In July 1925 Heisenberg published a paper that ushered in the new era of quantum mechanics. This epoch-making paper is generally regarded as being difficult to follow, partly because Heisenberg provided few clues as to how he arrived at his results. We give details of the calculations of the type that Heisenberg might have performed. As an example we consider one of the anharmonic oscillator problems considered by Heisenberg, and use our reconstruction of his approach to solve it up to second order in perturbation theory. The results are precisely those obtained in standard quantum mechanics, and we suggest that a discussion of the approach, which is based on the direct calculation of transition frequencies and amplitudes, could usefully be included in undergraduate courses on quantum mechanics.

Topics
Anharmonic oscillator, Perturbation theory
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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), … .
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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.
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We use ω rather than Heisenberg’s ν.
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We depart from the notation of Refs. 1 and 2, preferring that of Ref. 8, pp. 204–224. Our Xα(n) is Heisenberg’s aα(n).
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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.
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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.
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Reference 2, p. 264.
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Reference 2 p. 267.
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37.
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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.
Note that this means that, in x(t), all the terms which are of order λp arise from many different terms in Eq. (24).
49.
Except that Heisenberg relabeled most of the aα’s as aα(n).
50.
Because of an oversight, he wrote a0(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. (33343536). 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 4πm by πm, 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.
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It is essentially that given by L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Pergamon, Oxford, 1977), 3rd ed., pp. 67–68.
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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.
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See W. Heisenberg, Physics and Beyond (Allen & Unwin, London, 1971), p. 61.
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2004
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