Abstract
Between 1925 and 1927, Heisenberg participated prominently in the discovery, formulation and application of quantum mechanics. In this period he published 11 original scientific papers, three of which we shall mention here only briefly (they have been allocated to the next Group 4 of papers dealing with applications of quantum mechanics), while another one has been placed into the previous Group 2 (paper No. 12, p. 306). We shall now concentrate on the remaining seven papers, which played a crucial role in creating and establishing quantum mechanics and its physical interpretation. Occasionally, we shall also refer to the three reviews of the theory which Heisenberg wrote or coauthored at the same time and which have been included in Volume B of the Collected Works.
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References
R. Ladenburg: Die quantentheoretische Deutung der Zahl der Dispersionselektronen. Z. Phys. 4, 451–468 (1921). See the summary and English translation (“The quantum theoretical interpretation of the number of dispersion electrons”) in Sources of Quantum Mechanics, Edited with a Historical Introduction by B. L. van der Waerden(North Holland, Amsterdam 1967 ) pp. 10 – 11, 139–157.
N. Bohr, H. A. Kramers, J. C. Slater: The quantum theory of radiation. Philos. Mag. (6) 47, 785–802 (1924); reprinted in Sources of Quantum Mechanics, pp. 159–176. (The above quote appears on p. 163.) German version: Uber die Quantentheorie der Strahlung. Z. Phys. 24, 69 – 87 (1924).
For an English translation of the Kramers Heisenberg paper (“On the dispersion of radiation by atoms”) and for a reprint of Kramers preceding notes on dispersion theory (“The law of dispersion and Bohr’s theory of spectra”; “The quantum theory of dispersion”) see Sources of Quantum Mechanics, pp. 223–251, and pp. 177–180, 199–201. The quote is from p. 226.
See paper No. 1, p. 345.
J. H. Van Vleck: The absorption of radiation by multiply periodic orbits, and its relation to the correspondence principle and the Rayleigh-Jeans law. Part I. Some extensions of the correspondence principle. Part II. Calculation of absorption by multiply periodic orbits. Phys. Rev. (2) 24, 330–346, 347–365 (dated June 19, 1924, published in issue No. 4 of October 1924 ).
A. Einstein: Strahlungs-Emission und -Absorption nach der Quantentheorie. Verh. d. Deutsch. Phys. Ges. (2) 18, 318–323 (1916)
A. Einstein: Zur Quantentheorie der Strahlung. Mitt. Phys. Ges. Zürich 16, No. 18, 47–62 (1916)
A. Einstein: Phys. Z. 18, 121–128 (1917). English translation of the latter paper: “On the Quantum Theory of Radiation” in Sources of Quantum Mechanics, pp. 63–77.
The importance of Van Vleck’s paper was stressed by Jordan in a letter to Van der Waerden, dated December 1, 1961. (See Sources of Quantum Mechanics, pp. 17–18.)
See Sources of Quantum Mechanics, p. 25. The full text of Heisenberg’s letter can be found in Wolfgang Pauli: Wissenschaftlicher Briefwechsel/Scientific Correspondence, Band/Volume I: 1919–1929, ed. by A. Hermann, K. von Meyenn, V. F. Weisskopf (Springer, New York, Heidelberg, Berlin 1979) pp. 219–221.
The full text of Heisenberg’s letter to Kronig dealing with the reformulation of the (classical) Fourier series has been given by R. Kronig in his contribution (entitled “The turning point”) to the Pauli memorial volume Theoretical Physics in the Twentieth Century, ed. by M. Fierz, V. F. Weisskopf (Interscience, New York, 1960) pp. 5–37, especially pp. 23–24.
For a detailed story of Heisenberg’s discovery of quantum mechanics see also J. Mehra, M. Rechenberg: The Historical Development of Quantum Theory, Volume 2. The Discovery of Quantum Mechanics 1925 (Springer, New York, Berlin, Heidelberg 1982) especially Chapter IV.
From the letters and other documents, it is not completely obvious whether Heisenberg evaluated in Helgoland the anharmonic oscillator with the equation of motion (8), or that having an anharmonic term proportional to q3 in the equation of motion. In his published paper discussed below Heisenberg presented only the solution for the latter.
See Sources of Quantum Mechanics, p. 25. (The English translation is from Mehra and Rechenberg, The Discovery of Quantum Mechanics 1925, quoted in footnote 10, p. 263.)
