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

Moritz Förster1, Carsten Albert1,2, and Gesche Pospiech1

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Phys. Rev. Phys. Educ. Res. 22, 020122 – Published 28 August, 2026

DOI: https://doi.org/10.1103/zwh6-84p5

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.

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References (85)

  1. K. Krijtenburg-Lewerissa, H. J. Pol, A. Brinkman, and W. R. van Joolingen, Insights into teaching quantum mechanics in secondary and lower undergraduate education, Phys. Rev. Phys. Educ. Res. 13, 010109 (2017).
  2. A. Donhauser, P. Bitzenbauer, L. Qerimi, S. Heusler, S. Küchemann, J. Kuhn, and M. S. Ubben, Empirical insights into the effects of research-based teaching strategies in quantum education, Phys. Rev. Phys. Educ. Res. 20, 020601 (2024).
  3. P. Bitzenbauer, Quantum physics education research over the last two decades: A bibliometric analysis, Educ. Sci. 11, 699 (2021).
  4. W. Dür and S. Heusler, Visualization of the invisible: The qubit as key to quantum physics, Phys. Teach. 52, 489 (2014).
  5. H. R. Sadaghiani and J. Munteanu, Spin first instructional approach to teaching quantum mechanics in sophomore level modern physics courses, in 2015 Physics Education Research Conference Proceedings, College Park, MD (American Association of Physics Teachers, 2015), p. 287.10.1119/perc.2015.pr.067
  6. S. Montagnani, A. Stefanel, M. L. Mù Chiofalo, L. Santi, and M. Michelini, An experiential program on the foundations of quantum mechanics for final-year high-school students, Phys. Educ. 58, 035003 (2023).
  7. G. Pospiech, Quantum cryptography as an approach for teaching quantum physics, in Teaching-Learning Contemporary Physics, Challenges in Physics Education, edited by B. Jarosievitz and C. Sükösd (Springer International Publishing, Cham, 2021), pp. 19–31.
  8. E. Y. Weissman, A. Merzel, N. Katz, and I. Galili, Keep it secret, keep it safe: Teaching quantum key distribution in high school, Eur. Phys. J. Quantum Technol. 11, 64 (2024).
  9. G. Pospiech, Teaching quantum physics between quantum technology and general education, in Frontiers of Fundamental Physics FFP16, Springer Proceedings in Physics, edited by E. Aydiner, B. G. Sidharth, M. Michelini, and C. Corda (Springer International Publishing and Imprint Springer, Cham, 2024), Vol. 392, pp. 393–416.
  10. G. Pospiech, Die zweite Quantenrevolution: Quanteninformatik im Physikunterricht, in PhyDid B, Didaktik der Physik, edited by J. Grebe-Ellis and H. Grötzebauch (Deutsche Physikalische Gesellschaft, 2021), pp. 1–11.
  11. C. Albert, M. Förster, and G. Pospiech, Developing a quantum physics curriculum for lower secondary education: Insights into the design of a spin first teaching concept and first empirical findings, Eur. Phys. J. Quantum Technol. 12, 132 (2025).
  12. W. Dür and S. Heusler, The qubit as key to quantum physics part ii: Physical realizations and applications, Phys. Teach. 54, 156 (2016).
  13. M. Bondani, M. L. Chiofalo, E. Ercolessi, C. Macchiavello, M. Malgieri, M. Michelini, O. Mishina, P. Onorato, F. Pallotta, S. Satanassi, A. Stefanel, C. Sutrini, I. Testa, and G. Zuccarini, Introducing quantum technologies at secondary school level: Challenges and potential impact of an online extracurricular course, Physics 4, 1150 (2022).
  14. A. M. Kelly, T.-C. Wei, D. Schneble, and M. Darienzo, Exploratory factor analysis of a precollege quantum information science and technology survey: Exploring career aspiration formation and student interest, Eur. Phys. J. Quantum Technol. 12, 11 (2025).
