Abstract
It is usually claimed that the Laguerre polynomials were popularized by Schrödinger when creating wave mechanics; however, we show that he did not immediately identify them in studying the hydrogen atom. In the case of relativistic Dirac equations for an electron in a Coulomb field, Dirac gave only approximations, Gordon and Darwin gave exact solutions, and Pidduck first explicitly and elegantly introduced the Laguerre polynomials, an approach neglected by most modern treatises and articles. That Laguerre polynomials were not very popular before their use in quantum mechanics, probably because they had been little used in classical mathematical physics, is confirmed by the fact that, as we show, they had been rediscovered independently several times during the nineteenth century, in published or unpublished studies of Abel, Murphy, Chebyshev, and Laguerre.
Similar content being viewed by others
References
Akhieser, N.I. 1998. Function theory according to Chebyshev. In Mathematics of the XIXth century, vol. 3, ed. A.N. Kolmogorov and A.P. Yushkevich, 1–81. Basel: Birkhäuser.
Anonymous. 1952. Dr. F.B. Pidduck. Nature No. 4317. July 26: 141.
Askey, R. 1982. Comments to (Szegö, 1968). In Collected papers of G. Szegö, vol. 3, ed. R. Askey, 866–869. Boston: Birkhäuser.
Auvil P.B., Brown L.M. (1978) The relativistic hydrogen atom: a simple solution. American Journal of Physics 46: 679–681
Bacry H., Boon M. (1985) Lien entre certains polynômes et quelques fonctions transcendantes. Comptes Rendus de l’Academie des Sciences Paris I 301: 273–276
Bernkopf, M. 1991. Laguerre, Edmond Nicolas. In Biographical dictionary of mathematics, vol. 3, ed. C.C. Gillipsie, 1315–1317. New York: Charles Scribner’s Sons.
Bethe, H.A., and E.E. Salpeter. 1957. Quantum mechanics of one- and two-electrons systems. In Handbuch der Physik, vol. 35, ed. S. Flügge, 88–436. Berlin: Springer.
Bloch, F. 1976. Heisenberg and the early days of quantum mechanics. Physics Today. December: 23–27.
Brezinski, C., A. Draux, A.P. Magnus, P. Maroni, and A. Ronveaux, eds. 1985. Polynômes orthogonaux et applications (Bar-le-Duc, 1984). Lecture Notes in Mathematics No. 1171. Berlin: Springer.
Buchholz H. (1969) The confluent hypergeometric function. Springer, Berlin
Chatterji, S.D. 1997–1998. Cours d’analyse, 3 vol. Lausanne: Presses polytechniques et universitaires romandes.
Chebychev, P.L. 1858. Sur les fractions continues. J. Math. Pures Appl. 3 (2): 289–323. Oeuvres. St. Petersburg: Acad. Imp. Sciences. Reprint New York: Chelsea, 1962, vol. 1: 201–230.
Chebyshev, P.L. 1859. Sur le développement des fonctions à une seule variable. Bull. Phys.-Math. Acad. Imp. St. Pétersbourg 1: 193–200. Oeuvres. St. Petersburg: Acad. Imp. Sciences. Reprint New York: Chelsea, vol. 1: 501–508.
Courant R., Hilbert D. (1924) Methoden der Mathematischen Physik, vol. 1. Springer, Berlin
Darwin C.G. (1927) The electron as a vector wave. Proceedings of the Royal Society of London Series A 116: 227–253
Darwin C.G. (1928) The wave equation of the electron. Proceedings of the Royal Society of London Series A 118: 654–680
Davis L. Jr. (1939) A note on the wave equations of the relativistic atom. Physical Review 56: 186–187
de Broglie L. (1923a) Ondes et quanta. Comptes Rendus de l’Academie des Sciences Paris 177: 507–510
de Broglie L. (1923b) Quanta de lumière, diffraction et interférences. Comptes Rendus de l’Academie des Sciences Paris 177: 548–550
de Broglie L. (1923c) Les quanta, la théorie cinétique des gaz et le principe de Fermat. Comptes Rendus de l’Academie des Sciences Paris 177: 630–632
de Broglie L. (1934) L’électron magnétique (Théorie de Dirac). Hermann, Paris
Dehesa, J.S., F. Dominguez Adame, E.R. Arriola, and A. Zarzo. 1991. Hydrogen atom and orthogonal polynomials. In Orthogonal polynomials and their applications (Erice, 1990). IMACS Annals Computing and Applied Mathematics, vol. 9, 223–229. Basel: Baltzer.
