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Physical Review Physics Education Research

Teachers’ acceptance and understanding of a reduced Dirac formalism and the Bloch sphere

Anonymous, Carsten Albert, Gesche Pospiech

Published Jun 29, 2026• 0 citations
Publisher

Abstract

Mathematical descriptions are the focus of quantum physics education research, as quantum physics is inherently described by its mathematical structures. However, there is a lack of research regarding the role of mathematical descriptions for understanding fundamental principles of quantum physics in teacher preparation and training. In order to investigate how preservice teachers and in-service teachers assess and understand the mathematical formalism in quantum physics, an explorative interview study was conducted. Therefore, we developed a teaching-learning sequence employing a reduced Dirac formalism in the context of single-photon polarization serving as an exemplary two-state system. In that respect, the focus was on two mathematical representations. The ket notation was used as an algebraic representation in connection with the Bloch sphere as a graphical representation of quantum states. On that basis, semistructured and material-based interviews with N = 15 preservice teachers and N = 15 in-service teachers were conducted. The data suggest that the reduced Dirac formalism is well understood and accepted by both preservice teachers and in-service teachers. However, some in-service teachers raise critical concerns about the abstract nature of the mathematical formalism, pointing to a perceived lack of experimental or phenomenological connection. The ket notation is valued not only for its communicative clarity and its role as a language but also for its structural role. By comparison, the Bloch sphere is met with more cautious acceptance—yet its use in visualizing experiments and measurement processes is positively assessed.

DOI
10.1103/zwh6-84p5

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[This paper is part of the Focused Collection in Investigating and Improving Quantum Education through Research.] Evidence from physics education research suggests that the introduction of a formalistic description of quantum phenomena can be beneficial to student learning, particularly in terms of helping students move away from naive realist views of quanta or space-time descriptions of quantum phenomena. However, the mathematical requirements for mastering quantum formalism are typically far beyond the scope of secondary school education. Therefore, in previous research, we have proposed a reduced Dirac notation tailored to the capabilities of secondary school students, which is included in a novel teaching-learning sequence. However, the cognitive processes of students when confronted with an introduction to quantum physics via a reduced Dirac notation have not been thoroughly investigated: In particular, it seems important to investigate whether this approach is actually beneficial to students’ learning and to explore the learning difficulties students are likely to encounter in such a setting. This paper contributes to this research gap by reporting the findings of an acceptance survey: N = 14 learners participated in one-on-one interviews following a protocol consisting of four cyclical phases (information phase, acceptance evaluation, paraphrasing phase, and application phase). The findings of our study indicate that the instructional elements within our new teaching-learning sequence were generally well accepted by students but also revealed learning difficulties that students may encounter. Thus, on the one hand, the findings of this study pave the way for future research on students’ learning of quantum physics using Dirac notation. On the other hand, the in-depth findings reported here may guide teachers in designing learning environments that introduce Dirac notation at the upper secondary school and introductory university levels. Published by the American Physical Society 2024
Abstract This paper describes the design of a new teaching-learning sequence on quantum physics aimed at upper secondary school students. In this teaching-learning sequence, GeoGebra simulations and interactive screen experiments are used to investigate the behaviour of a single photon at beam splitter and single photon interference in a Michelson interferometer. We propose a minimal formalism using Dirac notation, which avoids complex numbers and elaborate vector calculus, to make a quantitative description of the quantum optics experiments accessible to secondary school students. With this new educational pathway, we take into account findings from physics education research, which suggest that the introduction of a mathematical formalism tailored to students’ abilities might help them to overcome naive-realist views of quanta or space-time descriptions of quantum phenomena, while at the same time facilitating a transition to a functional understanding of quantum models.
Abstract The teaching/learning of quantum mechanics via two-state systems is continuously spreading in secondary schools due to its promising results in physics education research. One possibility is the use of polarisation states of photons. This paper reports on a polarisation-based introduction to quantum mechanics in physics teacher education. A widely used school material prepares teacher trainees for their future work and also improves their conceptual knowledge. This part includes statistical calculations using only secondary school mathematics and a new formulation of the uncertainty relation, using only real numbers. The second step is to prepare the formalism of quantum mechanics using real two-dimensional vectors and matrices. Considering that students may not learn complex linear algebra, we offer a new way to introduce the complete formalism of two-state systems via circular polarisation providing a step-by-step exploration of complex quantum states. This points out the advantage of using complex linear algebra via a physical example, providing the opportunity to reach the elements of advanced quantum physics and quantum computing while deepening the physics background of the secondary school material.
Quantum mechanics is included in many curricula across countries because of its cultural value and technological application. In the last decades, two-state approaches to quantum mechanics became popular because of the age of quantum computers. This article presents an experiment with 24 Hungarian high school students on teaching/learning quantum mechanics according to Dirac’s approach to concepts and basic formalism developed in the context of light polarization. Tutorials, pre/post-tests, and oral interviews are the main monitoring tools used to collect data on the students’ learning path. From the qualitative and quantitative data analysis, learning progressions emerged in the phenomenology exploration and on the probabilistic nature of single quantum measurement. The students’ conceptions of quantum state are enriched, confirming the importance to focus educational approaches on fundamental topics. For one section of students, the complex relationship between quantum state and property remained problematic, but the students’ interpretations of a quantum state can be categorized. Two lines of reasoning emerged regarding the impossibility to attribute a trajectory to a quantum system, one more orthodox and one that seeks to avoid the probabilistic nature of the quantum world.
Research in physics education has revealed that students often have incomplete and incoherent ways to use the concept of photon. Most researchers in science education recommend photons to be taken in a realistic sense as existing entities of quantum objects. Furthermore, in science education literature it is often claimed that realistic stance to photons also is preferred by physicists. Taken that photons in quantum theory of light invariably refer to quantized degrees of the freedom of an electromagnetic field it is questionable how one should introduce photons as entities without inadvertently supporting naïve particle-realism about photons, a well-known situation in introductory teaching of quantum theory of light. Fewer researchers suggest that photons should be understood as photon states, quantized degrees of freedom of electromagnetic field, as in contemporary quantum optics and quantum theory of light. Here, by using a questionnaire (a Likert-scale survey), we explore pre-service physics teachers’ preferences to use expressions that can be associated with either quanton or photon state views about photons in context of single-photon double-slit experiment. The questionnaire is administered in two stages (in the beginning and after a teaching sequence) during a pre-service physics teachers’ course addressing quantum optics and technology. Based on the analysis of the responses, we find that at the beginning of the course, the responses are scattered but in the final survey, clusters corresponding to use of photon state-based expressions become more consistent.
Abstract Physics education research has revealed that students have multiple ways of understanding the concept of the photon and that students’ use of it is often incoherent and incomplete. In this study, we explore what kinds of statements pre-service physics teachers and university physics students choose on a multiple-choice questionnaire, designed to reveal students’ stances on three conceptual views of the photon (quantum state, quantum realistic, and naïve dualistic) in the context of quantum optical experiments. The questionnaire was conducted in a pre-post manner. The test group (pre-service physics teachers N = 13) attended an introductory quantum physics course which emphasised coherent use of natural language, and the control group (university physics students pre N = 33, post N = 25) attended a standard introductory course in quantum physics. Pre-test results show that both groups expressed incoherent use of the concept of the photon. The post-test results reveal that after teaching intervention, the test group showed significant improvement in the coherent use of the concept of the photon as a quantum state. The post-test results of the control group remained incoherent. The results show that paying attention to the coherent use of natural language in teaching quantum physics helps pre-service physics teachers to recognise and choose coherent expressions about the photon.