'Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik,' pp. 336-363 in Zeitschrift für Physik, 36. Band, 5 Heft, 27 March, 1926

First edition, journal issue in original printed wrappers, of Pauli's important derivation of the hydrogen spectrum from the new quantum mechanics, providing the first validation of Heisenberg's theory. As Pauli explains in the abstract: "It is shown that the Balmer terms of an atom with a single electron are yielded correctly by the new quantum mechanics and that the difficulties which arose in the old theory... disappear in the new theory." Heisenberg's epochal 'Über quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen,' the invention of quantum mechanics, had appeared just a few months earlier, in September 1925. "The first nontrivial and physically important application of quantum mechanics was made soon afterwards by Wolfgang Pauli [in the present paper], who calculated the stationary energy values of the hydrogen atom by the matrix method and found complete agreement with Bohr's 1913 formulas. From this moment, there was no longer any doubt about the correctness of the theory among physicists" (Max Born, Physics in My Generation (1956), p. 181). "The Heisenberg formulation of quantum mechanics is exceedingly easy to apply to problems with linear equations of motion, such as that of the harmonic oscillator, but the presence of the inverse square law in the attractive force of the hydrogen problem leads to mathematical problems of considerable complexity. This paper therefor added important support to the new mechanics. In his discussion of the hydrogen atom, he of course included the new degree of freedom to which he had previously drawn attention and referred to the new hypothesis of Uhlenbeck and Goudsmit, which identified this degree of freedom with an internal rotation or 'spin' of the electron, a hypothesis which by this time, after some initial controversy, was beginning to find acceptance" (Peierls, p. 179).

"Now the time had come to test the new formalism on the hydrogen atom. Dirac... had solved a two-dimensional hydrogen model ['Quantum mechanics and a preliminary investigation of the hydrogen atom,' Proc. Roy. Soc.,Vol. 110 (1926)]. But the real thing apparently was too difficult even for Born. Heisenberg wrote in his contribution to the Memorial Volume for Pauli: 'At that time I myself was somewhat unhappy that I did not succeed in deriving even the simple hydrogen atom.' Liberated from his work on the 'Quantentheorie' [Handbuch der Physik, Vol. 23, pp. 1-278] Pauli had gone into this problem at full pelt, and on 17 January 1926 he submitted for publication his paper 'On the hydrogen spectrum from the standpoint of the new quantum mechanics.' Already on 3 November 1925 Heisenberg had written to Pauli: 'I probably don't have to tell you how much I rejoiced at your new theory of hydrogen and how much I admire that you have made out this theory so far,' and he closed the letter with 'hearty congratulations for your theory'" (Enz, No Time to be Brief (2002), pp. 134-5).

"Besides Heisenberg, it was Niels Bohr who first learned about the results of Pauli's hydrogen calculation. Hendrik Kramers, who spent a few days in Hamburg in early November, had informed Bohr about it. Bohr immediately wrote to Pauli: 'To my great joy I heard from Kramers that you have succeeded in deriving the Balmer formula' (Bohr to Pauli, 13 November 1925). In the same letter Bohr asked for further details of the calculation and Pauli obliged without delay. Bohr and the theoreticians close to him found Pauli's success striking. Convinced that it was indeed true, Bohr wrote to Pauli: 'Kramers, Kronig and I, who have just gone once again with the greatest pleasure through your beautiful calculation on the hydrogen spectrum, send you many friendly greetings from Tisvilde (Bohr to Pauli, 5 December 1925).

"The great satisfaction which Heisenberg and Bohr expressed on Pauli's achievement arose for a reason which could be appreciated nowhere better than in Gottingen and Copenhagen. At these places the conclusion had been reached during the past three years that Bohr's apparently-so-successful theory of atomic systems, including his celebrated derivation of the Balmer formula, possessed no rational theoretical foundation whatsoever. However, Paulis admirable calculation of the hydrogen spectrum, performed on the basis of the matrix mechanics, which could be regarded as the complete logical and mathematical formulation of the quantum concepts developed in Copenhagen and Gottingen since 1923, now provided the necessary justification of the fundamental formula of Bohr's theory" (Mehra & Rechenberg, p. 181).

Van der Waerden, Sources of Quantum Mechanics, no. 16. Mehra & Rechenberg, The Historical Development of Quantum Theory, vol. III, 1982. Peierls, 'Wolfgang Ernst Pauli. 1900-1958,' Biographical Memoirs of Fellows of the Royal Society 5 (1960), pp. 174-192.