That I Cannot Conceive of After the Results of Your Dissertation: Fritz Reiche and the F-sum Rule

Jähnert, Martin

doi:10.1007/978-3-030-13300-9_6

Martin Jähnert^3,4

Part of the book series: Archimedes ((ARIM,volume 56))

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Abstract

This chapter discusses the work of Fritz Reiche on the quantum theory of radiation and his application of the correspondence principle in the context of dispersion theory. Following his private correspondence with Kramers in 1923 and 1924, I reconstruct Reiche’s attempts to determine transition probabilities on the basis of the correspondence principle and show how they led to the formulation of a relation among transition probabilities, which came to be known as the f-sum rule or the Thomas-Reiche-Kuhn sum rule.

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Notes

1.
My analysis of Reiche’s work with the correspondence principle is based first of all on his correspondence with Hendrik Antoon Kramers (AHQP 8b.9), in which he developed his arguments. It is complemented by Reiche’s published articles: Ladenburg and Reiche (1923, 1924), Reiche and Thomas (1925), Reiche (1926a,b, 1929), and Rademacher and Reiche (1927). Focusing on Reiche’s work, my reconstruction keeps the discussion of the general development of dispersion theory to a minimum. The quantum theory of dispersion developed by Kramers, Born, and Van Vleck is discussed only as far as it is necessary to understand Reiche’s work.
2.
These considerations dropped out of sight as Jammer, Darrigol and Duncan and Janssen analyzed the dispersion theories of Ladenburg and Reiche, Kramers, Born, Van Vleck, and Kramers and Heisenberg in detail and argued that these dispersion theories paved the way to Heisenberg’s Umdeutung. See Jammer (1966, 181–195), van der Waerden (1968, 10–18), Darrigol (1992, 224–234), Konno (1993), Duncan and Janssen (2007a, 581–597), and Jordi Taltavull (2013, 52–53). In addition, there are a few studies dedicated at least in part to Reiche and his scientific work: Benjamin Bederson and Valentin Wehefritz presented general biographical sketches which focus primarily on Reiche’s persecution in Nazi Germany and his late escape to the U.S. in 1941. See Bederson (2005) and Wehefritz (2002). Clayton Gearhart studied Reiche’s earlier work on the quantum theory of molecules and contextualized his textbook Die Quantentheorie. Ihr Ursprung und ihre Entwicklung. See Gearhart (2010, 2012). At any rate, these works did not discuss Reiche’s pathway to the f-sum rule.
3.
For general biographical information, see Bederson (2005) and also Wehefritz (2002).
4.
Jordi Taltavull (2013, 31–32). For a comprehensive discussion of dispersion in classical radiation theory, see Jordi’s forthcoming dissertation.
5.
Jordi Taltavull (2013, 33).
6.
Jordi Taltavull (2013, 34–48).
7.
Note that although they discussed the number of dispersion electrons and the dispersion constant, Ladenburg’s and Reiche’s considerations focused entirely on this comparison in the case of absorption until March 1924. Their argument did not involve Ladenburg’s central assertion that the number of absorbing systems was equal to the number of dispersing systems, and dispersion played no constructive role until they received Kramers’ quantum theory of dispersion.
8.
Planck (1921).
9.
Ladenburg (1921, 452).
10.
Ladenburg (1921, 453) and Einstein (1916).
11.
Note that I am following Ladenburg’s notation here. Einstein had assumed that the radiation density was classical and had not associated it with a particular transition. This difference did not play a role, as Ladenburg also assumed that his u _ki was equivalent to the radiation density u ₀ of the classical radiation field so that it cancelled out in his comparison.
12.
For the first quote, see Ladenburg (1921, 451). “Infolgedessen ist es […] diese W[ahrscheinlichkeit] der spontanen Rückgänge (multipliziert mit dem in jener Einsteinschen Beziehung auftretenden Verhältnisse der statistischen Gewichte der Quantenzustände), die an die Stelle jener Dispersionskonstante R tritt […]” For the second quote, see Ladenburg (1921, 468). “Die […] Zahl R der Dispersionselektronen hat nach der Quantentheorie die Bedeutung des Produktes $ N_i a_{ki}\frac {g_k}{g_i} \frac {mc^3}{8\pi e^2 \nu ^2_{ik}}$.”
13.
Ladenburg (1921, 454). “Indem wir die Gleichung (2) als die Definition für die experimentell bestimmbare Größe (2) ansehen, die natürlich in der Quantentheorie keine bestimmte Bedeutung hat, erhalten wir durch Gleichsetzen der beiden Gleichungen (2) und (6) die quantentheoretische Deutung der Dispersionskonstante R […].”
14.
Examples considered by Ladenburg were (a) the transition probabilities for excited hydrogen, which were obtained from the measurements on anomalous dispersion, and the decay experiments of Stark and Wien. See Stark (1916) and Wien (1919, 1921). Moreover, Ladenburg considered the transition probabilities for the sodium doublet obtained from intensity measurements and measurements on dispersion. See Ladenburg (1921, 456–464).
