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arXiv:1701.01411 · Physics Education Research Conference (PERC)

Student difficulties with representations of quantum operators corresponding to observables

Emily Marshman, Chandralekha Singh

Published Dec 29, 2016• 7 citations
arXivHTMLPDF

Abstract

Dirac notation is used commonly in quantum mechanics.However, many upper-level undergraduate and graduate students in physics have difficulties with representations of quantum operators corresponding to observables especially when using Dirac notation.To investigate these difficulties, we administered free-response and multiple-choice questions and conducted individual interviews with students in advanced quantum mechanics courses.We discuss the analysis of data on the common difficulties found. I.

DOI
10.1119/perc.2016.pr.049

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Dirac notation is used commonly in quantum mechanics.However, many upper-level undergraduate and graduate students in physics have difficulties with representations of quantum operators corresponding to observables especially when using Dirac notation.To investigate these difficulties, we administered free-response and multiple-choice questions and conducted individual interviews with students in advanced quantum mechanics courses.We discuss the analysis of data on the common difficulties found. I.
Dirac notation is often used in upper level quantum mechanics courses, but students struggle with this representation.To investigate the difficulties that advanced students (i.e., upper-level undergraduate and graduate students) have while translating state vectors in Dirac notation to wave functions in position and momentum representations, we administered free-response and multiple-choice questions and conducted individual interviews with students.We find that students display common difficulties with these topics.
Abstract Learning quantum mechanics (QM) is widely recognized as a challenging task, as it involves highly abstract concepts and requires a significant shift from classical physics reasoning. Developing expertise in QM requires students to represent physical concepts using multiple formalisms, such as Dirac notation and algebraic representations, and to move flexibly between them. This study investigates two main categories of student difficulties related to Dirac notation. The first concerns difficulties in expressing a generic state vector in the position and momentum representations. The second relates to challenges in writing the eigenvectors of the position and momentum operators in Dirac notation. To explore these issues, we administered questionnaires to third-year undergraduate physics students enrolled in a QM course at the Lyon 1 University. We present a detailed analysis of student responses, identify the principal difficulties encountered, and compare our findings with results reported in prior American studies. Our results show that many students struggle to establish a clear connection between Dirac notation and its functional algebraic representation. In particular, students find it easier to recognize state vectors and eigenvectors than to construct them explicitly. Additionally, common difficulties include confusion between bras (e.g. x | and p |) and the corresponding operators, as well as confusion between the Dirac delta distribution and the Kronecker delta. Finally, we propose some interpretations of our results in light of those reported in previous studies.