Schrödinger: Journal of Physics Education
Schrödinger: Journal of Physics Education

Advancing Physics and Physics Education Through Research and Innovation

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Schrödinger: Journal of Physics Education

Advancing Physics and Physics Education Through Research and Innovation

A Century of the Schrödinger Equation Foundations, Structure and Applications

Ridha Horchani
Sultan Qaboos University ROR
Oman
horchani@squ.edu.om
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  • Purpose of the study: This study aims to provide a comprehensive pedagogical review of the Schrödinger equation by integrating its physical derivations, mathematical structure, and applications to support advanced undergraduate and beginning graduate students in understanding quantum mechanics coherently.

    Methodology: Literature review, pedagogical synthesis, canonical model analysis, Hilbert-space formalism, self-adjoint operator framework, spectral theory approach, quantum mechanics textbooks and journal sources, mathematical physics methods, conceptual analysis, and visualization of wave packets and quantum phenomena were used as tools and methods in this study.

    Main Findings: The study shows that multiple derivations of the Schrödinger equation converge to a unified structure based on linear, unitary evolution with a self-adjoint Hamiltonian. Key quantum phenomena such as superposition, tunnelling, and quantization emerge consistently from canonical models, while mathematical conditions ensure physical consistency and probability conservation.

    Novelty/Originality of this study: This study uniquely integrates physical derivations, rigorous mathematical structure, and pedagogical organization into a single coherent framework. It bridges conceptual gaps between theory and application, offering a unified reference that enhances understanding of quantum mechanics and supports both self-study and instructional practices.

  • How to cite

    [1]
    R. Horchani, “A Century of the Schrödinger Equation Foundations, Structure and Applications”, Sch. Jo. Phs. Ed, vol. 7, no. 2, pp. 56–86, Apr. 2026, doi: 10.37251/sjpe.v7i2.2393.

