Quantum Physics in Wroclaw (1917-1945)

Helmut Rechenberg, Max-Planck-Institut für Physik, Munich

Quantum Physics in Wroclaw (1917-1945)

Introduction

Following his Ph.D exam in Göttingen (in summer 1906), several months of military service (from which he was released in spring 1907 because of asthma troubles) and finally a postgraduate study at Cambridge University (with Joseph John Thomson), Max Born returned in fall 1907 to the University of Wroclaw to get into active research in physics. He remembered then getting into the institute run by the new professors Otto Lummer and Ernst Pringsheim, who in spite of their different personalities – Lummer a blond Aryan full of life and flamboyant in temper, Pringsheim of Jewish origin, a quiet thinker, modest and elegant in manners  cooperated extremely well, behaving like an inseperable couple. Born also got acquainted with the assistants and research students at the physics institute, and he described them as follows:

“Lummer’s assistant, his right-hand man and friend was Erich Waetzmann ...

A nice friendly mild chap he would do anything to help anybody. He was on bad terms with one man only, his colleague Clemens Schaefer, a lecturer and also assistant. We others, mainly Ladenburg, Reiche, Loria and myself, tried to keep out as much as we could. I knew (Rudolf) Ladenburg from my

childhood. ... We quickly became friends. ... Fritz Reiche was one of the very few research pupils of Max Planck ... a tiny very delicate Jew who combined the typical humour of the Berliner with a deep melancholy and pessimism. ... I learned from him a great deal about radiation and quantum theory which he had studied at the source, in personal contact with Planck. ... The last of our group, Stanislaus Loria, was a Pole from Cracow ... highly educated who was proud of his nationality, a most charming man with fine features and perfect manners.“ 1

The four musketeers Born, Ladenburg, Loria and Reiche met regularly and discussed intensively about their laboratory work and especially about the new theories of relativity and the quantum, in which they all took deep interest. Born soon moved back to Göttingen and joined in fall 1908 Hermann Minkowski in investigating relativistic electrodynamics, while the others remained for a while in Wroclaw: Ladenburg until 1924, Reiche until 1912 – both went to Berlin, but Reiche returned in 1921 (before Ladenburg left). Loria came back to the

University of Cracow in 1910 as lecturer, but also he showed up again in

Wroclaw, however only after World War II, now as professor of physics in the

new Polish University.

I. Rotating Carousel and Point of Stability – The Successors of Pringheim and Lummer

On 28 June 1917 Ernst Pringsheim died very suddenly. Clemens Schaefer immediately took over his lecture course in the winter semester 1917/18 (on the theory of electricity and magnetism) and became his successor but changed three years later on an experimental chair at the University of Marburg. Then Erwin Schrödinger was invited, who gave only a short starring in summer 1921;

he was succeeded by Fritz Reiche. The take-over of power by the Nazi regime in Germany forced Reiche to leave his position in 1933, and in 1934 Erwin Fues – by the way the first contributor other than Schrödinger to wave mechanics – took over the theoretical professorship for the next decade. On the other hand, when Otto Lummer died in July 1925, Clemens Schaefer was immediately called back from Marburg to be the last experimental Ordinarius at the German Breslau University. We now summarize the curricula of the physics professors mentioned.

Clemens Schaefer was born on 24 March 1878 in Remscheid, Rhineland, and started to study physics and mathematics at the University of Bonn, changing after three semesters to Berlin, where in 1900 he obtained his Ph.D.

with Heinrich Rubens at the TH (Technische Hochschule) Charlottenburg. In 1903 he became Privatdozent (lecturer) at the University of Breslau; he was promoted further in 1910 to an associate professorship, and in 1917 he took over the theoretical chair. Following an absence from 1920 to 1926 as director of the physics institute at the University of Marburg, he returned to Wroclaw on the same position. Driven westwards after World War II, he served as a professor of physics at the University of Cologne (1946-1950). He died on 9 July 1968 in Cologne.

For Erwin Schrödinger, born in Vienna on 12 August 1887 and graduated in 1910 at the University of Vienna (where he also obtained his Habilitation in 1913). After military service in World War II at the Italian front he occupied associate professorships at the University of Jena and the TH Stuttgart. The position at Wroclaw was his first full professorship, a big step on the academic ladder carrying him further to the University of Zurich (1921-1927) and in 1927 on Max Planck’s chair at the University of Berlin. In 1933 he left voluntarily Nazi Germany, staying for three years at Oxford University and then returning as a professor to his home country (at the University of Graz 1936-1938). On being dismissed after the Austrian Anschluß to the Third Reich, he escaped via Belgium to Ireland, where he assumed a leading position in the newly founded Institute for Advanced Studies in Dublin (1940-1956). After his retirement he returned to the just freed Austrian Republic and got a professorship at the University of Vienna, where he died on 4 January 1961.

Fritz Reiche, his successor in Woclaw, was born on 4 July 1883 in Berlin

and studied first at the University of Munich phyics (with Wilhelm Conrad

Röntgen), chemistry (with Adolf von Baeyer) and mathematics (with Ferdinand

von Lindemann). He then continued at the University of Berlin and obtained his Ph.D. with Planck in 1907, who recommended him to Lummer in Wroclaw for further experimental studies. Returning in 1912 as Privatdozent to Berlin and since 1918 as scientific collaborator of Fritz Haber’s Kaiser Wilhelm-Institut für physikalische und Elektrochemie, he stayed until 1921. Upon having been fired from his Wroclaw professorship in 1933, he spent a year as visting professor at the German University of Prague and then lived privately in Berlin. In 1941 he managed to emigrate (with the help of Planck) to the United States, where he taught at several colleges of New York, before receiving finally an adjunct professorship at New York University (1946-1958). He died on 15 January 1958 in New York.