  15. G. Pospiech and M. Förster, Quanteninformatik in der Lehrerbildung mit Transfer in den berufsorientierenden Unterricht, in PhyDid B—Didaktik der Physik—Beiträge zur DPG-Frühjahrstagung Greifswald, edited by H. Grötzebauch and S. Heinicke (Deutsche Physikalische Gesellschaft, 2024).
  16. A. Merzel et al., The core of secondary level quantum education: A multi-stakeholder perspective, Eur. Phys. J. Quantum Technol. 11, 27 (2024).
  17. P. Bitzenbauer, Quantenoptik an Schulen: Studie Im Mixed-Methods Design zur Evaluation des Erlanger Unterrichtskonzepts zur Quantenoptik, Studien zum Physik- und Chemielernen (Logos Verlag, Berlin, 2020), Vol. 303.
  18. M. Michelini, R. Ragazzon, L. Santi, and A. Stefanel, Proposal for quantum physics in secondary school, Phys. Educ. 35, 406 (2000).
  19. G. Pospiech, A. Merzel, G. Zuccarini, E. Weissman, N. Katz, I. Galili, L. Santi, and M. Michelini, The role of mathematics in teaching quantum physics at high school, in Teaching-Learning Contemporary Physics, Challenges in Physics Education, edited by B. Jarosievitz and C. Sükösd (Springer International Publishing, Cham, 2021), pp. 47–70.
  20. G. Pospiech and R. A. S. Karam, Role of mathematics in physics from multiple perspectives, in The International Handbook of Physics Education Research, edited by M. F. Taşar and P. Heron (AIP Publishing Books 2023, Melville, New York, 2023).
  21. G. Pospiech, An educational perspective on the connections between physics and mathematics, in Physics Teacher Education, Challenges in Physics Education, edited by J. Borg Marks and P. Galea (Springer Nature Switzerland and Imprint Springer, Cham, 2023), pp. 39–53.
  22. R. Karam, Framing the structural role of mathematics in physics lectures: A case study on electromagnetism, Phys. Rev. ST Phys. Educ. Res. 10, 010119 (2014).
  23. M. Pietrocola, Mathematics as a structural language of physical thought, in Connecting Research in Physics Education with Teacher Education, edited by M. Vicentini and E. Sassi (International Commission on Physics Education, 2008), ISBN [Amazon][WorldCat].
  24. A. R. P. de Ataíde and I. M. Greca, Epistemic views of the relationship between physics and mathematics: Its influence on the approach of undergraduate students to problem solving, Sci. Educ. 22, 1405 (2013).
  25. E. F. Redish and E. Kuo, Language of physics, language of math: Disciplinary culture and dynamic epistemology, Sci. Educ. 24, 561 (2015).
  26. M.-A. Geyer and W. Kuske-Janßen, Mathematical representations in physics lessons, in Mathematics in Physics Education, edited by G. Pospiech, M. Michelini, and B.-S. Eylon (Springer, Cham, 2019), pp. 75–102.
  27. J. K. Gilbert, Visualization: An emergent field of practice and enquiry in science education, in Visualization: Theory and Practice in Science Education, edited by J. K. Gilbert, M. Reiner, and M. Nakhleh (Springer Science, Dordrecht, 2008), pp. 3–24.
  28. S. Ainsworth, The functions of multiple representations, Comput. Educ. 33, 131 (1999).
  29. S. Ainsworth, The educational value of multiple-representations when learning complex scientific concepts, in Visualization: Theory and Practice in Science Education, edited by J. K. Gilbert, M. Reiner, and M. Nakhleh (Springer Science, Dordrecht, 2008), pp. 191–208.
  30. F. Hennig, K. Tóth, J. Veith, and P. Bitzenbauer, Mathematical sense making of quantum phenomena using dirac notation: Its effect on secondary school students’ functional thinking about photons, Eur. Phys. J. Quantum Technol. 11, 61 (2024).
  31. C. Singh, Student understanding of quantum mechanics at the beginning of graduate instruction, Am. J. Phys. 76, 277 (2008).
