Representations for a Spins-First Approach to Quantum Mechanics

Corinne Manogue, Oregon State University Elizabeth Gire, University of Memphis David McIntyre, Oregon State University Janet Tate, Oregon State University

Submitted Abstract to PERC 2011

In the Paradigms in Physics Curriculum at Oregon State University, we take a spins-first approach to quantum mechanics using a java simulation of successive Stern-Gerlach experiments to explore the postulates. The experimental schematic is a diagrammatic representation that we use throughout our discussion of quantum measurements. With a spins-first approach, it is natural to start with Dirac bra-ket language for states, observables, and projection operators. We also use explicit matrix representations of operators and ask students to translate between the Dirac and matrix languages. The projection of the state onto a basis is represented with a histogram. When we subsequently introduce wave functions, the wave function attains a natural interpretation as the continuous limit of these discrete histograms or of a projection of a Dirac ket onto position or momentum eigenstates. We are able to test the students’ facility with moving between these representations in later modules.

  • Spins First: we use the experimental schematic as a diagrammatic representation.
  • Dirac Bra-ket language
    • students use spontaneously for wave functions
    • allows you to bypass explicit spatial integrals
    • Ethan's example on last Tuesday of 2011 periodic potentials: ask student to write Hamiltonian in matrix form. Students write matrix with entries $\langle 1|H|1\rangle$, etc. Then give alpha, beta and zero names to entries of approximately the same size. Allows you to see symmetries in the Hamiltonian explicitly. Do we have video of this?
  • Introduce the idea that $\langle n|\Psi\rangle$ is a coefficient of a vector, eigenfunction expansions: make this a central EARLY idea in quantum. Bra-ket notation makes this representation easy and natural. What evidence do we have that students get this?
    • Janet's representation of these coefficients as a histogram. Really emphasize this/who else does it?
  • Different kinds of operators: rotations (classical), observalbles, projections,
  • bra-ket, abstract symbols, and explicit matrix representations
    • Activities where we explicitly ask students to move back and forth between representations, paired activities where they do the same calculations in different representations.
    • explicit matrix multiplication (matrices with just underscores) vs. Henri and David Roundy's representations with $A_{ij}$.
  • Wave functions are then an example of coefficients of vectors
    • $\psi(x)=\langle x\vert \psi\rangle$
    • Talk explicitly about the transition from discrete to continuous representations.
    • Don Mountcastle's issue about dimension differences in this limit. How much do we need to worry about this?
    • histograms become wave function graphs.

Research questions:

  • Are students doing sense-making with these representations?
    • Look for video from 1-2 waves talking about wave functions.
  • What representations do students use to make sense out of quantum systems?
    • Look for changes in representation compared to the problem statement.
    • What kinds of discussions do students have when we are asking them to translate between representations?

Possible approaches:

  • Pick one activity: look for evidence of student understanding
  • Grounded approach: let the research question arise from the data set.
  • Ampere's Law approach:
    • here are some of the issues
    • here are some things that you might do
    • here are affordances of the activities tha tmight help students come to a new understanding.
  • Formative assessment: assess student understanding as they are doing the activities.
  • Use “standard” quantum assessments
    • Chandralekha Singh
    • Sam McKagan
    • We should arrange a session where participants bring in quantum questions and we try to devise new parts to an assessment. People to invite: Chandralekha Singh, Sam McKagan, Mark Haugan, Liz Gire, Stamatis Vokos, David McIntyre, Corinne Manogue, (others??)

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