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    1. B. C. Hall, Quantum Theory for Mathematicians. New York, NY, USA: Springer, 2013.
    2. Y. Aharonov, S. Popescu, and J. Tollaksen, “The quantum mechanics of complex phases,” Phys. Today, vol. 74, no. 12, pp. 44–50, Dec. 2021.
    3. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. I: Functional Analysis, rev. ed. New York, NY, USA: Academic Press, 1981.
    4. M. Born, “Zur Quantenmechanik der Stoßvorgänge,” Z. Phys., vol. 37, no. 12, pp. 863–867, 1926. DOI: https://doi.org/10.1007/BF01397477
    5. L. E. Ballentine, Quantum Mechanics: A Modern Development, 2nd ed. Singapore: World Scientific, 2014. DOI: https://doi.org/10.1142/9038
    6. S. J. Gustafson and I. M. Sigal, Mathematical Concepts of Quantum Mechanics, 3rd ed. Cham, Switzerland: Springer, 2020. DOI: https://doi.org/10.1007/978-3-030-59562-3
    7. G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed. Providence, RI, USA: American Mathematical Society, 2014.
    8. S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, 2nd ed. Cambridge, U.K.: Cambridge University Press, 2017.
    9. S. Weinberg, Lectures on Quantum Mechanics, 2nd ed. Cambridge, U.K.: Cambridge University Press, 2015. DOI: https://doi.org/10.1017/CBO9781316276105
    10. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems. Oxford, U.K.: Oxford University Press, 2002.
    11. T. Maudlin, Philosophy of Physics: Quantum Theory. Princeton, NJ, USA: Princeton University Press, 2019.
    12. M. Carlesso and A. Bassi, “Collapse models: Main properties and the state of the art of the experimental tests,” in Advances in Open Systems and Fundamental Tests of Quantum Mechanics. Cham, Switzerland: Springer, 2022, pp. 1–13. DOI: https://doi.org/10.1007/978-3-030-31146-9_1
    13. J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory, vol. 5. New York, NY, USA: Springer, 2000.
    14. G. B. Folland, Quantum Field Theory: A Tourist Guide for Mathematicians. Providence, RI, USA: American Mathematical Society, 2008. DOI: https://doi.org/10.1090/surv/149
    15. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed. Cambridge, U.K.: Cambridge University Press, 2010.
    16. J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed. Cambridge, U.K.: Cambridge University Press, 2020. DOI: https://doi.org/10.1017/9781108587280
    17. D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, 3rd ed. Cambridge, U.K.: Cambridge University Press, 2018.
    18. J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 2nd ed. Cambridge, U.K.: Cambridge University Press, 2011.
    19. R. Shankar, Principles of Quantum Mechanics, 2nd ed. New York, NY, USA: Springer, 1994. DOI: https://doi.org/10.1007/978-1-4757-0576-8
    20. C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics. New York, NY, USA: John Wiley & Sons, 1977.
    21. M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II: Fourier Analysis, Self-Adjointness. New York, NY, USA: Academic Press, 1975.
    22. B. C. Hall, Quantum Theory for Mathematicians. New York, NY, USA: Springer, 2013. DOI: https://doi.org/10.1007/978-1-4614-7116-5
    23. G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed. Providence, RI, USA: American Mathematical Society, 2014.
    24. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems. Oxford, U.K.: Oxford University Press, 2002.
    25. W. H. Zurek, “Decoherence, einselection, and the quantum origins of the classical,” Rev. Mod. Phys., vol. 75, no. 3, pp. 715–775, 2003, doi: 10.1103/RevModPhys.75.715. DOI: https://doi.org/10.1103/RevModPhys.75.715
    26. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals. New York, NY, USA: McGraw-Hill, 1965.
    27. J. von Neumann, Mathematical Foundations of Quantum Mechanics. Princeton, NJ, USA: Princeton University Press, 1955.
    28. P. A. M. Dirac, The Principles of Quantum Mechanics. Oxford, U.K.: Oxford University Press, 1930.
    29. E. Schrödinger, “Quantisierung als Eigenwertproblem (Erste Mitteilung),” Ann. Phys. (Leipzig), vol. 79, pp. 361–376, 1926. DOI: https://doi.org/10.1002/andp.19263840404
    30. E. Schrödinger, “Quantisierung als Eigenwertproblem (Zweite Mitteilung),” Ann. Phys. (Leipzig), vol. 79, pp. 489–527, 1926. DOI: https://doi.org/10.1002/andp.19263840602
    31. W. Heisenberg, “Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen,” Z. Phys., vol. 33, pp. 879–893, 1925. DOI: https://doi.org/10.1007/BF01328377
    32. E. Madelung, “Quantentheorie in hydrodynamischer Form,” Z. Phys., vol. 40, pp. 322–326, 1927. DOI: https://doi.org/10.1007/BF01400372
    33. D. Bohm, “A suggested interpretation of the quantum theory in terms of ‘hidden’ variables. I,” Phys. Rev., vol. 85, no. 2, pp. 166–179, 1952, doi: 10.1103/PhysRev.85.166. DOI: https://doi.org/10.1103/PhysRev.85.166
    34. E. Schrödinger, “Über das Verhältnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der meinen,” Ann. Phys. (Leipzig), vol. 384, pp. 734–756, 1926. DOI: https://doi.org/10.1002/andp.19263840804
    35. R. Clifton, J. Bub, and H. Halvorson, “Characterizing quantum theory in terms of information-theoretic constraints,” Found. Phys., vol. 33, no. 11, pp. 1561–1591, 2003, doi: 10.1023/A:1026056716397. DOI: https://doi.org/10.1023/A:1026056716397
    36. G. Chiribella, G. M. D’Ariano, and P. Perinotti, “Informational derivation of quantum theory,” Phys. Rev. A, vol. 84, no. 1, Art. no. 012311, 2011, doi: 10.1103/PhysRevA.84.012311. DOI: https://doi.org/10.1103/PhysRevA.84.012311
    37. L. Hardy, “Quantum theory from five reasonable axioms,” arXiv:quant-ph/0101012, 2001.
    38. R. de la Madrid, “The role of the rigged Hilbert space in quantum mechanics,” Eur. J. Phys., vol. 26, no. 2, pp. 287–312, 2005, doi: 10.1088/0143-0807/26/2/007. DOI: https://doi.org/10.1088/0143-0807/26/2/008
    39. E. N. Economou, Green’s Functions in Quantum Physics, 3rd ed. Berlin, Germany: Springer, 2006. DOI: https://doi.org/10.1007/3-540-28841-4
    40. W. B. Case, “Wigner functions and Weyl transforms for pedestrians,” Am. J. Phys., vol. 76, no. 10, pp. 937–946, 2008, doi: 10.1119/1.2957889. DOI: https://doi.org/10.1119/1.2957889
    41. B.-G. Englert, Lectures on Quantum Mechanics, Vol. 2: Simple Systems. Singapore: World Scientific, 2006. DOI: https://doi.org/10.1142/6093-vol2
    42. O. Penrose and L. Onsager, “Bose-Einstein condensation and liquid helium,” Phys. Rev., vol. 104, no. 3, pp. 576–584, 1956, doi: 10.1103/PhysRev.104.576. DOI: https://doi.org/10.1103/PhysRev.104.576
    43. C. N. Yang, “Concept of off-diagonal long-range order and the quantum phases of liquid He and of superconductors,” Rev. Mod. Phys., vol. 34, no. 4, pp. 694–704, 1962, doi: 10.1103/RevModPhys.34.694. DOI: https://doi.org/10.1103/RevModPhys.34.694
    44. G. Lindblad, “On the generators of quantum dynamical semigroups,” Commun. Math. Phys., vol. 48, no. 2, pp. 119–130, 1976, doi: 10.1007/BF01608499. DOI: https://doi.org/10.1007/BF01608499
    45. V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, “Completely positive dynamical semigroups of N-level systems,” J. Math. Phys., vol. 17, no. 5, pp. 821–825, 1976, doi: 10.1063/1.522979. DOI: https://doi.org/10.1063/1.522979
    46. L. S. Schulman, Techniques and Applications of Path Integration. New York, NY, USA: Wiley, 1981. DOI: https://doi.org/10.1063/1.2914703
    47. F. Laloë, Do We Really Understand Quantum Mechanics? Cambridge, U.K.: Cambridge University Press, 2012. DOI: https://doi.org/10.1017/CBO9781139177160