Understanding Heisenberg's "magical" paper of July 1925: A new look at the calculational details

American Journal of Physics

Volumn 72, Issue 11, 2004, Pages 1370-1379

  Aitchison, Ian J R ^a   MacManus, David A ^b   Snyder, Thomas M ^c  

^a UNIVERSITY OF OXFORD   (United Kingdom)

^b Tripos Receptor Research Ltd   (United Kingdom)

^c Lincoln Land Community College   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 8744311637     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1775243     Document Type: Article

References (64)

1. W. Heisenberg, "Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen," Z. Phys. 33, 879-893 (1925).
2. This is the title of the English translation, which is paper 12 in Ref. 3, pp. 261-276. We shall refer exclusively to this translation, and to the equations in it as (H1), (H2), ... .
3. Sources of Quantum Mechanics, edited by B. L. van der Waerden (North-Holland, Amsterdam, 1967). A collection of reprints in translation.
4. M. Born and P. Jordan, "Zur Quantenmechanik," Z. Phys. 34, 858-888 (1925), paper 13 in Ref. 3.
5. P. A. M. Dirac, "The fundamental equations of quantum mechanics," Proc. R. Soc. London, Ser. A 109, 642-653 (1926), paper 14 in Ref. 3.
6. M. Born, W. Heisenberg, and P. Jordan, "Zur Quantenmechanik II," Z. Phys. 35, 557-615 (1926), paper 15 in Ref. 3.
7. S. Weinberg, Dreams of a Final Theory (Pantheon, New York, 1992), pp. 53-54. Weinberg goes on to say that "Perhaps we should not look too closely at Heisenberg's first paper ... ." We will not follow his suggestion here.
8. S.-I. Tomonaga, Quantum Mechanics: Old Quantum Theory (North-Holland, Amsterdam, 1962), Vol. 1.
9. M. Jammer, The Conceptual Development of Quantum Mechanics (McGraw-Hill, New York, 1966).
10. E. MacKinnon, "Heisenberg, models and the rise of matrix mechanics," Hist. Stud. Phys. Sci. 8, 137-188 (1977).
11. J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory (Springer-Verlag, New York, 1982), Vol 2.
12. J. Hendry, The Creation of Quantum Mechanics and the Bohr-Pauli Dialogue (Reidel, Dordrecht, 1984).
13. T.-Y. Wu, Quantum Mechanics (World Scientific, Singapore, 1986).
14. M. Taketani and M. Nagasaki, The Formation and Logic of Quantum Mechanics (World Scientific, Singapore, 2002), Vol. 3.
15. G. Birtwistle, The New Quantum Mechanics (Cambridge U.P., Cambridge, 1928).
16. M. Born, Atomic Physics (Dover, New York, 1989).
17. J. Lacki, "Observability, Anschaulichkeit and abstraction: A journey into Werner Heisenberg's science and philosophy," Fortschr. Phys. 50, 440-458 (2002).
18. J. Mehra, The Golden Age of Theoretical Physics (World Scientific, Singapore, 2001), Vol. 2.
19. Of these the most detailed are Ref. 3, pp. 28-35, Ref. 8, pp. 204-224, Ref. 10, pp. 161-188, and Ref. 11, Chap. IV.
20. Reference 7, p. 53.
21. Reference 2, p. 262.
22. All quotations are from the English translation of Ref. 2.
23. We use ω rather than Heisenberg's ν.
24. α(n).
25. The reader may find it helpful at this point to consult Ref. 26, which provides a clear account of the connection between the classical analysis of an electron's periodic motion and simple quantum versions. See also J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975), 2nd ed., Sec. 9.2.
26. W. A. Fedak and J. J. Prentis, "Quantum jumps and classical harmonics," Am. J. Phys. 70, 332-344 (2002).
27. -|α|↔X(n,n+|α|) would not be correct because ω(n,n+|α|) is not the same, in general, as ω(n - |α|,n).
