RSS feeds
РусскийEnglish
Issue 1, 2026
Volume year page
Issues Authors PACS Subscription For authors

Issues

 / 

2026

 / 

January

←  →

From the history of physics. The 100th anniversary of quantum mechanics


Old quantum mechanics by Bohr and Sommerfeld from modern perspective

†  a, ‡  b, §  c
a Department of Mathematics, College of Arts and Sciences, Howard University, 2141 6th St NW, Washington, DC, 20059, USA
b Landeshauptstadt München, Referat für Bildung und Sport, Seminar TMG Universität, Drygalski Allee 2, 81477, München, 11794-3600, Germany
c School of Mathematical and Statistical Sciences, Arizona State University, P.O. Box 871804, Tempe, AZ, 85287-1804, USA

We review Bohr's atomic model and its extension by Sommerfeld from a mathematical perspective of wave mechanics. The derivation of quantization rules and energy levels is revisited using semiclassical methods. Sommerfeld-type integrals are evaluated by elementary techniques, and connections with the Schrödinger and Dirac equations are established. Historical developments and key transitions from classical to quantum theory are discussed to clarify the structure and significance of the old quantum mechanics.

Typically, an English full text is available in about 1 month from the date of publication of the original article.

Keywords: Bohr's atomic model, Bohr—Sommerfeld quantization rule, Schrödinger equation, Dirac equation, WKB method, fine structure formula, Sommerfeld puzzle, computer algebra system: Mathematica, transuranic elements
PACS: 01.65.+g, 02.70.−c, 03.65.−w (all)
DOI: 10.3367/UFNe.2025.08.040014
URL: https://ufn.ru/en/articles/2026/1/e/
Citation: Barley K, Ruffing A, Suslov S K "Old quantum mechanics by Bohr and Sommerfeld from modern perspective" Phys. Usp. 69 (1) (2026)

Received: 3rd, July 2025, 15th, August 2025

Оригинал: Барли К, Руффинг А, Суслов С К «Старая квантовая механика Бора и Зоммерфельда с современной точки зрения» УФН 196 83–103 (2026); DOI: 10.3367/UFNr.2025.08.040014

References (116) ↓ Cited by (1) Similar articles (20)