See the English translation of paper 3: “Quantum Theoretical Re-Interpretation of Kinematic and Mechanical Relations”, in Sources of Quantum Mechanics, pp. 261–276, especially pp. 261–262.
In Sources of Quantum Mechanics, pp. 30–32, van der Waerden has shown, following a suggestion by Th. Kuhn, that (n, n-a) and the analogous (n, n + a) are the same as the e and a which determine the emission and absorption of radiation by atoms in the paper of Kramers and Heisenberg. As we have seen, these e and a are vector amplitudes of virtual oscillators, multiplied by -e; but Heisenberg does not say this, avoiding the expression “virtual oscillator”.
As we have mentioned earlier, the idea of using such arrays of terms a (n, m) · exp[i w(n, m) t] to describe quantum mechanical variables occurred already in Heisenberg’s letter to Kronig dated June 5, 1925.
Heisenberg used this quantum condition already in Helgoland to obtain energy conservation for the anharmonic oscillator.
For a proof of this equivalence see Sources of Quantum Mechanics, p. 31, and Mehra and Rechenberg: The Discovery of Quantum Mechanics 1925, quoted in footnote 10, pp. 242–247.
As has been mentioned above, Heisenberg presented the full solution of the anharmonic oscillator, Eq. (13), in his paper. Interestingly enough, in section 3, he began with an anharmonic oscillator, described by Eq. (8), such as he had considered in former letters, e. g., the one to Kronig dated June 5 or the one addressed to Pauli on June 24. The main reason for taking the oscillator with Eq. (13) was that its solution involved fewer terms. The classical Fourier series for the position x involved only terms with the period ?t, 3 wt, 5 5wt,…, which implied a similar simplification for the quantum theoretical reformulation; in the latter, only amplitudes a(n,n-1), a(n,n-3), a(n,n-5), etc., occurred.
See Max Born: My Life- Recollections of a Nobel Laureate( Charles Scribner’s Sons, New York 1978 ) pp. 217 – 218.
M. Born: My Life, p. 218.
Heisenberg to Pauli, September 18, 1925; see Pauli: Wissenschaftlicher Briefwechsel I, p. 239.
After having read a letter in which Heisenberg explained his perturbation theory, Born came to the conclusion that it was not correct and developed his own scheme, based on Eq. (15). The equivalence of Heisenberg’s and Born’s definition has been shown in Sources of Quantum Mechanics, p. 49.
An English translation of the Three-Men-Paper (“On Quantum Mechanics II”) can be found in Sources of Quantum Mechanics, pp. 321–385.
See Sources of Quantum Mechanics, pp. 43 – 57, and J. Mehra, H. Rechenberg: The Historical Development of Quantum Theory, Volume 3. The Formulation of Matrix Mechanics and Its Modifications 1925–1926(Springer, New York, Berlin, Heidelberg 1982 ) pp. 321 – 385.
W. Heisenberg: Uber quantentheoretische Kinematik und Mechanik. Math. Ann. 95, 683–705 (received December 21, 1925, published 1926), reprinted as No. 3 in Volume B of Collected Works, pp. 29–51.
W. Pauli: Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik. Z. Phys. 36, 336–363 (received January 17, 1926, published in issue No. 5 of March 27, 1926); English translation: “On the Hydrogenspectrum from the Standpoint of the New Quantum Mechanics”, in Sources of Quantum Mechanics, pp. 387–415. Pauli applied a very clever trick to avoid the use of angle variables in the hydrogen problem. (Angle variables could not be treated in the matrix scheme of Born, Heisenberg and Jordan.) At about the same time Paul Dirac in Cambridge was able to extend the matrix scheme to include angle variables as well. His paper, “Quantum mechanics and a preliminary investigation of the hydrogen atom”, Proc. Roy. Soc. (London) A110, 561–579 (1926), was received on January 22, 1926 and published in the issue of March 1, 1926.
E. Schrödinger: Quantisierung als Eigenwertproblem (Erste Mitteilung). Ann. Phys. (4) 79, 361–379 (1926), received January 27, 1926;
E. Schrödinger: Quantisierung als Eigenwertproblem “(Zweite Mitteilung)”, ibid., 489–527, received February 23, 1926
E. Schrödinger: Quantisierung als Eigenwertproblem “(Dritte Mitteilung”), Ann. Phys. (4) 80, 437–490 (1926), received May 10, 1926
E. Schrödinger: Quantisierung als Eigenwertproblem “(Vierte Mitteilung)”, Ann. Phys. (4) 81, 109–139 (1926), received June 21, 1926
E. Schrödinger: Uber das Verhaltnis der Heisenberg-Born-Jordanschen Quantenmechanik zu der meinen. Ann. Phys. (4) 79. 734–756 (1926), received March 18, 1926.