Dieudonné, J. 1985. Fractions continuées et polynomes orthogonaux dans l’oeuvre de E.N. Laguerre. In Polynômes orthogonaux et applications (Bar-le-Duc, 1984). Lecture Notes in Mathematics No. 1171, 1–15. Berlin: Springer.
Dirac P.A.M. (1928) The quantum theory of the electron I. Proceedings of the Royal Society of London A 117: 610–624
Dirac P.A.M. (1930) The principles of quantum mechanics. Clarendon Press, Oxford
Durán J., Grünbaum F.A. (2004) Orthogonal matrix polynomials satisfying second-order differential equations. International Mathematics Research Notices 10: 461–484
Durán J., Grünbaum F.A. (2006) P A M Dirac meets M G Krein: matrix orthogonal polynomials and Dirac’s equation. Journal of Physics A: Mathematical and General 39: 3655–3662
Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G. (1953) Higher transcendental functions, 3 vol. McGraw-Hill, New York
Favard, J. 1960–1963. Cours d’analyse de l’École Polytechnique, 3 vol. Paris: Gauthier-Villars.
Fejér, L. 1908. Sur une méthode de M. Darboux. Comptes Rendus de l’Académie des sciences Paris 147: 1040–1042. Leopold Fejér Gesammelte Arbeiten, vol. I, ed. P. Turán, 444–445. Basel: Birkhäuser.
Fejér, L. 1909. On the determination of asymptotic values (Hungarian). Mat. és Term. Értesitó 27: 1–33. German translation in Leopold Fejér Gesammelte Arbeiten, vol. I. Basel-Birkhäuser, 1970: 474–502.
Feynman R.P., Leighton R.B., Sands M. (1966) The Feynman lectures on physics. Quantum mechanics. Addison Wesley, Boston
Frank, Ph., and R. von Mises. 1925–1935. Die Differential- und Integralgleichungen der Mechanik und Physik. Braunschweig: Vieweg. Vol. 1, 1st ed. Vol. 1, 1925. Vol. 2 1928. 2nd Vol. 1, 1930. Vol. 2, 1935.
Gautschi, W. 1981. A survey of Gauss-Christoffel quadrature formula. In E.B. Christoffel. The influence of his work on mathematics and the physical sciences, ed. P.L. Butzer and F. Fehér, 72–147. Basel: Birkhäuser.
Gerber J. (1969) Geschichte der Wellenmechanik. Archive for History of Exact Sciences 5: 349–416
Gordon W. (1928) Die Energieniveaus der Wassenstoffatoms nach der Diracschen Quantentheorie des Elektrons. Z. f. Phys. 48: 11–14
Goursat, É. 1910–1915. Cours d’analyse mathématique, 3 vol. Paris: Gauthier-Villars.
Hadamard, J. 1927–1930. Cours d’analyse de l’École Polytechnique, 2 vol. Paris: Hermann.
Hendriksen, E., and H. van Rossum. 1985. Semi-classical orthogonal polynomials. In Orthogonal polynomials and applications (Bar-le-Duc 1984). Lecture Notes in Mathematics No. 1171, 354–361. Berlin: Springer.
Henrici, P. 1974–1986. Applied and computational complex analysis, 3 vol. New York: Wiley-Interscience.
Hermite, Ch., H. Poincaré, É. and Rouché, eds. 1898–1905. Oeuvres de Laguerre, 2 vol. Paris: Gauthier-Villars. Reprinted New York: Chelsea, 1972.
Hill E.L., Landshoff R. (1938) The Dirac electron theory. Reviews of Modern Physics 10: 87–132
Hille, E., J. Shohat, and J.L. Walsh. 1940. A bibliography of orthogonal polynomials. Bulletin of the National Research Council No. 103. Washington, DC.
Humbert P. (1922) Monographie des polynômes de Kummer. Ann. de Math. 1(5): 81–92
Ince, E.L.(1927). Ordinary differential equations. London: Longmans. Reprint New York: Dover, 1958.
