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Abstract Title: Mathematical representations in quantum mechanics instruction
Abstract: Quantum mechanics involves the use of several mathematical representations of varying levels of familiarity to students, including Dirac notation, that are central to linking the mathematical formalism to the physical concepts and phenomena. This session highlights research in physics and mathematics education on: the structural features of these representations and their relationship to student reasoning; student fluency with and between representations; and student choice of representation to use for a particular problem, including how some of these choices depend on the instructional sequence (i.e., spins first or position first).
Abstract Type: Talk Symposium

Author/Organizer Information

Primary Contact: John Thompson
University of Maine

Symposium Specific Information

Moderator: John Thompson
Presentation 1 Title: Quantum notations as computational tools
Presentation 1 Authors: Elizabeth Gire, Oregon State University; Ed Price, California State University San Marcos
Presentation 1 Abstract: The formalism of quantum mechanics includes a rich collection of representations for describing quantum systems, including functions, graphs, matrices, histograms of probabilities, and Dirac notation. The varied features of these representations affect how computations are performed. For example, identifying probabilities of measurement outcomes for a state described in Dirac notation may involve identifying expansion coefficients by inspection, but if the state is described as a function, identifying those expansion coefficients often involves performing integrals. We have identified four structural features of quantum notations: individuation, degree of externalization, compactness, and symbolic support for computational rules. We will discuss how these structural features may or may not support student reasoning.
Presentation 2 Title: Investigating Students' Meta-Representational Competence with Matrix Notation and Dirac Notation
Presentation 2 Authors: Megan Wawro and Kevin Watson, Virginia Polytechnic Institute and State University; Warren Christensen, North Dakota State University
Presentation 2 Abstract: In this report we share analysis regarding students' meta-representational competence (MRC) that is expressed as they engage in solving quantum mechanics problems that involve linear algebra concepts. The particular characteristic of MRC that is the focus of this analysis is students' critiquing and comparing the adequacy of representations, specifically matrix notation and Dirac notation, and judging their suitability for various tasks (diSessa, 2004). With data from semi-structured individual interviews, we created categories of types of MRC elicited during students' work on an expectation value problem and a normalization problem. During the presentation will we share examples of the various MRC categories from our data and explore the relationship between strong MRC and a student's understanding and symbolization of linear algebra concepts within quantum mechanics problems.
Presentation 3 Title: Mathematization of Matrices in Quantum Mechanics
Presentation 3 Authors: Gina Passante, California State University Fullerton
Presentation 3 Abstract: Matrices play an important role in spins-first quantum mechanics (QM) instruction.  In this instructional paradigm students are first introduced QM using two-state systems with a heavy reliance on bra-ket notation and linear algebra.  The quantum mechanical state vectors can be represented by two-dimensional column vectors and operators can be represented by 2x2 matrices.  In this work, we explore student fluidity with matrix representations in quantum mechanics.  In particular, we look at student use, generation, and interpretation of vectors and matrices.  The data is drawn from student responses to questions in the context of two-state spin systems and degenerate perturbation theory.
Presentation 4 Title: Student use of different mathematical representations for expectation values of physical observables
Presentation 4 Authors: Homeyra Sadaghiani, California State Polytechnic University, Pomona
Presentation 4 Abstract: As part of ongoing research to understand how students relate quantum mechanics concepts and formalisms, we are investigating student approach to calculating and interpreting quantum mechanical expectation values for physical observables in two different teaching paradigms: Spins First and Position First. More specifically, we are comparing the range of the various mathematical representations students use for formalizing the concept of expectation value and whether or not there is a more recurrent mathematical representation among students in the two paradigms. Analyzing students' written responses to a series of open-ended research questions and informal focus group discussions with students from each paradigm has given us some insight into the frequency and commonality of the various mathematical formalisms students use.