Investigating Student Difficulties with Dirac Notation

Singh, Chandralekha; Marshman, Emily

doi:10.1119/perc.2013.pr.074

Cited by 20 publications

(21 citation statements)

References 10 publications

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“…Other difficulties with Dirac notation: In the investigation described in Ref. [73], some students also incorrectly claimed that one can always exchange the bra and ket states in the Dirac notation without changing its value if the operator sandwiched between them is a Hermitian operator corresponding to an observable, i.e., x|Q|Ψ = Ψ|Q|x ifQ is Hermitian. While some of them correctly reasoned that the eigenvalues of a Hermitian operator are real, they erroneously concluded that this implies that one can exchange the bra and ket states without complex conjugation if the scalar product involves sandwiching a Hermitian operator.…”

Section: Difficulties With Dirac Notation and Issues Related To Quantmentioning

confidence: 99%

Review of student difficulties in upper-level quantum mechanics

Singh

Marshman

2015

Phys. Rev. ST Phys. Educ. Res.

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143

133

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Learning advanced physics, in general, is challenging not only due to the increased mathematical sophistication but also because one must continue to build on all of the prior knowledge acquired at the introductory and intermediate levels.In addition, learning quantum mechanics can be especially challenging because the paradigms of classical mechanics and quantum mechanics are very different. Here, we review research on student reasoning difficulties in learning upper-level quantum mechanics and research on students' problem-solving and metacognitive skills in these courses. Some of these studies were multi-university investigations. The investigations suggest that there is large diversity in student performance in upper-level quantum mechanics regardless of the university, textbook, or instructor, and many students in these courses have not acquired a functional understanding of the fundamental concepts. The nature of reasoning difficulties in learning quantum mechanics is analogous to reasoning difficulties found via research in introductory physics courses. The reasoning difficulties were often due to over-generalizations of concepts learned in one context to another context where they are not directly applicable. Reasoning difficulties in distinguishing between closely related concepts and in making sense of the formalism of quantum mechanics were common. We conclude with a brief summary of the research-based approaches that take advantage of research on student difficulties in order to improve teaching and learning of quantum mechanics.Students taking upper-level quantum mechanics often develop survival strategies for performing reasonably well in their course work. For example, they become proficient at solving algorithmic problems such as the time-independent Schrödinger equation with a complicated potential energy and boundary conditions. However, research suggests that they often struggle to make sense of the material and build a robust knowledge structure. They have difficulty mastering concepts and applying the formalism to answer qualitative questions, e.g., questions related to the properties of Review of Student Difficulties in Upper-Level Quantum Mechanics

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Section: Difficulties With Dirac Notation and Issues Related To Quantmentioning

confidence: 99%

Review of student difficulties in upper-level quantum mechanics

Singh

Marshman

2015

Phys. Rev. ST Phys. Educ. Res.

Self Cite

143

133

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“…b) Inconsistency in identifying a quantum state in position representation Students in quantum mechanics courses often display inconsistent reasoning in their responses to consecutive questions. For example, a conceptual, multiple-choice survey was administered to 39 upper-level students to determine the extent to which they use appropriate problem-solving, reasoning, and self-regulatory skills while solving quantum mechanics problems [45]. In addition, think-aloud interviews were conducted with 23 students to observe how they reasoned about the quantum mechanics problems.…”

Section: Inconsistent And/or Context-dependent Reasoning In Quantum Mmentioning

confidence: 99%

Framework for understanding the patterns of student difficulties in quantum mechanics

Marshman

Singh

2015

Phys. Rev. ST Phys. Educ. Res.