  1. Schrödinger E "Quantisierung als Eigenwertproblem" 79 361 (1926), Vol. 384 po novoi numeratsii tomov na saite Wiley Online Library; Per. na angl. yaz., Schrödinger E "Quantisation as a problem of proper values (Part I)" Collected Papers On Wave Mechanics (AMS Chelsea Publishing) Vol. 302 (Providence, RI: Chelsea Publ. Co., 1926) p. 1-12; https://bookstore.ams.org/chel-302
  2. Planck M, The Nobel Prize in Physics 1918 was awarded to Max Karl Ernst Ludwig Planck "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta", https://www.nobelprize.org/prizes/physics/1918/summary/
  3. Einstein A, The Nobel Prize in Physics 1921 was awarded to Albert Einstein "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect", https://www.nobelprize.org/prizes/physics/1921/summary/
  4. Rutherford E, The Nobel Prize in Chemistry 1908 was awarded to Ernest Rutherford "for his investigations into the disintegration of the elements, and the chemistry of radioactive substances", https://www.nobelprize.org/prizes/chemistry/1908/summary/
  5. Bohr N, The Nobel Prize in Physics 1922 was awarded to Niels Henrik David Bohr "for his services in the investigation of the structure of atoms and of the radiation emanating from them", https://www.nobelprize.org/prizes/physics/1922/summary/
  6. Aaserud F, Heilbron J L Love, Literature, And The Quantum Atom: Niels Bohr’s 1913 Trilogy Revisited (Oxford: Oxford Univ. Press, 2013); https://academic.oup.com/book/6847
  7. Bohr N Niels Bohr Collected Works (Ed. L Rosenfeld) Vol. 2 Work On Atomic Physics (1912-1917, Ed. U Hoyer) (Amsterdam: North-Holland Publ. Co., 1981)
  8. Eckert M Arnold Sommerfeld: Science, Life And Turbulent Times. 1868-1951 (New York: Springer, 2013)
  9. El’yashevich M A "Razvitie Nil’som Borom kvantovoi teorii atoma i printsipa sootvetstviya (Raboty N. Bora 1912-1923 gg. po atomnoi fizike i ikh znachenie)" Usp. Fiz. Nauk 147 253-301 (1985); El’yashevich M A "Niels Bohr’s development of the quantum theory of the atom and the correspondence principle (his 1912-1923 work in atomic physics and its significance)" Sov. Phys. Usp. 28 879-909 (1985)
  10. Firme C L Quantum Mechanics: Detailed Historical, Mathematical And Computational Approaches (Boca Raton, FL: CRC Press, Taylor and Francis Group, 2022)
  11. Kragh H Niels Bohr And The Quantum Atom: The Bohr Model Of Atomic Structure 1913-1925 (Oxford: Oxford Univ. Press, 2012)
  12. Kragh H (Ed.) Niels Bohr: On The Constitution Of Atoms And Molecules (Classic Texts in the Sciences) (Cham: Birkhäuser, Springer Nature Switzerland AG, 2022)
  13. Mehra J, Rechenberg H The Quantum Theory Of Planck, Einstein, Bohr And Sommerfeld: Its Foundation And The Rise Of Its Difficulties 1900-1925 (The Historical Development Of Quantum Theory, Vol. 1, Pt. 2) 1st ed. (New York: Springer-Verlag, 1982); https://link.springer.com/book/9780387951751
  14. Migdal A B "Nil’s Bor i kvantovaya fizika" Usp. Fiz. Nauk 147 303-342 (1985); Migdal A B "Niels Bohr and quantum physics" Sov. Phys. Usp. 28 910-934 (1985)
  15. Milant’ev V P "Sozdanie teorii Bora i ee razvitie (k 90-letiyu borovskoi teorii atoma)" Usp. Fiz. Nauk 174 209-215 (2004); Milant’ev V P "Creation and development of Bohr’s theory (on the 90th anniversary of the Bohr theory of the atom)" Phys. Usp. 47 197 (2004)