15.
Ladenburg (1921, 468). Note that Ladenburg’s consistency check was limited to showing that the different values for transition probabilities were of the same order of magnitude.
16.
Ladenburg and Reiche (1923). For the following correspondence argument, see Ladenburg and Reiche (1923, 586–587).
17.
Ladenburg and Reiche (1923, 586). “Ebenso entspricht nach Einsteins Annahmen die Wirkung einer äußeren Strahlung auf ein Quantenatom derjenigen, die ein klassischer Oszillator durch eine auffallende Welle erfährt. Wenn deren Frequenz sich von der Eigenfrequenz des Oszillators nur wenig oder gar nicht unterscheidet, besteht die Reaktion des Oszillators in einer Vermehrung oder einer Verminderung seiner Energie, je nach Phasenunterschied zwischen der äußeren Welle und der Bewegung des Oszillators. In Analogie hierzu nimmt Einstein an, daß das Atom im Zustand i eine durch den Faktor b _ik charakteristische Wahrscheinlichkeit besitzt, unter Aufnahme der Energie hν aus der auffallenden Welle in den höheren Zustand k überzugehen (‘positive Einstrahlung’), und daß ein Atom im Zustand k eine andere Wahrscheinlichkeit (b _ki) besitzt, unter dem Einfluss der äußeren Welle in den Zustand i zurückzukehren (‘negative Einstrahlung’).”
18.
Ladenburg and Reiche (1923, 588). “Der Faktor […] kann bei nicht zu hoher Strahlungsdichte am Ort der Atome und bei mäßiger Temperatur der Gasschicht praktisch gleich 1 gesetzt werden.” The only discussion of problems with Ladenburg’s formula is given by Duncan and Janssen (2007a, 586–587). They argue that Ladenburg and Reiche “derived a result for emission consistent with the correspondence principle (i.e., merging with the classical result in the limit of high quantum numbers), but their attempts to derive similar results for absorption and dispersion were unconvincing.” They diagnosed that the reason for this failure was that they “did not limit their ‘correspondence’ arguments to the regime of high quantum numbers […]. These problems invalidate many of the results purportedly derived from the correspondence principle in their paper.” The approximation responsible for Ladenburg and Reiche’s result, however, is not a result of an unwarranted extrapolation beyond the high quantum number limit. As will be discussed in the following, it was in conflict with expectations on this limit, and it was this conflict that became the source for the revision of the argument.
19.
That Reiche was the driving force behind the theoretical calculations may be seen from a letter by Reiche to Kramers, 9 May 1923 (AHQP 8b.9). In it, Reiche discussed the details of the calculations. Ladenburg had been interested in Kramers’ dissertation before and asked for a separate copy of the dissertation, which was not available in Breslau. He had already mentioned that the transition probabilities could be determined from the correspondence principle in his 1921 paper. Actual calculations based on Kramers’ Zwischenbahn, however, did not appear in his work and are not part of the correspondence between Ladenburg and Kramers. For Ladenburg’s interest in the correspondence principle, see Ladenburg to Kramers, 26 June 1920 (AHQP 8a.9) and Ladenburg (1921, 455).
20.
Ladenburg and Reiche (1923, 586). “Das Korrespondenzprinzip knüpft an die Tatsache an, daß im Gebiet großer Quantenzahlen n […] die bei einem Übergang n ^′→ n ^′′ ausgestrahlte Schwingungszahl übereinstimmt mit der harmonischen Komponente (n ^′− n ^′′) ω in der Fourierzerlegung der Bewegung des Elektrons, wobei ω die Umlaufzahl ist. […] Bei großen Schwingungszahlen, d.h. kleinen Quantenzahlen ist eine solche Übereinstimmung natürlich nicht möglich, da dort die Umlaufzahlen in benachbarten Bahnen ganz verschieden sind, während die Schwingungszahl ν stets von der Energiedifferenz beider Bahnen abhängt. Es lässt sich jedoch leicht zeigen, daß auch in diesem Falle ν als ein bestimmter Mittelwert über die entsprechenden harmonischen Komponenten (n ^′− n ^′′) ω der Anfangs- und Endbahn sowie einer kontinuierlichen Reihe gedachter Zwischenbahnen darstellbar ist. Wir sagen mit Bohr, daß die beim Übergang n ^′→ n ^′′ ausgesandte Schwingungszahl ν mit der harmonischen Komponente (n ^′− n ^′′) ω in der Bewegung des Elektrons ‘korrespondiert’.”
21.
Ladenburg and Reiche (1923, 586–587).
22.
Ladenburg and Reiche (1923, 586). “Wir sehen in dieser Analogie der klassischen und quantenmäßigen Emissions- und Absorptionsgesetze die Folge des Bohrschen Korrespondenzprinzips.”
23.
Reiche and Ladenburg preferred the arithmetic mean value $n+\frac {1}{2}$ for its simplicity, although they knew that Kramers’ dissertation had advocated the more complicated logarithmic mean value $n=\frac {1}{e}\frac {(n+1)^{n+1}}{n^n}$. See Ladenburg and Reiche (1923, 589) and Reiche to Kramers, 9 May 1923 (AHQP 8b.9).
24.
Ladenburg and Reiche cited Planck (1921, 179). Planck had compared the “number of those molecules N _n lying in the elementary region n, which perform an act of emission in a certain time τ:”
$$\displaystyle \begin{aligned} N_nA_n\tau \end{aligned}$$