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Compared with introductory physics, relatively little is known about the development of expertise in advanced physics courses, especially in the case of quantum mechanics. Here, we describe a framework for understanding the patterns of student reasoning difficulties and how students develop expertise in quantum mechanics. The framework posits that the challenges many students face in developing expertise in quantum mechanics are analogous to the challenges introductory students face in developing expertise in introductory classical mechanics. This framework incorporates both the effects of diversity in upper-level students' prior preparation, goals, and motivation in general (i.e., the facts that even in upper-level courses, students may be inadequately prepared, have unclear goals, and insufficient motivation to excel) as well as the "paradigm shift" from classical mechanics to quantum mechanics. The framework is based on empirical investigations demonstrating that the patterns of reasoning, problem-solving, and self-monitoring difficulties in quantum mechanics bear a striking resemblance to those found in introductory classical mechanics. Examples from research in quantum mechanics and introductory classical mechanics are discussed to illustrate how the patterns of difficulties are analogous as students learn to unpack the respective principles and grasp the formalism in each knowledge domain during the development of expertise. Embracing such a framework and contemplating the parallels between the difficulties in these two knowledge domains can enable researchers to leverage the extensive literature for introductory physics education research to guide the design of teaching and learning tools for helping students develop expertise in quantum mechanics. I. INTRODUCTIONA solid grasp of the fundamental principles of quantum physics is essential for many scientists and engineers. However, quantum physics is a technically difficult and abstract subject. The subject matter makes instruction quite challenging, and even capable students constantly struggle to develop expertise and master basic concepts.In order to help students develop expertise in quantum mechanics, one must first ask how experts compare to novices in terms of their knowledge structure and their problem-solving, reasoning, and metacognitive skills. According to Sternberg [1], some of the characteristics of an expert in any field include: 1) having a large and well organized knowledge structure about the domain; 2) spending more time in determining how to represent problems than searching for a problem strategy (i.e., more time spent analyzing the problem before implementing the solution); 3) working forward from the given information in the problem and implementing strategies to find the unknowns; 4) developing representations of problems based on deep, structural similarities between problems; 5) efficient problemsolving; when under time constraints, experts solve problems more quickly than novices, and 6) accurately predicting the difficulty in sol...

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“…Students typically encounter both Dirac notation and wave functions, and are expected to be able to solve problems with both. Marshman and Singh [22][23][24] identified various difficulties that students have with Dirac notation and with translating quantum states from Dirac notation to wave function representation, some of which appear to impact student ability to apply the Born rule.…”

Section: Introductionmentioning

confidence: 99%

Investigating how students relate inner products and quantum probabilities

Wan

Emigh

Shaffer

2019

Phys. Rev. Phys. Educ. Res.

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The Born rule, which describes the formalism for determining probabilities, is one of the most fundamental postulates in quantum mechanics. This paper presents results from an investigation into how students apply the Born rule to determine probabilities for energy and position measurements. The investigation includes two stages with independent methods: a quantitative analysis of student written work and a qualitative analysis of student individual interviews. The data from written tasks suggest that after instruction many students have not developed a coherent model for determining probabilities that they can apply to observables regardless of whether the eigenvalues are discrete or continuous. Moreover, many students seem to lack a functional understanding of quantum states and inner products that allows them to translate between Dirac notation and wave function representation. These results motivate student interviews, which allow us to probe student reasoning in depth. Prior research suggests that various features of each notation used in quantum mechanics may have an impact on how students perform computations. We postulate that the features of quantum notations may also interact with student sensemaking. Therefore, we analyze student interviews through the lens of the structural features of quantum notations framework. In particular, we discuss how different structural features may facilitate or hinder student sensemaking about concepts relevant to determining probabilities. The results from both quantitative and qualitative data suggest that unsuccessfully differentiating between a wave function and its associated state vector in Dirac notation may be a primary barrier for students to develop a model for determining probabilities for discrete and continuous cases.

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Investigating Student Difficulties with Dirac Notation

Cited by 20 publications

References 10 publications

Review of student difficulties in upper-level quantum mechanics

Review of student difficulties in upper-level quantum mechanics

Framework for understanding the patterns of student difficulties in quantum mechanics

Investigating how students relate inner products and quantum probabilities

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