  16. Reed B C The Bohr Atom: A Guide (Bristol: IOP Publ., 2020)
  17. Steinhauser M O Quantenmechanik Für Naturwissenschaftler: Ein Lehr- Und Übungsbuch Mit Zahlreichen Aufgaben Und Lösungen (Berlin: Springer, 2022)
  18. Sommerfeld A "Zur Quantentheorie der Spektrallinien" Ann. Physik 356 1-94 (1916)
  19. Sommerfeld A "Zur Feinstruktur der Wasserstofflinien. Geschichte und gegenwärtiger Stand der Theorie" Naturwissenschaften 28 417-423 (1940)
  20. Wilson W "LXXXIII. The quantum-theory of radiation and line spectra" London Edinburgh Dublin 29 795-802 (1915)
  21. Videoroliki razlichnykh prepodavatelei ob atome Bora: "Bohr Model in Brief: The planetary model, its connection to emission spectra and quantized electrons", https://www.youtube.com/watch?v=xytRf3fs_gY; "What is the Bohr model of the atom?", https://www.youtube.com/watch?v=_Gt7mo8SNkA; "How Niels Bohr created the quantum atom", https://www.youtube.com/watch?v=xINR4MoqYVc; "This star almost broke Bohr’s atomic model", https://www.youtube.com/watch?v=BcX1aYrLct4; "Sommerfeld and the fine structure of the atom", https://www.youtube.com/watch?v=H4ZnVpiQTDw; obrazovatel’nye videoroliki, posvyashchennye atomnoi modeli Bora i svyazannym s nei kontseptsiyam: "Why don’t electrons fall onto the nucleus?", https://www.youtube.com/watch?v=cf7t-tZnNuE
  22. Michelson A A, The Nobel Prize in Physics 1907 was awarded to Albert Abraham Michelson "for his optical precision instruments and the spectroscopic and metrological investigations carried out with their aid", https://www.nobelprize.org/prizes/physics/1907/summary/
  23. Albert A Michelson, Wikipedia, online accessed 29 May 2025, https://en.wikipedia.org/w/index.php?title=Albert_A._Michelson&oldid=1293056736
  24. Biedenharn L C "The "Sommerfeld Puzzle" revisited and resolved" Found. Phys. 13 13-34 (1983)
  25. Kragh H "The fine structure of hydrogen and the gross structure of the physics community, 1916-26" Hist. Stud. Phys. Sci. 15 67-125 (1985)
  26. Thomson J J, The Nobel Prize in Physics 1906 was awarded to Joseph John Thomson "in recognition of the great merits of his theoretical and experimental investigations on the conduction of electricity by gases", https://www.nobelprize.org/prizes/physics/1906/summary/
  27. Granovskii Ya I "Formula Zommerfel’da i teoriya Diraka" Usp. Fiz. Nauk 174 577 (2004); Granovskii Ya I "Sommerfeld formula and Dirac’s theory" Phys. Usp. 47 523 (2004)
  28. Sommerfeld A Atomic Structure And Spectral Lines (New York: Dutton, 1923), translated from the 3rd German ed.; https://archive.org/details/atomicstructures0000somm/page/n7/mode/2up; in German, Sommerfeld A Atombau Und Spektrallinien Bd. 1, 2. Aufl. (Braunschweig: F. Vieweg and Sons, 1921); archive.org/details/atombauundspekt00sommgoog/page/478/mode/2up
  29. Darwin C G "The wave equations of the electron" Proc. R. Soc. London A 118 654-680 (1928)
  30. Gordon W "Die Energieniveaus des Wasserstoffatoms nach der Diracschen Quantentheorie des Elektrons" Z. Phys. 48 11-14 (1928)
  31. Dirac P A M "The quantum theory of the electron" Proc. R. Soc. London A 117 610 (1928)
  32. Dirac P A M "The evolution of the physicist’s picture of nature" Sci. Am. 208 (5) 45 (1963)