  32. G. Zhu and C. Singh, Surveying students’ understanding of quantum mechanics in one spatial dimension.
  33. C. Singh and E. Marshman, Review of student difficulties in upper-level quantum mechanics, Phys. Rev. ST Phys. Educ. Res. 11, 020117 (2015).
  34. S. DeVore and C. Singh, Interactive learning tutorial on quantum key distribution, Phys. Rev. Phys. Educ. Res. 16, 010126 (2020).
  35. P. Hu, Y. Li, and C. Singh, Challenges in addressing student difficulties with time-development of two-state quantum systems using a multiple-choice question sequence in virtual and in-person classes, Eur. J. Phys. 43, 025704 (2022).
  36. P. Hu, Y. Li, and C. Singh, Challenges in addressing student difficulties with basics and change of basis for two-state quantum systems using a multiple-choice question sequence in online and in-person classes, Eur. J. Phys. 44, 065703 (2023).
  37. P. Hu, Y. Li, and C. Singh, Challenges in addressing student difficulties with quantum measurement of two-state quantum systems using a multiple-choice question sequence in online and in-person classes, Phys. Rev. Phys. Educ. Res. 19, 020130 (2023).
  38. P. Hu, Y. Li, and C. Singh, Investigating and improving student understanding of the basics of quantum computing, Phys. Rev. Phys. Educ. Res. 20, 020108 (2024).
  39. K. T. Hahn and E. Gire, Waving arms around to teach quantum mechanics, Am. J. Phys. 90, 778 (2022).
  40. A. Merzel, E. Y. Weissman, N. Katz, and I. Galili, Mathematical structures in quantum physics education for high school students: Unveiling the power of dirac notation for conceptual and problem-solving proficiency, Phys. Rev. Phys. Educ. Res. 20, 020134 (2024).
  41. F. Hennig, K. Tóth, M. Förster, and P. Bitzenbauer, A new teaching-learning sequence to promote secondary school students’ learning of quantum physics using dirac notation, Phys. Educ. 59, 045007 (2024).
  42. M. Michelini and A. Stefanel, A path to build basic quantum mechanics ideas in the context of light polarization and learning outcomes of secondary students, J. Phys.: Conf. Ser. 1929, 012052 (2021).
  43. M. Michelini, A. Stefanel, and K. Tóth, Implementing dirac approach to quantum mechanics in a Hungarian secondary school, Educ. Sci. 12, 606 (2022).
  44. L. Qerimi, S. Malone, E. Rexigel, S. Mehlhase, J. Kuhn, and S. Küchemann, Exploring the mechanisms of qubit representations and introducing a new category system for visual representations: Results from expert ratings, Eur. Phys. J. Quantum Technol. 12, 45 (2025).
  45. M. Michelini, S. Faletic, and G. Pospiech, Work group 3 position paper: Teacher education and teaching/learning quantum physics, J. Phys.: Conf. Ser. 2297, 012015 (2022).
  46. A. Woitzik, K. Mecke, and G. Düchs, Das Lehramtsstudium Physik in Deutschland, Studien der Deutschen Physikalischen Gesellschaft e.V (Deutsche Physikalische Gesellschaft e.V and Institut der Deutschen Wirtschaft, Bad Honnef and Köln, 2023).
  47. A. J. Heine, Was Ist Theoretische Physik? Eine Wissenschaftstheoretische Betrachtung Und Rekonstruktion Von Vorstellungen Von Studierenden Und Dozenten über Das Wesen der Theoretischen Physik, Studien zum Physik- und Chemielernen (Logos Verlag, Berlin, 2018), Vol. 255.
  48. M. Schöne, Die verbesserung der lehramtsausbildung im bereich moderner physik am Beispiel der quantentheorie: Konzeption und evaluation eines fachdidaktischen seminars (2019).
  49. J. Breuer, Quantenmechanik in der Lehramtsausbildung–Vorstellungen und Ziele von Hochschullehrenden, Die Hochschullehre 159 (2020).
  50. K. Tóth, Integrating dirac approach to quantum mechanics into physics teacher education, in Proceedings of the TIM20-21 Physics Conference, AIP Conference Proceedings (AIP Publishing, Timisoara, Romania, 2023), p. 050011.