Ivory J., Jacobi C.G.J. (1837) Sur le développement de \({(1-2xz+z^2)^{-1/2}}\) . J. Math. Pures Appl. 2: 105–106
Jammer M. (1966) The conceptual development of quantum mechanics. Mc-Graw-Hill, New York
Jordan, C. 1887. Cours d’analyse de l’École Polytechnique. 3 vol. Paris: Gauthier-Villars. 2e éd, 1893.
Kragh H.S. (1981) The genesis of Dirac’s relativistic theory of electrons. Archive for History of Exact Sciences 24: 31–67
Kragh H.S. (1982) Erwin Schrödinger and the wave equation: the crucial phase. Centaurus 26: 154–197
Kragh H.S. (1990) Dirac. A scientific biography. Cambridge University Press, Cambridge
Krein M.G. (1949) Infinite j-matrices and a matrix moment problem (Russian). Doklady Akademii Nauk SSSR 69: 125–128
Kubli F. (1970) Louis de Broglie und die Entdeckung der Materiewellen. Archive for History of Exact Sciences 7: 26–68
Kummer E.E. (1836) Ueber die hypergeometrische Reihe F(α,β,γ). Journal für reine Und angewandte Mathematik 15(39–83): 127–172
Lagrange, J.L. 1762. Solution de différents problèmes de calcul intégral. Miscellanea Taurinensia 3 (1762–1765). Oeuvres, vol. 1, 471–668. Paris: Gauthier-Villars.
Laguerre, E.N. (1879a). Sur l’intégrale \({\int_{x}^\infty\frac{e^{-x}\,dx}{x}}\) . Bull. Soc. Math. France 7: 428–437. Oeuvres, vol. I, 428–437.
Laguerre, E.N. 1879b. Sur la réduction en fractions continues d’une fonction qui satisfait à une équation linéaire du premier ordre à coefficients rationnels. Bull. Soc. Math. France 8: 21–27. Oeuvres, vol. I, 438–444.
Laguerre, E.N. 1884. Sur la réduction en fractions continues d’une fraction qui satisfait à une équation linéaire du premier ordre à coefficients rationnels. Comptes Rendus de l’Academie des Sciences Paris 98: 209–212. Oeuvres, vol. I, 445–448.
Laguerre, E.N. 1885. Sur la réduction des fractions continues d’une fraction qui satisfait à une équation différentielle linéaire du premier ordre dont les coefficients sont rationnels. J. Math. Pures Appl. 1 (4): 135–165. Oeuvres, vol. II, 685–711.
Landau L., Lifchitz E. (1972) Théorie quantique relativiste. Première partie. Mir, Moscou
Laurent, H. 1885–1991. Traité d’analyse, 7 vol. Paris: Gauthier-Villars.
Lavrentiev M., Shabat B. (1977) Méthodes de la théorie des fonctions d’une variable complexe. Mir, Moscou
Lebedev N.N. (1965). Special functions and their applications. Prentice-Hall. Reprint New York: Dover, 1972.
Mehra J., Rechenberg H. (1987) The historical development of quantum theory. Vol. 5: Erwin Schrödinger and the rise of wave mechanics, 2 vol. Springer, New York
Milne A. (1915) On the roots of the confluent hypergeometric equation. Proceedings of the Edinburgh Mathematical Society 33: 48–64
Moore W. (1989) Schrödinger: life and thought. Cambridge University Press, Cambridge
Murphy, R. 1833–1835. On the inverse method of definite integrals, with physical applications. Transactions of the Cambridge Philosophical Society 4: 355–408; 5: 13–148, 315–393.
Nikiforov A., Ouvarov V. (1976) Éléments de la théorie des fonctions spéciales. Mir, Moscou
Nikiforov A., Uvarov V. (1988) Special functions of mathematical physics. Birkhäuser, Basel
Pauli W. (1927) Zur Quantenmechanik des magnetischen Elektrons. Z. f. Physik 43: 601–623
Perron, O. 1929. Die Lehre von den Kettenbrücken, 2nd ed. Leipzig: Teubner. Reprint New York: Chelsea.
Picard, É. 1891–1897. Traité d’analyse, 3 vol. Paris: Gauthier-Villars.
Pidduck F.D. (1910) On the propagation of a disturbance in a fluid under gravity. Proceedings of the Royal Society of London A 83: 347–356
Pidduck F.D. (1929) Laguerre polynomials in quantum mechanics. Journal of the London Mathematical Society 4(1): 163–166
Pollaczek, F. 1956. Sur une généralisation des polynômes de Jacobi. Mémorial des sciences mathématiques No. 131. Paris: Gauthier-Villars.