28. Conventional notation, subsequent to Ref. 4, would replace n - α by a second index m, say. We prefer to remain as close as possible to the notation of Heisenberg's paper.
29. Reference 2, p. 264.
30. Reference 2, p. 265.
31. Reference 2, p. 266.
32. Reference 2 p. 267.
33. This step apparently did not occur to him immediately. See Ref. 11, p. 231.
34. Actually not quite. We have taken the liberty of changing the order of the arguments in the first terms in the braces; this (correct) order is as given in the equation Heisenberg wrote before Eq. (H20).
35. W. Thomas, "Über die Zahl der Dispersionelektronen, die einem stationären Zustande zugeordnet sind (Vorläufige Mitteilung)," Naturwissenschaften 13, 627 (1925).
36. W. Kuhn, "Über die Gesamtstärke der von einem Zustande ausgehenden Absorptionslinien," Z. Phys. 33, 408-412 (1925), paper 11 in Ref. 3.
37. W. Heisenberg, as discussed in Ref. 11, pp. 243 ff.
38. Reference 9, p. 193; Φ is any function defined for stationary states.
39. M. Born, "Über Quantenmechanik," Z. Phys. 26, 379-395 (1924), paper 7 in Ref. 3.
40. Reference 9, p. 202.
41. H. A. Kramers and W. Heisenberg, "Über die Streuung von Strahlen durch Atome," Z. Phys. 31, 681-707 (1925), paper 10 in Ref. 3.
42. For considerable further detail on dispersion theory, sum rules, and the "discretization" rules, see Ref. 8, pp. 142-147, 206-208, and Ref. 9, Sec. 4.3.
43. See Ref. 3, p. 37.
44. Reference 2, p. 268.
45. For an interesting discussion of the possible reasons why he chose to try out his method on the anharmonic oscillator, see Ref. 11, pp. 232-235; and Ref. 3, p. 22. Curiously, most of the commentators-with the notable exception of Tomonaga (Ref. 8)-seem to lose interest in the details of the calculations at this point.
46. The (n,n - α) matrix element, in the standard terminology.
47. Equation (22) is not in Heisenberg's paper, although it is given by Tomonaga, Ref. 8, Eq. (32.20′).
48. p arise from many different terms in Eq. (24).
49. α(n).
50. 0(n) in place of a (n,n) in Eq. (33).
51. MacKinnon (Ref. 10) suggests how, in terms of concepts from the "virtual oscillator" model, Eq. (28) may be transformed into Eqs. (33)-(36). We do not agree with MacKinnon (Ref. 10, footnote 62) regarding "mistakes" in Eqs. (35) and (36).
52. In the version of the quantum condition that Heisenberg gave just before Eq. (H20), he unfortunately used the same symbol for the transition amplitudes as in (H16)-see Eq. (16)-but replaced 4 πm by cm, not explaining where the factor 1/4 came from [see Eq. (42)]; he also omitted the λ's.
53. Heisenberg omitted the superscripts.
54. See, for example, L. I. Schiff, Quantum Mechanics (McGraw-Hill, New York, 1968), 3rd ed., p. 72.
55. Reference 2, p. 272.
56. This is the justification suggested in Ref. 4, p. 297 of paper 13 in Ref. 3.
57. It is essentially that given by L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1977), 3rd ed., pp. 67-68.
58. L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, Oxford, 1976), 3rd ed., pp. 86-87.
59. This equation corresponds to Eq. (88) of Ref. 4, in which there appears to be a misprint of 17/30 instead of 11/30.
60. See Ref. 4, p. 305.
61. Reference 57, p. 136.
62. Reference 2, pp. 272-3.
63. See W. Heisenberg, Physics and Beyond (Allen & Unwin, London, 1971), p. 61.
64. W. Pauli, "Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik," Z. Phys. 36, 336-363 (1926), paper 16 in Ref. 3.