  33. Akhiezer A I, Berestetskii V B Quantum Electrodynamics (Interscience Monographs and Texts in Physics and Astronomy) Vol. 11 (New York: Interscience Publ., 1965); Per. s russk. yaz., Akhiezer A I, Berestetskii V B Kvantovaya Elektrodinamika (M.: Fizmatgiz, 1959)
  34. Berestetskii V B, Lifshitz E M, Pitaevskii L P Relativistic Quantum Theory (Oxford: Pergamon Press, 1971); Per. s russk. yaz., Berestetskii V B, Lifshits E M, Pitaevskii L P Relyativistskaya Kvantovaya Teoriya (M.: Fizmatgiz, 1968)
  35. Davydov A S Quantum Mechanics (Oxford: Pergamon Press, 1965); Per. s russk. yaz., Davydov A S Kvantovaya Mekhanika (M.: Fizmatgiz, 1963)
  36. Landau L D, Lifshitz E M Quantum Mechanics: Non-Relativistic Theory 3rd ed. (Oxford: Pergamon Press, 1977); Per. s russk. yaz., Landau L D, Lifshits E M Kvantovaya Mekhanika. Nerelyativistskaya Teoriya 3-e izd., pererab. i dop. (M.: Nauka, 1977)
  37. Rose M E Relativistic Electron Theory (New York: John Wiley and Sons, 1961); https://ia801405.us.archive.org/22/items/in.ernet.dli.2015.148057/2015.148057.Relativistic-Electron-Thoery.pdf
  38. Schiff L I Quantum Mechanics (International Ser. in Pure and Applied Physics) 3rd ed. (New York: McGraw-Hill Book Co., 1968); https://physicsgg.files.wordpress.com/2021/01/schiff-l.i.-quantum-mechanics-mgh-1949t417s.pdf
  39. Barli K, Vega-Guzman Kh, Ruffing A, Suslov S K "Otkrytie relyativistskogo uravneniya Shredingera" Usp. Fiz. Nauk 192 100 (2022); Barley K, Vega-Guzmán J, Ruffing A, Suslov S K "Discovery of the relativistic Schrödinger equation" Phys. Usp. 65 90 (2022)
  40. Ellis L, Ellis I, Koutschan C, Suslov S K "On potentials integrated by the Nikiforov—Uvarov method" Applications And Q -Extensions Of Hypergeometric Functions (Contemporary Mathematics) Vol. 819 (Eds H S Cohl, R S Costas-Santos, R S Maier) (Providence, RI: American Mathematical Society, 2025) p. 43; https://bookstore.ams.org/view?ProductCode=CONM/819
  41. Goren B, Barley K, Suslov S K Proc. Of The Second Intern. Workshop On Quantum Nonstationary Systems (Eds A Dodonov, C C H Ribeiro) (São Paulo: LF Editorial, 2024) p. 333-384, Ch. 19; http://www.cif.unb.br/eventos/proceedings-qns2
  42. Kryuchkov S I, Lanfear N A, Suslov S K "The role of the Pauli—Lubański vector for the Dirac, Weyl, Proca, Maxwell and Fierz—Pauli equations" Phys. Scr. 91 035301 (2016); Kryuchkov S I, Lanfear N A, Suslov S K arXiv:1510.05164
  43. Kryuchkov S I, Suslov S K, Vega-Guzmán J M "The minimum-uncertainty squeezed states for atoms and photons in a cavity" J. Phys. B 46 104007 (2013)
  44. Suslov S K "The ’Somerfield’s Puzzle’ and its extensions" Proc. of the Second Intern. Workshop on Quantum Nonstationary Systems (Eds A Dodonov, C C H Ribeiro) (São Paulo: LF Editorial, 2024) p. 43-59; https://lfeditorial.com.br/produto/proceedings-of-the-second-international-workshop-on-quantum-nonstationary-systems/
  45. Suslov S K, Trey B "The Hahn polynomials in the nonrelativistic and relativistic Coulomb problems" J. Math. Phys. 49 012104 (2008)
  46. Suslov S K, Vega-Guzmán J M, Barley K "An introduction to special functions with some applications to quantum mechanics" Orthogonal Polynomials. 2nd AIMS - Volkswagen Stiftung Workshop, Douala, Cameroon, 5-12 October, 2018. Conf. Proc. (Tutorials, Schools, and Workshops in the Mathematical Sciences, Eds M Foupouagnigni, W Koepf) (Cham: Birkhäuser, 2020) p. 517-628
