2013 Physics Education Research Conference Proceedings 2014
DOI: 10.1119/perc.2013.pr.074
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Investigating Student Difficulties with Dirac Notation

Abstract: Quantum mechanics is challenging even for advanced undergraduate and graduate students. Dirac notation is a convenient notation used extensively in quantum mechanics. We have been investigating the difficulties that the advanced undergraduate and graduate students have with Dirac notation. We administered written free response and multiple-choice questions to students and also conducted semi-structured individual interviews with 23 students using a think-aloud protocol to obtain a better understanding of the r… Show more

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Cited by 21 publications
(21 citation statements)
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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%
“…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%
“…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%
“…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%