Letter from W. Pauli to P. Jordan, dated April 12, 1926. See B. L. van der Waerden’s article, “From Matrix Mechanics and Wave Mechanics to Unified Quantum Mechanics”, in: The Physicist’s Conception of Nature, ed. by J. Mehra (Reidel, Dordrecht, Boston 1973 ), pp. 276 – 293.
The same conclusion was reached independently a little later by Paul Dirac in his paper “On the theory of quantum mechanics”, Proc. Roy. Soc. London A112, 661–677 (received August 26, 1926, published in issue No. A762 of October 1, 1926).
W. Heisenberg: Quantenmechanik. Die Naturwissenschaften 14, 989–994 (published in the issue of November 5, 1926); reprinted as No. 4 in Volume B of Collected Works, pp. 52–57.
E. Schrodinger: “Quantisierung als Eigenwertproblem (Vierte Mitteilung)”, quoted Ref. 27, in particular Sect. 7.
M. Born: Zur Quantenmechanik der Stofivorgange. Z. Phys. 37, 863–867 (received June 25, 1926, published in issue No. 12 of July 10,1926)
M. Born: Quantenmechanik der Stofivorgange. Z. Phys. 38, 803–827 (received July 21, 1926, published in issue No. 11/12 of September 14, 1926 ).
The generalized scheme, the socalled “transformation theory”, is contained in the papers of P.A.M. Dirac: The physical interpretation of the quantum dynamics. Proc. Roy. Soc. London A113, 621–641 (received December 2, 1926, published in issue No. A765 of January 1, 1927), and of P. Jordan: Uber eine neue Begrtindung der Quantenmechanik. Z. Phys. 40, 809–838 (received December 18, 1926, published in issue No. 11/12 of January 18, 1927) Dirac had been in Copenhagen since September 1926 and had had the opportunity of discussing physics with Bohr and Heisenberg. The aim of his paper was to extend Heisenberg’s considerations on quantum fluctuations (paper No. 6, p. 472).
An English translation of this paper, “The Physical Content of Quantum Kinematics and Mechanics”, can be found in: Quantum Theory and Measurement, ed. by J. A. Wheeler, W. H. Zurek (Princeton University Press, Princeton, N.J. 1983 ) pp. 62 – 84.
W. Heisenberg: Physics and Beyond. Encounters and Conversations (Harper - Row, New York, Evanston, London 1971) pp. 77–78. (Original German publication: Der Teil und das Ganze. Gesprache im Umkreis der Atomphysik, Piper, Munich 1969; reprinted in Collected Works, Volume C III, especially pp. 111–112).
N. Bohr: The quantum postulate and the recent development of atomic theory. Nature 121, 580–590 (presented on September 16, 1927, at the International Conference of Physics at Como and as a report at the Fifth Solvay Conference in October 1927. published in the Supplements to the issue of April 14, 1928). German version: Das Quantenpostulat und die neuere Entwicklung der Atomphysik. Die Naturwissenschaften 16, 245–257 (published in the issue of April 13, 1928 ).
M. Born, W. Heisenberg: “La mecanique des quanta”, in “lectrons et Photons. Rapports et Discussions du Conseil de Physique Tenu a Bruxelles du 24 au 29 Octobre 1927 sous les Auspices de I”lnstitut International de Physique Solvay, ed. by Institut International de Physique Solvay (Gauthier-Villars, Paris 1928) pp. 143–181; reprinted as paper No. 7 in Volume B of Collected Works, pp. 58–96.
W. Heisenberg: The Physical Principles of the Quantum Theory, translated into English by C. Eckardt, F. C. Hoyt (The University of Chicago Press, 1930); reprinted in Volume B of Collected Works, pp. 117–166.
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van der Waerden, B.L., Rechenberg, H. (1985). Quantum Mechanics (1925–1927). In: Blum, W., Rechenberg, H., Dürr, HP. (eds) Original Scientific Papers Wissenschaftliche Originalarbeiten. Werner Heisenberg Gesammelte Werke Collected Works, vol A / 1. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61659-4_23
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