  47. Ruffing A "Mathematische Methoden der Quantenmechanik", Lektsii, prochitannye v Myunkhenskom un-te (Technische Univ. München) - 2001-2014
  48. Mehra J, Rechenberg H The Historical Development Of Quantum Theory Vol. 1, Pt. 1 The Quantum Theory Of Planck, Einstein, Bohr And Sommerfeld: Its Foundation And The Rise Of Its Difficulties 1900-1925 (New York: Springer-Verlag, 1982); https://link.springer.com/book/9780387951744
  49. Einstein A "Autobiographical note" Albert Einstein: Philosopher-Scientist (Library of Living Philosophers, Vol. 7) Vol. 1 (Ed. P A Schilpp) (La Salle, IL: Open Court Publ. Co, 1970) p. 46-47
  50. Gantmacher F Lectures In Analytical Mechanics (Moscow: Mir Publ., 1975); Per. s russk. yaz., Gantmakher F R Lektsii Po Analiticheskoi Mekhanike (M.: Nauka, 1966)
  51. Goldstein H, Poole C, Safko J Classical Mechanics (San Francisco, CA: Addison Wesley, 2002)
  52. Nikiforov A F, Uvarov V B Special Functions Of Mathematical Physics. A Unified Introduction With Applications (Basel: Birkhäuser, 1988)
  53. Varshalovich D A, Moskalev A N, Khersonskii V K Quantum Theory Of Angular Momentum (Singapore: World Scientific, 1988)
  54. Merzbacher E Quantum Mechanics 3rd ed. (New York: John Willey and Sons, 1998)
  55. Berry M V, Mount K E "Semiclassical approximations in wave mechanics" Rep. Prog. Phys. 35 315 (1972)
  56. Ghatak A K, Gallawa R L, Goyal I C Modified Airy Function And WKB Solutions To The Wave Equation (NIST Monograph) Vol. 176 (Boulder, CO: National Institute of Standards and Technology, 1991)
  57. Barley K K, Ruffing A, Suslov S K "Old quantum mechanics by Bohr and Sommerfeld from modern perspective", Supplementary files
  58. Landau L D, Lifshitz E M The Classical Theory Of Fields 4th ed. (Oxford: Pergamon Press, 1975); Per. s russk. yaz., Landau L D, Lifshits E M Teoriya Polya (M.: Nauka, 1973)
  59. Sommerfeld A Wave-Mechanics 2nd ed. (New York: E.P. Dutton and Co. Inc., 1938), Suppl. Vol. to "Atomic Structure and Spectral Lines"
  60. Schrödinger E, The Nobel Prize in Physics 1933 was awarded jointly to Erwin Schrödinger and Paul Adrien Maurice Dirac "for the discovery of new productive forms of atomic theory", https://www.nobelprize.org/prizes/physics/1933/summary/
  61. Dirac P A M, The Nobel Prize in Physics 1933 was awarded jointly to Erwin Schrödinger and Paul Adrien Maurice Dirac "for the discovery of new productive forms of atomic theory", https://www.nobelprize.org/prizes/physics/1933/dirac/facts/
  62. Schrödinger E "The continuous transition from micro- to macromechanics" Collected Papers On Wave Mechanics (Providence, RI: Chelsea Publ. Co.: AMS Chelsea Publ., 2010) p. 41-44; Translated from German, Schrödinger E Naturwissenschaften 28 664-666 (1926)
  63. Schrödinger E "An undulatory theory of the mechanics of atoms and molecules" Phys. Rev. 28 1049-1070 (1926)
  64. von Meyenn K (Ed.) Eine Entdeckung Von Ganz Außerordentlicher Tragweite: Schrödingers Briefwechsel Zur Wellenmechanik Und Zum Katzenparadoxon (Berlin: Springer, 2011)
  65. de Broglie L, The Nobel Prize in Physics 1929 was awarded to Prince Louis-Victor Pierre Raymond de Broglie "for his discovery of the wave nature of electrons", https://www.nobelprize.org/prizes/physics/1929/summary/