with the number of molecules whose “state variables g decreases by r _a in the time τ:”
$$\displaystyle \begin{aligned} NW(g)r_a. \end{aligned}$$

In this equation, N is the total number of molecules and W(g) is the “Verteilungsdichte” [distribution density]. r _a is the energy emitted per frequency in the time τ according to classical radiation theory. Planck had calculated its value for a quantized harmonic oscillator as:
$$\displaystyle \begin{aligned} r_a=\frac{8\pi^2}{3}\frac{e^2\nu^2 hn}{mc^3}\tau. \end{aligned}$$
25.
Planck gave his arguments in terms of the numbers of systems that made a transition or radiated classically, Ladenburg and Reiche’s argument was based on considerations of the radiation energy of an individual system.
26.
Ladenburg and Reiche (1923, 589).
27.
In the specific case, this means that the respective classical number of dispersion electrons is associated with the number of atoms in the ground state. This interpretation follows from the definitions of the transition process. While Ladenburg and Reiche did not dwell on this when making their calculations, they identified the number of quantum systems N _i=0 with “a singular spatial quantum oscillator in the lowest quantum state” in a later part of their paper. (Ladenburg and Reiche 1923, 591, my emphasis).
28.
Ladenburg and Reiche (1923, 590). “[Es ist zu beachten, daß] die R Oszillatoren auf die N Raumpunkte gleichsam aufgeteilt werden müssen.”
29.
Ladenburg and Reiche (1923, 590). “Wir können diese Aufteilung […] formal so ausdrücken, daß an jedem der N Raumpunkte ein klassischer Oszillator von der Ladung xe und der Masse xm zu denken ist, wobei, wie oben $x=\frac {R}{N}$ ist, oder mit anderen Worten: jeder dieser N ‘Ersatzoszillatoren’ soll unter dem Einfluß der äußeren Welle ein elektrisches Moment annehmen, dessen Amplitude x mal so groß ist wie die eines klassischen Oszillators von der Ladung e und der Masse m.”
30.
Ladenburg and Reiche (1923, 591, emphasis in the original). “[…] die Wahrscheinlichkeit der möglichen Quantenübergänge […] ist ein Maß nicht nur für den Betrag der Quantenabsorption sondern auch für den Betrag der Zerstreuung und Dispersion […]. Dabei nehmen wir nicht etwa an, daß die Übergänge unter Einfluß der Welle ν wirklich zustande kommen. Vielmehr müssen wir uns nach dem Korrespondenzprinzip vorstellen […], dass diese Wahrscheinlichkeit für das Zustandekommen der Quantenübergänge durch die Amplitude der mit ν ₀ korrespondierenden harmonischen Komponente der Bewegung, also durch die Konfiguration des Atoms bestimmt ist, und es ist begreiflich, dass diese Amplitude nicht nur die Häufigkeit der wirklichen Quantenübergänge […] sondern auch die Reaktion des Atoms auf die Wellen beliebiger Schwingungszahl regelt.”
31.
Reiche to Kramers, 9 May 1923 (AHQP 8b.9, emphasis in the original). “Wenn es sich z. B. um ein in der x-y=Ebene, etwa in einer Keplerbahn umlaufendes Elektron handelt (wie z.B. beim ungestörten H=Atom), […] muss man dann nicht, um a _ki für einen bestimmten Übergang zu berechnen a _ki hν gleichsetzen derjenigen sekundlichen Strahlung, die ein Elektron bei der korrespondierenden harmonischen Kreisbewegung […] nach der klassischen Elektrodyanmik ausstrahlt? […] Ich stelle diese Frage deshalb, weil aus Ihrer Dissertation S. 46 […] hervorzugehen scheint, dass man a _ki hν vergleichen müsse mit der Strahlung eines linear schwingenden Elektrons, die ja nur halb so gross ist. (Sie sprechen dort von einem Elektron ‘performing a simple harmonic vibration […]’).”
