give students a few minutes to reflect on the following statement:

“The wave function given by represents a lack of knowledge about the state of the system. The system is definitely in either the ground state or the first excited state. The wave function simply tells you that the probability is 1/2 that the system is really in the ground state and 1/2 that it is really in the first excited state.” Do you agree or disagree with this statement?

What are some different experiments that you might perform on these systems? Would all of them result in distinguishable results?

Why is it insufficient to investigate only a single particle from each ensemble?

What is the underlying mathematics that gives an experimental difference between these ensembles? [Emphasize that this illustrates a fundamental “quantum-ness” that does not arise from classical descriptions of particles.]

Challenge: If students have learned about the time evolution of wave functions, ask whether there are any differences in how each probability density depends on time?

Bring students back to the SWBQ and the student quote they considered before starting the worksheet. What answers would students give now?

Similar exercises can be performed using spin states and Stern-Gerlach analyzers in different directions. Online resources that support this can be found at http://www.st-andrews.ac.uk/physics/quvis/simulations_html5/sims/superposition/superposition-mixed-states.html.