  66. Mehra J, Rechenberg H Erwin Schrödinger And The Rise Of Wave Mechanics Pt. 2 The Creation Of Wave Mechanics; Early Response And Applications 1925-1926 (The Historical Development of Quantum Theory) Vol. 5/2 1st ed. (New York: Springer-Verlag, 1987); https://link.springer.com/book/9780387951805
  67. Moore W Schrödinger: Life And Thought (Ser. Canto Classics) (Cambridge: Cambridge Univ. Press, 2015)
  68. Brillouin L "La mécanique ondulatoire de Schrödinger une méthode générale de résolution par approximations successives" C. R. Acad. Sci. 183 24-26 (1926); Brillouin L https://gallica.bnf.fr/ark:/12148/bpt6k3136h/f24
  69. Kramers H A "Wellenmechanik und halbzahlige Quantisierung" Z. Phys. 39 828-840 (1926)
  70. Wentzel G "Eine Verallgemeinerung der Quantenbedingungen für die Zwecke der Wellenmechanik" Z. Phys. 38 518-529 (1926)
  71. Olver F, Daalhuis A, Lozier D, Schneider B, Boisvert R, Clark C, Mille B, Cohl H, McClain M, NIST Digital Library of Mathematical Functions (2025), release 1.2.4 of 2025-03-15, https://dlmf.nist.gov/
  72. Nikiforov A F, Novikov V G, Uvarov V B Quantum-Statistical Models Of Hot Dense Matter. Methods For Computation Opacity And Equation Of State (Progress in Mathematical Physics) Vol. 37 (Basel: Birkhäuser, 2005)
  73. Langer R E "On the asymptotic solutions of ordinary differential equations, with an application to the Bessel functions of large order" Trans. Am. Math. Soc. 33 23-64 (1931)
  74. Langer R E "On the connection formulas and the solutions of the wave equation" Phys. Rev. 51 669-676 (1937)
  75. Kragh H "Erwin Schrödinger and the wave equation: The crucial phase" Centaurus 26 154-197 (1982)
  76. Kragh H "Equation with the many fathers. The Klein—Gordon equation in 1926" Am. J. Phys. 52 1024-1033 (1984)
  77. Joas C, Lehner C "The classical roots of wave mechanics: Schrödinger’s transformations of the optical-mechanical analogy" Stud. Hist. Philos. Sci. B 40 338-351 (2009)
  78. Bethe H A, Salpeter E Quantum Mechanics Of One- And Two-Electron Atoms (Mineola, NY: Dover Publ., 2008)
  79. Good R H (Jr.) "The generalization of the WKB method to radial wave equations" Phys. Rev. 90 131-137 (1953)
  80. Lu P "Derivation of Sommerfeld—Dirac fine-structure formula by WKB method" Phys. Rev. A 1 1283-1285 (1970)
  81. Mehra J (Ed.) The Discovery Of Electron Spin Vol. 1 (Singapore: World Scientific, 2001) p. 585-611, Ch. 17
  82. Stern O, The Nobel Prize in Physics 1943 was awarded to Otto Stern "for his contribution to the development of the molecular ray method and his discovery of the magnetic moment of the proton", https://www.nobelprize.org/prizes/physics/1943/summary/
  83. Petrov S V "Oshibalsya li Zommerfel’d? (K istorii poyavleniya spina v relyativistskikh volnovykh uravneniyakh)" Usp. Fiz. Nauk 190 777 (2020); Petrov S V "Was Sommerfeld wrong? (To the history of the appearance of spin in relativistic wave equations)" Phys. Usp. 63 721 (2020)
  84. Heisenberg W, The Nobel Prize in Physics 1932 was awarded to Werner Karl Heisenberg "for the creation of quantum mechanics, the application of which has, inter alia, led to the discovery of the allotropic forms of hydrogen", https://www.nobelprize.org/prizes/physics/1932/summary/
  85. Heisenberg W "Ausstrahlung von Sommerfelds Werk in der Gegenwart" Phys. Blätter 24 530-537 (1968)