  51. P. A. M. Dirac, The Principles of Quantum Mechanics, [reprint the fourth edition St. John’s College, Cambridge 11 May 1957] ed. (Snowball Publishing, Illinois, 1957).
  52. G. J. Wiener, S. M. Schmeling, and M. Hopf, The technique of probing acceptance as a tool for teachers’ professional development: A PCK study, J. Res. Sci. Teach. 55, 849 (2018).
  53. P. Bitzenbauer and J.-P. Meyn, A new teaching concept on quantum physics in secondary schools, Phys. Educ. 55, 055031 (2020).
  54. M. Schneider, B. Rittle-Johnson, and J. R. Star, Relations among conceptual knowledge, procedural knowledge, and procedural flexibility in two samples differing in prior knowledge, Dev. Psychol. 47, 1525 (2011).
  55. S. Wörner, S. Becker, S. Küchemann, K. Scheiter, and J. Kuhn, Development and validation of the ray optics in converging lenses concept inventory, Phys. Rev. Phys. Educ. Res. 18, 020131 (2022).
  56. R. R. Skemp, Relational understanding and instrumental understanding, Math. Teach. Middle Sch. 12, 88 (2006).
  57. G. Pospiech, Physik verstehen—mathematisierung und aufgabenkultur, in Verhandlungen der Frühjahrstagung der DPG Berlin, edited by V. Nordmeier (Deutsche Physikalische Gesellschaft, 2008).
  58. M. Förster and G. Pospiech, Zur rolle mathematischer repräsentationen für das verständnis quantenphysikalischer prinzipien, in PhyDid B—Didaktik der Physik—Beiträge zur DPG-Frühjahrstagung Greifswald, edited by H. Grötzebauch and S. Heinicke (Deutsche Physikalische Gesellschaft, 2024).
  59. K. Tóth, S. Faletič, M. Michelini, G. Pospiech, A. Betti, M. Nicolini, M. Parmiggiani, J. Veith, and P. Bitzenbauer, Investigating the effect of two-state approaches on students’ understanding of quantum measurement: A quasiexperimental field study, Phys. Rev. Phys. Educ. Res. 21, 020142 (2025).
  60. P. Bitzenbauer, S. Faletič, M. Michelini, K. Tóth, and G. Pospiech, Design and evaluation of a questionnaire to assess learners’ understanding of quantum measurement in different two-state contexts: The context matters, Phys. Rev. Phys. Educ. Res. 20, 020136 (2024).
  61. M. Waitzmann, Wirkung eines quantenoptischen Realexperiments auf das physikalische Reasoning (Institutionelles Repositorium der Leibniz Universität Hannover, Hannover, 2023).
  62. W. Dür and S. Heusler, Was man vom einzelnen Qubit über Quantenphysik lernen kann, PhyDid A—Physik und Didaktik in Schule und Hochschule 1 (2012).
  63. W. Dür and S. Heusler, Was man von zwei Qubits über Quantenphysik lernen kann: Verschränkung und Quantenkorrelationen, PhyDid A—Physik und Didaktik in Schule und Hochschule 11 (2014).
  64. P. Hu, Y. Li, R. S. K. Mong, and C. Singh, Student understanding of the bloch sphere, Eur. J. Phys. 45, 025705 (2024).
  65. E. Rexigel, J. Bley, A. Arias, L. Qerimi, S. Küchemann, J. Kuhn, and A. Widera, Investigating the use of multiple representations in university courses on quantum technologies, Eur. Phys. J. Quantum Technol. 12, 22 (2025).
  66. E. Marshman and C. Singh, Improving student understanding of dirac notation by using analogical reasoning in the context of a three-dimensional vector space, in 2020 Physics Education Research Conference Proceedings (American Association of Physics Teachers, 2020), pp. 309–314.
  67. E. Marshman, A. Maries, and C. Singh, Using multiple representations to improve student understanding of quantum states, Phys. Rev. Phys. Educ. Res. 20, 020152 (2024).