32.
Kramers to Reiche, 15 May 1923 (BSC 15.1). “Mit ihren Ansichten, die gestellten Fragen betreffend, bin ich im grossen Ganzen ganz einig; an den betreffenden Punkten ist die Diskussion in meiner Dissertation leider etwas knapp gefasst.”
33.
Ladenburg and Reiche (1923, 594). As Reiche argued, Kramers’ way of estimating the values for different transitions and the experimental conditions in the Stark effect were responsible for the discrepancies.
34.
Thomas (1924) and Fues (1922a,b). One might ask here how Thomas’ work related to Sommerfeld and Heisenberg’s work on multiplets, discussed in Chap. 4. Thomas referenced their work and knew that there actually was a precessional motion of the electronic orbit around the axis of the total angular momentum of the atom. Due to the central field approximation, he argued, his method was “not sufficient” to deal with Sommerfeld and Heisenberg’s precessional motion and its underlying dynamics. See Thomas (1924, 186).
35.
Thomas (1924, 195).
36.
For Reiche’s assessment see footnote 1 in Thomas (1924, 195).
37.
Reiche to Kramers, 28 December 1923 (AHQP 8b.9). “Auf dem Wege über Herrn Pauli und Herrn Minkowski erfuhren Herr Ladenburg und ich vor wenigen Tagen von einem Einwand, den Sie gegen die Formeln 7 und 8 unserer gemeinsamen Arbeit im Bohrheft der Naturwissenschaften geäussert haben und der sich darauf bezog, dass man, zum mindesten für grossen Quantenzahlen n, ein Übereinstimmen der Zahl R der klassischen Oszillatoren mit der Zahl N der Quantenoszillatoren erwarten müsse. Mit diesem Einwand haben Sie vollkommen Recht.”
38.
Reiche to Kramers, 28 December 1923 (AHQP 8b.9). “Die Sache klärt sich, wie wir glauben, folgendermassen auf: wählt man als ‘Quantenatome’, die man mit den R klassischen (harmonischen, räumlichen) Oszillatoren vergleicht, speziell räumliche, harmonische Quantenoszillatoren, so muss man in Gleichung 4^a unserer Arbeit rechts eine Summe über alle Quantenzustände schreiben; denn wegen der strengen Harmonizität sprechen alle Quantenoszillatoren, auf welcher Energiestufe sie sich auch befinden, auf ν ₀ an.”
39.
Reiche to Kramers, 28 December 1923 (AHQP 8b.9). Note that Reiche obtained this expression in a somewhat pedestrian but essentially equivalent way. Introducing the same values for the statistical weights, he obtained the transition probability b _n+1,n from the transition probability b _n,n+1 using the relations of Einstein’s radiation theory. The transition probability b _n,n−1 then followed in the same way from the expression for b _n+1,n by considering the argument n − 1 instead of n.
40.
Reiche to Kramers, 28 December 1923 (AHQP 8b.9). “Man gelangt aber offensichtlich zu diesem Resultat nur dann, wenn man, wie es oben geschehen ist, für die Übergangswahrscheinlichkeit den Wert (vergl. Gleichung 2^a):
$$\displaystyle \begin{aligned} a_{n+1,n}=\frac{n+1}{\tau} \left(\text{wo}~\frac{1}{\tau}=\frac{8\pi^2e^2\nu_o^2}{3mc^3}\right) \end{aligned}$$