  86. Yourgrau W, Mandelstam S Variational Principles In Dynamics And Quantum Theory 3rd ed. (Philadelphia, PA: W.B. Saunders Co., 1968)
  87. Schrödinger E "Preface" Collected Papers On Wave Mechanics 3rd ed. (Montréal: Minkowski Institute Press, 2021) p. ix-xiii
  88. Schrödinger E "Quantisation as a problem of proper values" Collected Papers On Wave Mechanics (AMS Chelsea Publishing Ser.) Vol. 302 (Providence, RI: Chelsea Publ. Co., 2010) p. 13-40, Pt. II; https://bookstore.ams.org/chel-302
  89. Einstein A Letters On Wave Mechanics: Correspondence With H.A. Lorentz, Max Planck, And Erwin Schrödinger (New York: Philosophical Library, Open Road, 2011)
  90. Fock V "Zur Schrödingerschen Wellenmechanik" Z. Phys. 38 242-250 (1926)
  91. Wessels L "Schrödinger’s route to wave mechanics" Stud. Hist. Phil. Sci. A 10 311-340 (1979)
  92. Bloch F "Heisenberg and the early days of quantum mechanics" Phys. Today 29 (12) 23-27 (1976)
  93. Rozental S (Ed.) Niels Bohr: His Life And Work As Seen By His Friends And Colleagues (Amsterdam: North-Holland Publ. Co., 1968) p. 73
  94. Koutschan C, Suazo E, Suslov S K "Fundamental laser modes in paraxial optics: from computer algebra and simulations to experimental observation" Appl. Phys. B 121 315-336 (2015)
  95. Jackson J D Classical Electrodynamics 2nd ed. (New York: John Wiley and Sons, 1975)
  96. Ewald P P "Ein Buch über mathematische Physik: Courant—Hilbert" Naturwissenschaften 13 384-387 (1925)
  97. Courant R, Hilbert D Methoden Der Mathematischen Physik (Grundlehren der mathematischen Wissenschaften) Vol. 12 (Berlin: Springer, 1924)
  98. Oganessian Y "Discovery of the island of stability for super heavy elements" IPAC2017: Proc. of the 8th Intern. Particle Accelerator Conf., Copenhagen, Denmark, 14-19 May, 2017 (Eds G Arduini, M Lindroos, J Pranke, V R W Schaa, M Seidel) (Copenhagen, Denmark: Eur. Spallation Source (ESS), MAXIV and Aarhus Univ., 2017) p. 4848-4851; https://proceedings.jacow.org/ipac2017/
  99. Barley K, Ruffing A, Suslov S K "Oganesson versus uranium hydrogen-like ions from the viewpoint of old quantum mechanics" arXiv:2509.06249
  100. Gumberidze A, Stöhlker Th, Banaś D, Beckert K, Beller P, Beyer H F, Bosch F, Hagmann S, Kozhuharov C, Liesen D, Nolden F, Ma X, Mokler P H, Steck M, Sierpowski D, Tashenov S "Quantum electrodynamics in strong electric fields: The ground-state Lamb shift in hydrogenlike uranium" Phys. Rev. Lett. 94 223001 (2005)
  101. Gumberidze A, Stöt;hlker Th, Banaś D, Beckert K, Beller P, Beyer H F, Bosch F, Cai X, Hagmann S, Kozhuharov C, Liesen D, Nolden F, Ma X, Mokler P H, Steck M, Sierpowski D, Tashenov S, Warczak A, Zou Y "Precision tests of QED in strong fields: experiments on hydrogen- and helium-like uranium" J. Phys. Conf. Ser. 58 87-92 (2007)
  102. Flerov G N, Karamian S A "The search for superheavy elements in nature: foundations and prospects" Atti Del Convegno Mendeleeviano, Periodicità E Simmetrie Nella Struttura Elementare Della Materia. Torino-Roma, 15-21 Settembre, 1969/Convegno Mendeleeviano. Torino-Roma, 1969 (Ed. M Verde) (Turin, Italy: Vincenzo Bona, 1971) p. 73-96
  103. Mohr P J, Plunien G, Soff G "QED corrections in heavy atoms" Phys. Rep. 293 227-369 (1998)