  68. T. Wan, P. J. Emigh, and P. S. Shaffer, Investigating how students relate inner products and quantum probabilities, Phys. Rev. Phys. Educ. Res. 15, 010117 (2019).
  69. M. Wawro, K. Watson, and W. Christensen, Students’ metarepresentational competence with matrix notation and dirac notation in quantum mechanics, Phys. Rev. Phys. Educ. Res. 16, 020112 (2020).
  70. E. Gire and E. Price, Structural features of algebraic quantum notations, Phys. Rev. ST Phys. Educ. Res. 11, 020109 (2015).
  71. C. von Aufschnaiter, Laborstudien zur Untersuchung von Lernprozessen, in Methoden in der naturwissenschaftsdidaktischen Forschung, edited by D. Krüger, I. Parchmann, and H. Schecker (Springer Spektrum, Berlin, Heidelberg, 2014), pp. 81–105.
  72. W. Jung, Probing acceptance: A technique for investigating learning difficulties, in Research in Physics Learning, IPN, edited by R. Duit, F. Goldberg, and H. Niedderer (Keil, 1992), pp. 278–295.
  73. F. Hennig, K. Tóth, J. Veith, and P. Bitzenbauer, Introducing quantum physics concepts and dirac notation at the secondary school level: Insights into student reasoning from an acceptance survey, Phys. Rev. Phys. Educ. Res. 20, 020147 (2024).
  74. C. Haagen-Schützenhöfer, Lehr- und Lernprozesse im anfangsoptikunterricht der Sekundarstufe I: Habilitations-schrift (2016).
  75. S. Zloklikovits and M. Hopf, Elektromagnetische Strahlung in der Sek. 1 unterrichten, in Tagungsband der Jahrestagung der Gesellschaft für Didaktik der Chemie und Physik (Gesellschaft für Didaktik der Chemie und Physik, 2019).
  76. T. Dresing and T. Pehl, Praxisbuch Interview, Transkription & Analyse: Anleitungen und Regelsysteme für qualitativ Forschende, 8th ed. (Marburg, 2018).
  77. U. Kuckartz and S. Rädiker, Qualitative Inhaltsanalyse: Methoden, Praxis, Computerunterstützung: Grundlagentexte Methoden, 5th ed., Grundlagentexte Methoden (Beltz Juventa, Weinheim and Basel, 2022).
  78. A. Meyermann and M. Porzelt, Hinweise zur Anonymisierung Qualitativer daten.
  79. U. Kuckartz and S. Rädiker, Qualitative Content Analysis: Methods, Practice and Software, 2nd ed. (SAGE, Los Angeles, London, New Delhi, Singapore, Washington DC, and Melbourne, 2023).
  80. P. Mayring, Qualitative Inhaltsanalyse: Grundlagen und Techniken, 11th ed., Beltz Pädagogik (Beltz, Weinheim, 2010).
  81. J. R. Landis and G. G. Koch, The measurement of observer agreement for categorical data, Biometrics 33, 159 (1977).
  82. J. Breuer, Implementierung fachdidaktischer Innovationen durch das Angebot materialgestützter Unterrichtskonzeptionen. Fallanalysen zum Nutzungsverhalten von Lehrkräften am Beispiel des Münchener Lehrgangs zur Quantenmechanik, Studien zum Physik- und Chemielernen (2021), Vol. Band 314.
  83. L. Rozenblit and F. Keil, The misunderstood limits of folk science: An illusion of explanatory depth, Cogn. Sci. 26, 521 (2002).
  84. K. Tóth, M. Michelini, and P. Bitzenbauer, From light polarization to quantum physics: Supporting lower secondary school students’ transition from gestalt to functional thinking, Eurasia J. Math. Sci. Technol. Educ. 20, em2449 (2024).
  85. J. D. Gifford and N. D. Finkelstein, Categorical framework for mathematical sense making in physics, Phys. Rev. Phys. Educ. Res. 16, 020121 (2020).

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Phys. Rev. Phys. Educ. Res. 22, 020122– Published 28 August, 2026
  • Received 17 June 2025
  • Accepted 29 June 2026
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