benutzt, der ohne Mittelwertbildung über die verschiedenen zwischen Anfangszustand (n + 1) und Endzustand (n) liegenden Zwischenzustände, berechnet ist, in welchem vielmehr nur der Anfangszustand auftritt.”
41.
Reiche to Kramers, 28 December 1923 (AHQP 8b.9). “Erhebt man also die Beziehung $ R=\sum \limits _{i=0}^\infty N_i $ zur Forderung, so scheint mir daraus direkt zu folgen, dass man beim harmonischen Oszillator für a _n+1,n die obige Formel für beliebige Quantenzahlen anzunehmen, also in diesem speziellen Fall von einer Mittelung abzusehen hat.”
42.
Reiche to Kramers, 28 December 1923 (AHQP 8b.9). “Jedoch vermisse ich selbst in der obigen Forderung das korrespondenzmässige Element, nämlich den Grenzübergang zu hohen Quantenzahlen. Wie denken Sie darüber?”
43.
Reiche to Kramers, 28 December 1923 (AHQP 8b.9). “Es fällt mir übrigens eben noch folgende Möglichkeit ein: wenn man von vornherein nur Quantenoszillatoren im n ^ten Zustand betrachtet und sie mit R klassischen Oszillatoren von einer ganz bestimmten Energie vergleicht, so würden wohl nur die erzwungenen Übergänge n → n + 1 und n → n − 1 in Betracht kommen. Dann erblickt man:
$$\displaystyle \begin{aligned} R\frac{\pi e^2}{m}=h\nu_0(N_n b_{n,n+1}-N_n b_{n,n-1}). \end{aligned}$$

Unter Benutzung der oben angebenen (ungemittelten) Werte […] folgt:
$$\displaystyle \begin{aligned} R=N_n\text{.''} \end{aligned}$$
44.
Reiche to Kramers, 28 December 1923 (AHQP 8b.9). “die Korrespondenzforderung würde lauten:
$$\displaystyle \begin{aligned} \lim\limits_{n=\infty}\left(\frac{n+3}{n+1}f(n)-f(n-1)\right)=3. \end{aligned}$$

Dies ist offenbar für alle f(n) erfüllt, die, für grosse n, die Form f(n) ∼ n annehmen, also z.B. für den arithmetischen Mittelwert $f(n)=n+\frac {1}{2}$, oder für den logarithmischen: $f(n)=\frac {1}{e}\frac {(n+1)^{n+1}}{n^n}$.”
45.
In light of this attempt to preserve as much of Kramers’ approach as possible, it is worth noting that there was a different line of defense, which Reiche did not consider. For the spatial harmonic oscillator, as we have seen, the equality between the number of classical oscillators and the total number of quantum oscillators did not hinge on the transition probabilities alone but rather on the cancellation of the factors arising from the transition probability and the statistical weights. This was not the case for the linear oscillator. Here, the statistical weights did not play a role so that difference expressions in Reiche’s relation only depended on the transition probabilities calculated from the correspondence principle so that the difference of the transition probabilities in two adjacent states was independent of n for both the initial state n + 1 and the Zwischenbahn $n+\frac {1}{2}$. Considering the spatial and the linear harmonic oscillator, one could thus have concluded that the correspondence approach failed for degenerate systems but remained valid for nondegenerate ones or that the calculation of the statistical weights needed to be amended. This was done by Edwin Kemble in his work on band spectra at the same time; Kemble (1924, 1925a,b). For Kemble, the solution to the problem was to construct an average value for the statistical weight of the degenerate system by considering it as the sum of nondegenerate systems. While he took the linear harmonic oscillator into consideration, Reiche did not discuss this point but rather developed his own attempt to save the Zwischenbahn model, which was certainly not the only possible response to the inconsistency he had encountered. See Reiche to Kramers, 28 December 1923 (AHQP 8b.9).
46.
This reception can be reconstructed from a letter from Reiche to Kramers on 9 April 1924 (AHQP 8b.9).
47.
Reiche to Kramers, 9 April 1924 (AHQP 8b.9) Reiche only gave the Fourier series for a multiply periodic system and the classical dispersion formula. For reconstructions of this classical derivation see Darrigol (1992, 225–228) and Duncan and Janssen (2007b, 646–652), as well as Konno (1993, 122–123) for a shorter exposition of Kramers’ derivation.
48.
For the details of this translation procedure see Duncan and Janssen (2007b, 635–637). A discussion of the problems encountered by Kramers in developing the formula is given in Konno (1993).
49.
This point allows an interesting side remark on the reconstruction of Kramers’ work. As is well known, Kramers did not mention the replacements of differentials by differences in his first note to Nature and only briefly mentioned it in his second note. See Kramers (1924a,b). As Slater and with him Dresden and Konno have stressed, Kramers already had the central idea of translating differentials into differences in January 1924. See Slater (1975, 15), Dresden (1987, 155) and Konno (1993, 125). The self-evidence of Reiche’s recapitulation of the argument strengthens this point and suggests that Kramers had communicated a short description of his derivation based on the replacement of differentials by differences in his letter to Reiche.
50.
Reiche to Kramers, 9 April 1924 (AHQP 8b.9, my emphasis). “Ich habe mir, im Anschluss an Epsteins Arbeit, unter Benutzung der Born-Paulischen Methode, den klassischen Ausdruck für P, den Sie in Ihrem Brief angeben, leicht abgeleitet und mir auch den korrespondenzmässigen Übergang zu der Quantenformel ohne weiteres klar machen können.”
51.
Reiche to Kramers, 9 April 1924 (AHQP 8b.9). Reiche also discussed questions concerning Reiche and Ladenburg’s arguments on the equality of the total scattering energy [Gesamtstreuung] and absorption, which Kramers felt was not valid. Defending his and Ladenburg’s former considerations, he considered the polarization to be complex and incorporated a damping term into the classical expression for the polarization. He then considered the imaginary part associated with absorption according to classical radiation theory, and made the transition to the corresponding quantum formula to reproduce his former results within Kramers’ approach. For the present reconstruction, this argument did not play a role, as Reiche did not connect it with the determination of transition probabilities at the time.
52.
Reiche to Kramers, 9 April 1924 (AHQP 8b.9). “Die Bemerkung in Ihrem Briefe, dass man aus dem Grenzfall ν = 0, also eines konstanten elektrischen Feldes, Aufschluss über die Werte der A gewinnt, habe ich leider bisher nicht ganz verstanden.”
53.
Ibid.
54.
Moreover, due to Kramers’ derivation of the dispersion formula, the central model in Reiche’s consideration was no longer the degenerate spatial harmonic oscillator but the nondegenerate linear oscillator. Thereby the statistical weights, which had been central in Reiche’s previous considerations, came to play a secondary role.
55.
In principle, this ground-state argument would have been possible already on the basis of Reiche’s earlier considerations on absorption. Since he conceived the relation between transition probabilities as a consistency check for transition probabilities, this possibility did not occur to him.
56.
Reiche’s calculations are mentioned but not discussed in a letter from Ladenburg to Kramers, 31 May 1924 (AHQP 8.9). Otherwise, the AHQP does not contain letters from Reiche to Kramers after April 1924 or any other material pertaining to his subsequent work. The Reiche papers at the AIP contain little material dating from the 1920s. Reiche’s dismissal from Breslau in 1933 and his escape from Germany in 1941, which has been described by Bederson (2005), make it likely that his notebooks and correspondence are no longer extant. Willy Thomas, as reported by Reiche in his oral history interview with Kuhn, died young of tuberculosis. His papers, if they existed, could not be traced. See interview with Fritz Reiche by Thomas S. Kuhn on 9 May 1962, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA, www.aip.org/history-programs/niels-bohr-library/oral-histories/4841-3. [Accessed on 21 March 2019: 11:38]
57.
For the negotiations between Copenhagen and Breslau, see Ladenburg to Kramers, 31 May 1924, Kramers to Ladenburg, 3 June 1924, Kramers to Ladenburg, 5 June 1924 and Ladenburg to Kramers, 8 June 1924 (all given in AHQP 8b.9). Pointing out that Reiche had begun to work on hydrogen, Ladenburg assured Kramers that they would cease to pursue the issue if Kramers or Bohr wanted to work on the problem simultaneously. Kramers took up the offer and asserted that it was his and Bohr’s “intention to think this point […] through ourselves, or to give it to some of the gents at the institute.” Kramers to Ladenburg, 3 June 1924 (AHQP 8b.9). Kramers eventually did not consider the problem himself prior to the advent of matrix mechanics and offered the problem to Werner Kuhn, who arrived in Copenhagen from Zurich in the spring 1925. Kuhn’s approach to the problem differed remarkably from the one of Reiche and Thomas. Reiche and Thomas arrived at the f-sum rule following a derivation from the general framework of multiply periodic systems and the new correspondence techniques. Kuhn, on the other hand, argued solely on the basis of the dispersion formula in the limit of high frequencies of the external radiation.
58.
Thomas (1925) and Reiche and Thomas (1925).
59.
See Assmus (1993) for a more detailed discussion of post-doctoral education in the U.S. before World War II.
60.
See Duncan and Janssen (2007a, 561).
61.
This is also suggested by Van Vleck’s own comments on the f-sum rule in his 1926 Bulletin for the National Research Council “Quantum Principles and Line Spectra,” in which he claimed to have anticipated the f-sum rule in his 1924 paper. Here, Van Vleck did not mention any personal communication with Reiche and Thomas and acknowledged that their work had developed an argument that was well beyond the scope of his own considerations on the subject. See Van Vleck (1926).