  104. Shabaev V M "Two-time Green’s function method in quantum electrodynamics of high-Z few-electron atoms" Phys. Rep. 356 119-228 (2002); Shabaev V M physics/0009018
  105. Shabaev V M "Kvantovaya elektrodinamika tyazhelykh ionov i atomov: status i perspektivy" Usp. Fiz. Nauk 178 1220 (2008); Shabaev V M "Quantum electrodynamics of heavy ions and atoms: current status and prospects" Phys. Usp. 51 1175 (2008)
  106. Suslov S K "Expectation values in relativistic Coulomb problems" J. Phys. B 42 185003 (2009)
  107. Chapman K "The oganesson odyssey" Nature Chem. 10 796 (2018)
  108. Giuliani S A, Matheson Z, Nazarewicz W, Olsen E, Reinhard P-G, Sadhukhan J, Schuetrumpf B, Schunck N, Schwerdtfeger P "Colloquium: Superheavy elements: Oganesson and beyond" Rev. Mod. Phys. 91 011001 (2019)
  109. Kaygorodov M Y, Skripnikov L V, Tupitsyn I I, Eliav E, Kozhedub Y S, Malyshev A V, Oleynichenko A V, Shabaev V M, Titov A V, Zaitsevskii A V "Electron affinity of oganesson" Phys. Rev. A 104 012819 (2021)
  110. Kragh H From Transuranic To Superheavy Elements: A Story Of Dispute And Creation (SpringerBriefs in History of Science and Technology) (Cham: Springer, 2018)
  111. Oganessian Yu Ts, Utyonkov V K, Lobanov Yu V, Abdullin F Sh, Polyakov A N, Sagaidak R N, Shirokovsky I V, Tsyganov Yu S, Voinov A A, Moody K J, Patin J B, Shaughnessy D A, Stoyer M A, Stoyer N J, Wilk P A, Kenneally J M, Landrum J H, Wild J F, Gulbekian G G, Bogomolov S L, Gikal B N, Mezentsev A N, Iliev S, Subbotin V G, Sukhov A M, Subotic K, Zagrebaev V I, Vostokin G K, Itkis M G, Lougheed R W "Synthesis of the isotopes of elements 118 and 116 in the 249Cf and 245Cm+48Ca fusion reactions" Phys. Rev. C 74 44602 (2006)
  112. Oganessian Yu Ts, Abdullin F Sh, Alexander C, Binder J, Boll R A, Dmitriev S N, Ezold J, Felker K, Gostic J M, Grzywacz R K, Hamilton J H, Henderson R A, Itkis M G, Miernik K, Miller D, Moody K J, Polyakov A N, Ramayya A V, Roberto J B, Ryabinin M A, Rykaczewski K P, Sagaidak R N, Shaughnessy D A, Shirokovsky I V, Shumeiko M V, Stoyer M A, Stoyer N J, Subbotin V G, Sukhov A M, Tsyganov Yu S, Utyonkov V K, Voinov A A, Vostokin G K "Production and decay of the heaviest nuclei 293, 294117 and 294118" Phys. Rev. Lett. 109 162501 (2012)
  113. Ceulemans A, Thyssen P "The ’chemical mechanics’ of the periodic table" Mendeleev To Oganesson: A Multidisciplinary Perspective On The Periodic Table (Eds E Scerri, G Restrepo) (New York: Oxford Univ. Press, 2018) p. 104-121
  114. Powers R T "Frequencies of Radial Oscillation and Revolution as Affected by Features of a Central Potential" Atti Del Convegno Mendeleeviano, Periodicità E Simmetrie Nella Struttura Elementare Della Materia. Torino-Roma, 15-21 Settembre, 1969/Convegno Mendeleeviano. Torino-Roma, 1969 (Ed. M Verde) (Turin, Italy: Vincenzo Bona, 1971) p. 235-242
  115. Wheeler J A "From Mendeléev’s atom to the collapsing star" Philosophical Foundations Of Science: Proc. Of Section L, 1969, American Association For The Advancement Of Science (Boston Studies in the Philosophy of Science) Vol. 11 (Eds R J Seeger, R S Cohen) (Dordrecht: Springer, 1974) p. 257-301, Ch. 15
  116. Pomeranchuk I, Smorodinsky J "On the energy levels of systems with Z > 137" J. Phys. USSR 9 97-100 (1945); https://ufn.ru/ufn13/ufn13_4/Russian/pomeranchuk.pdf
© 1918–2026 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions