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We discuss an investigation of student difficulties with Dirac notation in the context of a three-dimensional vector space and the validation and evaluation of a Quantum Interactive Learning Tutorial (QuILT) to improve student understanding of these concepts. We find that many upper-level undergraduate students in quantum mechanics courses have difficulties with Dirac notation even in the context of a three-dimensional vector space after traditional instruction. The QuILT uses analogical reasoning and builds on students’ prior knowledge of three-dimensional vectors in the familiar context of introductory mechanics to help students build a coherent understanding of Dirac notation in three dimensions before transitioning to the quantum mechanical context. We summarize the development of the QuILT and findings from its evaluations. We thank the National Science Foundation for support.


AAPT SM 20

Improving Student Understanding Of Dirac Notation by Using Analogical Reasoning in the Context of a Three-Dimensional Vector Space

upper-division physics

dirac notation

quantum mechanics

The 2020 AAPT Virtual Summer Meeting brought together physics educators from around the globe to learn about the latest in physics theory, research methodology and education from leaders in their field. The program featured plenary speakers, sessions, workshops and many special activities.

For more information please visit the 2020 AAPT Virtual Summer Meeting page at: https://www.aapt.org/Conferences/SM2020/index.cfm.


The 2020 AAPT Virtual Summer Meeting brought together physics educators from around the globe to learn about the latest in physics theory, research methodology and education from leaders in their field. 

technical paper

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Student Difficulties with Operators Corresponding to Observables in Dirac Notation

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Here we focus on an investigation that suggests that, even though Dirac notation is used extensively, many advanced undergraduate and Ph.D. students in physics have difficulty expressing the identity operator and other Hermitian operators corresponding to physical observables in Dirac notation. We first describe the difficulties students have with expressing the identity operator and other Hermitian operators corresponding to observables in Dirac notation. We then discuss how the difficulties found via written surveys and individual interviews were used as a guide in the development and validation of a Quantum Interactive Learning Tutorial (QuILT) to help students develop a good grasp of these concepts. We also discuss the effectiveness of the QuILT based on in-class evaluations. We thank the National Science Foundation for support.

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Students’ self-efficacy, interest and identity in physics can influence their learning, performance and career decisions. Therefore, we investigated the impact of learning environment on male and female students’ physics self-efficacy, interest and identity in calculus-based introductory physics. Findings can be useful in creating equitable and inclusive learning environments in which all students can thrive. We thank the National Science Foundation for support.

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Improving Student Understanding Of Dirac Notation by Using Analogical Reasoning in the Context of a Three-Dimensional Vector Space

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