62.
Van Vleck (1924b). See Duncan and Janssen (2007b, 640–643) for a detailed reconstruction of Van Vleck’s calculations leading to this result.
63.
Note that Van Vleck considered an individual system, whereas Planck discussed the absorption of R systems.
64.
Van Vleck (1924b, 359).
65.
Van Vleck (1924b, 359).
66.
Van Vleck (1924b, 359–360).
67.
Reiche and Thomas (1925, 520–521).
68.
Reiche and Thomas (1925, 514–515). The energy emitted by a number of N _a degenerate systems in the transition from an upper state to a lower state: $S=N_a \bar {a_a} h\nu _a$ had to be identical to the energy S ^∗ emitted by the same number of nondegenerate systems making transitions from a g _a-fold upper level to the g-fold lower level. To establish the total energy, Reiche and Thomas first considered the transitions from the g _a-fold upper levels to one of the lower levels given by $S^*=\frac {N_a}{g_a}\sum \limits _ma_ah\nu _a$ and summed up these transitions according to the “Ornstein-Burger-Dorgelo-sum rules,” which implied that the energy radiated in a transition to one lower state was identical for all g-sublevels. In the transition from the f-sum rule for nondegenerate systems to the f-sum for degenerate system one thus had to take the sum of all the transition probability a _a of the nondegenerate system and replace them by: $\sum \limits _m a_a=\frac {g_a}{g}\bar {a_a}$ For the inverse transition, on the other hand, the situation was different. Instead of $\frac {N_a}{g_a}$-systems making the transition from the upper state, one had now to consider $\frac {N}{g}$-systems making the transition from one of the lower states thus radiating the energy $S^*=\frac {N}{g}\sum \limits _ma_eh\nu _e$ so that taking the Ornstein-Burger-Dorgelo-sum rule into account the transition probability of the degenerate system was identical to the sum of the transition probabilities of nondegenerate systems. $\sum \limits _m a_e=\bar {a_e}$ Combining these two results, Reiche and Thomas obtained their f-sum rule for degenerate systems.
69.
Reiche and Thomas (1925, 511). “Wir wollen im folgenden für die Gesamtheit der f (und damit der Ü[bergangs]w[ahrscheinlichkeiten]), die einem stationären Zustand zugeordnet sind, einen Satz ableiten, der keine mechanischen Symbole mehr enthält und der, wenn er sich in dieser Form bewährt, ein Permanenzgesetz der Ü.W. genannt werden könnte.”
70.
Reiche (1926b).
71.
Reiche (1926a, 1929) and Rademacher and Reiche (1927).
72.
Such a comparison was made by Duncan and Janssen in the case of John H. Van Vleck. As they concluded, Van Vleck was “on the verge of Umdeutung” but did not take the next step. Asking why he did not, they argue that Van Vleck “was too wedded to the orbits of the Bohr-Sommerfeld theory to discard them.” (Duncan and Janssen 2007b, 665). Following their assessment, the absence of any reference to a program of finding a new quantum mechanics, and Van Vleck’s own recollections that he would have to have been a lot more “perceptive” to come up with such a new theory, I am inclined to draw a more radical conclusion. Like other physicists discussed in this book, there was no next step for Van Vleck to take as he did not think about dispersion as the key for a new quantum mechanics, but as a particular physical problem to be solved within a given framework.

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Institut für Philosophie, Literatur-, Wissenschafts- und Technikgeschichte Fachbereich Wissenschaftsgeschichte, Technische Universität Berlin, Berlin, Germany
Martin Jähnert
Max Planck Institute for the History of Science, Berlin, Germany
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Jähnert, M. (2019). That I Cannot Conceive of After the Results of Your Dissertation: Fritz Reiche and the F-sum Rule. In: Practicing the Correspondence Principle in the Old Quantum Theory. Archimedes, vol 56. Springer, Cham. https://doi.org/10.1007/978-3-030-13300-9_6

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