Rainville E.O. (1960) Special functions. McMillan, New York
Rechenberg H. (1988) Erwin Schrödinger and the creation of wave mechanics. Acta Physica Polonica B19: 683–695
Ronveaux A., Mawhin J. (2005) Rediscovering the contributions of Rodrigues on the representation of special functions. Expositiones Mathematicae 23: 361–369
Rose M.E. (1961) Relativistic electron theory. Wiley, New York
Rouché É. (1886) Edmond Laguerre, sa vie et ses travaux. Journal de l’École Polytechnique 56: 213–271
Schiff L.I. (1968) Quantum mechanics, 3rd ed. McGraw-Hill, New York
Schlesinger, L. 1900. Einführung in die Theorie der Differentialgleichungen mit einer unabhängigen Variabeln. Sammlung Schuber No. XIII, G.J. Göschensche Verlag.
Schrödinger E. (1926a) Quantisierung als Eigenwertproblem. (Erste Mitteilung). Annalen der Physik 79(4): 361–376
Schrödinger E. (1926b) Quantisierung als Eigenwertproblem. (Zweite Mitteilung). Annalen der Physik 79(4): 489–527
Schrödinger E. (1926c) Quantisierung als Eigenwertproblem. (Dritte Mitteilung: Störungstheorie, mit Anwendung auf den Starkeeffekt der Balmerlinien). Annalen der Physik 80(4): 437–490
Schrödinger, E. 1927. Abhandlungen zur Wellenmechanik. Leipzig: Barth. English translation Collected papers on wave mechanics. London: Blackie & Son, 1928. French translation Mémoires sur la mécanique ondulatoire. Paris: Félix Alcan, 1933.
Slater L.J. (1960) Confluent hypergeometric functions. Cambridge University Press, Cambridge
Smirnov V. (1969–1984) Cours de mathématiques supérieures, 4 vol. Mir, Moscou
Sokhotskii, Yu.W. 1873. On definite integrals and functions used in series expansions (Russian). PhD thesis, St. Petersburg.
Sommerfeld A., Runge J. (1911) Anwendung der Vektorrechnung auf die Grundlagen der geometrischen Optik. Ann. f. Phys. 35(4): 277–298
Sonine N. (1880) Recherches sur les fonctions cylindriques et le développement des fonctions continues en séries. Math. Ann. 16: 1–80
Steffens K.G. (2006) The history of approximation theory. From Euler to Bernstein. Birkhäuser, Basel
Sylow, L., Lie, S. (eds) (1881) Oeuvres complètes de Niels Hendrik Abel, 2 vol. Christiana, Grondhal & Son
Szegö G. (1959) Orthogonal polynomials, 2nd ed. Providence, American Mathematical Society
Szegö G. 1968. An outline of the history of orthogonal polynomials. In Proceedings of the conference on orthogonal expansions and their continuous analogues, 3–11. Carbondale: Southern Illinois University Press. Collected Papers. Vol. 3, 1982: 857–865.
Turán, P. 1970. Bemerkungen. In Leopold Fejér Gesammelte Arbeiten, vol. I, ed. P. Turán, 502–503. Basel: Birkhäuser.
Valiron, G. 1942–1945. Cours d’analyse mathématique, 2 vol. Paris: Masson.
Weber, H. 1900–1901. Die partiellen Differential-Gleichungen der Mathematischen Physik, 2 vol. Braunschweig: Vieweg.
Wessels L. (1979) Schrödinger’s route to wave mechanics. Studies in History and Philosophy of Science 10: 311–340
Weyl, H. 1928. Gruppentheorie und Quantenmechanik. Leipzig: Hirzel. English transl. London: Methuen, 1931. Reprint New York: Dover.
Whittaker E., Watson N. (1902) Modern analysis. Cambridge University Press, Cambridge
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Jeremy Gray.
Rights and permissions
About this article
Cite this article
Mawhin, J., Ronveaux, A. Schrödinger and Dirac equations for the hydrogen atom, and Laguerre polynomials. Arch. Hist. Exact Sci. 64, 429–460 (2010). https://doi.org/10.1007/s00407-010-0060-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00407-010-0060-3