Sturm und Zwang in Copenhagen
From September 1922, Pauli spent a year in Copenhagen, and helped Bohr to explore, among other things, the mysteries of the anomalous Zeeman
[163] "Atomrnystik" is in Heisenberg to Pauli, 19 Nov. 1921, PB , no. 16; Bohr [1922c], [391]; Bohr 1923e; BCW 4: [647]n.
[164] Pauli's position was inferred from Heisenberg to Pauli, 25 Nov. 1921, PB , no. 17, and 17 Dec. 1921, PB , no. 18; Landé 1922: in this article Landé managed to do without the passivity of the core; Pauli to Bohr, 11 Feb. 1924, PB , no. 73.
effect. He first tried his best to find a multiperiodic model of this effect: one that would necessarily involve some extramechanical property of the core but that would nevertheless retain the regular quantum theory of multiperiodic systems with integral quantum numbers. As Bohr put it in March 1923, after the failure of Pauli's attempt, "It was a desperate attempt to remain true to the integral quantum numbers; we hoped to find in the very paradoxes an indication of the path along which one should search for a solution of the anomalous Zeeman effect."[165]
Confronted with this failure, Bohr was pressed to locate the precise type of departure from ordinary mechanics needed to account for the success of Heisenberg's model. To this end he proposed the notion of unmechanischer Zwang , a form of nonmechanical stress occurring in the interaction between the atomic core and outer electrons. In order to be fundamental, this notion had to be independent of particular models, and of the specific labeling of multiplet terms favored by Sommerfeld and Landé. Thus Bohr reasoned in terms of the a priori statistical weights of the nk states, for they had a direct empirical meaning, as the total number of terms in a magnetic field corresponding to a given value of n and k .[166]
Consider the case of alkali doublets. On the one hand, the multiplicity associated with nk had to be 2(2k - 1), because, according to Landé, there were 2k choices of m corresponding to i = k , and 2k - 2 choices of m corresponding to i = k - 1. On the other hand, for a vanishing coupling between core and outer electron, the statistical weight of the core had to be one in order to account for the diamagnetism of the corresponding noble gas;[167] and the multiplicity of the outer electron would be that given by the Sommerfeld atom: 2k (Bohr and Sommerfeld excluded the value m = 0 on the grounds that the corresponding orbit is adiabatically connected to an orbit passing through the nucleus).[168] The resulting total
[165] Bohr to Landé, 3 Mar. 1923, AHQP. At the end of 1922 Pauli sent a manuscript to Pauli containing an analysis of the anomalous Zeeman effect with integral quantum numbers only (see Heisenberg to Sommerfeld, 4 Jan. 1923, AHQP), but he soon surrendered to Heisenberg's objections (see Heisenberg to Pauli, 21 Feb. 1923, PB ) and admitted with Bohr a half-integral j .
[166] See the excellent account m Serwer 1977.
[167] Indeed, the absence of paramagnetism implies a vanishing net magnetic moment of the atom, and therefore no magnetic splitting of its energy. This result seemed to be at variance with Bohr's second atomic theory, which gave a unit angular momentum to all noble gases (due to the heliumlike K-shell, the other shells being saturated and symmetrical). Bohr avoided the contradiction by admitting a violation of ordinary mechanics already m the diamagnetic behavior of noble gases (cf. n. 148).
[168] Interestingly, the exclusion of m = 0 led to the correct number of Zeeman components for the levels of the hydrogen atom: to a nk level in the Sommerfeld atom corresponds a definite value of J = k - ½ m the modern theory, which gives 2J + 1 = 2k magnetic sublevels.
multiplicity was thus 1 × 2k = 2k . This result was incompatible with the existence of a multiperiodic model of the interaction between core and electron, since in such a model an adiabatic variation of the coupling strength would have conserved the total statistical weight of an nk term. [169]
In March 1923 Bohr concluded:
The coupling of the series electron to the atomic core is subject to a stress [Zwang ] which is not analogous to the effect of an external field, but which forces the atomic core to adopt two different orientations in the atom, instead of the single orientation possible in a constant external field, while, at the same time, as a result of the same stress, in the atomic assemblage the outer electron can only assume 2k - 1 orientations in an external field instead of 2k .[170]
This way of splitting the multiplicity 2(2k - 1) into two factors was of course suggested by Heisenberg's Rumpf model. However, as Bohr and Pauli noticed, it explained why the S-states (k = 1) of alkali atoms were singlets instead of doublets, a fact for which Heisenberg had no satisfactory explanation. Indeed, in this case Heisenberg's model still gives two orientations for the Rumpf in the outer electron's field, and therefore a doublet (in absence of external field). Instead Bohr's Zwang gives the multiplicity 2(2k - 1) = 2, to be attributed to a singlet with double magnetic splitting. [171]
More fundamentally, Bohr's Zwang was in perfect harmony with his previous analysis of the origins of the failure of the helium atom. In both cases the stability of the relevant dynamic structure eluded ordinary mechanics and requested a nonmechanical stress. This lack of mechanical stability implied a violation of the adiabatic theorem, which in turn created a gap in the definition of energy (in the helium case) or in the definition of statistical weights (in the alkali doublet case).
Altogether, Bohr did not think that the anomalous Zeeman effect made the situation worse than it already was as a result of the failure of the helium atom. He concluded without any sign of a disturbance: "Under these circumstances [the necessity of an unmechanical Zwang ], we must presume that the coupling between the series electrons and the atomic core cannot be directly described according to the quantization rules of multi-periodic systems." Pauli reacted quite differently, identifying this failure of general principles as a personal failure, as appears in one of his letters to Landé (May 1923): "I am very depressed that I have not been able to
[169] Bohr 1923e, 274-277; Bohr [1923f], [525]-[530].
[170] Bohr 1923e, 276.
[171] Ibid., 279.
find a satisfactory explanation of these dumbfoundingly simple regularities [of the anomalous Zeeman effect] in terms of a model.[172]