The last theory Ordinarius at the University of Breslau, Erwin Fues, was born on 17 January 1893 in Stuttgart on 17 January 1893, where he also died exactly 77 years later on 17 January 1920. He studied physics from 1912 to 1914 at the universities of Berlin and Munich, continuing (after two years of military service) in Tübingen (1916-1918) and again Munich, receiving his Ph.D. with Sommerfeld in 1919. In 1922 he joined Paul Ewald at the TH

Stuttgart and became there a Privatdozent in 1924. From 1925 to 1927 he held a fellowship of the International Education Board in Zurich (with Schrödinger) and Copenhagen (with Bohr). In 1929 he was called to occupy a chair in applied mathematics at the TH Hanover – there he established a seminar in theoretical physics. From Hanover and Göttingen, where he took over in 1933/34 the lecture courses for the displaced Max Born, he was called in 1934 as professor of theoretical physics to Wroclaw and changed in 1943 to the University

Vienna. After World War II he obtained in 1947 a chair at the TH Stuttgart (and a honorary professorship in Munich).

II. Teaching and Research of the Professors (1917-1945)

II.1 Schaefer’s theoretical and expermental lectures and textbooks; his work on infrared spectra

Of all the Wroclaw professors treated here, Clemens Schaefer must be considered as the most versatile one, not only because he investigated both experimental and theoretical topics and lectured on them during his long

professional career. Indeed, he started out as an experimentalist until obtaining

his Habilitation in 1903: at that time he came to Wroclaw to support the old

Oskar Emil Meyer in his laboratory work and teaching. Notably, he announced

lecture courses on theoretical topics (such as Maxwell’s electrodynamics in the

summer semester 1904, and mechanics in the winter semester 1904/05). Ernst

Pringsheim, who arrived in fall of 1905 to take over the new position as theory

Ordinarius, established a systematic series of five successive courses, starting

with mechanics of mass points, then continua mechanics, electrodynamics,

optics and finally thermodynamics. Schaefer complemented this series by

lecturing on special topics; he later occasionally alternated with Pringsheim in giving the standard courses. From this practice his textbooks Einführung in die theoretische Physik (Introduction to Theoretical Physics) emerged, of which Volume I (on mechanics) was published first in 1914 (by Walter de Gruyter, Berlin), Vol. II (on heat theory and kinetic theory of matter) in 1920, and finally Vol. III, Part 1 (on electrodynamics and optics) in 1932 and Part II (on quantum theory) in 1937. As a director of the experimental physics institute in Marburg and Wroclaw, Schaefer was of course obliged to deliver the big introductory courses in this field; they were converted into a second series of textbooks, written together with his former student and associate Ludwig Bergmann: the Lehrbuch der Experimentalphysik, of which Vol. I was dealing with mechanics, acoustics and heat (Walter de Gruyter, Berlin, published first in 1943), Vol. II with electricity and magnetism (published in 1953), and Vol. III (in parts) with optics and atomic physics (published since 1956).

Schaefer contributed on various topics of theoretical and experimental

research. In his early Wroclaw times he studied, e.g., in some detail the passage of electromagnetic waves through a lattice of wires or the validity of the

Lorentz-Einstein relativistic mass formula. Since 1920 he returned to

investigations on infrared spectroscopy (the topic of his Habilitation): thus he studied the absorption spectra of molecules and molecular groups in crystals, measured the reflection power of infrared radiation from solids and determined the eigenfrequencies, especially the optically inactive ones, from combination oscillations. The monograph on the infrared spectrum, composed with Frank Matossi (his Breslau assistant from 1928 to 1938), summarized an essential part of Schaefers research revealing the structure of molecules and crystals 2.

Another set of experimental investigations he carried out between 1933 and 1939 jointly with Ludwig Bergmann, namely, the diffraction of light from ultrasonic waves; the data thus obtained led to a comfortable determination of the elastic constants of crystals and glasses.

II.2 Fritz Reiche and the quantum theory

Also the scientific work of Fritz Reiche, the other long-resident professor in Wroclaw, extended over theoretical and experimental problems. During his early assistant years he helped Otto Lummer to edit the unpublished lectures of the optician Ernst Abbe on the image formation in a microscope and

collaborated with him to prove experimentally the fundamental optical cosine-

law of Johannes Lambert. Further he supported Rudolf Ladenburg in examining

the energy distribution of spectral lines emitted and absorbed by atoms. After his

return to Berlin he worked on a wide range of quantum-theoretical questions,

publishing papers on paramagnetism of atoms, the quantization of the spinning

top, the explantation of X-ray spectra of atoms and the specific heats of solids,

often in collaboration with the local experimentalists and theoreticians, such as

James Franck, Hartmut Kallmann, Ladenburg and Adolf Smekal. In his

pioneering book on Die Quantentheorie in ihrer Entwicklung he provided the first summary of all known aspects of the theory developed since Planck’s discovery in 1900 3. When in the following years this “old quantum theory“

increasingly failed to account for the observed atomic phenomena, e.g., in the spectrum of many-electron atoms, Reiche helped to create with his friend Rudolf Ladenberg and his student Willy Thomas the dispersion-theoretical approach in the quantum theory of atomic spectra, which constituted one of the paths leading directly to the new atomic theory, i.e., quantum mechanics. The Nazi times interrupted his research and teaching until he escaped to America; there he turned to new problems connected with classical fluid dynamics.

Perhaps we should add here a word on his predecessor Erwin Schrödinger, who did not deal in his short Wroclaw period at all with problems of quantum and atonic theory (as he had done before in Vienna and Stuttgart and would do in Zurich and Berlin afterwards). But he accepted a request of Otto Lummer to write the extended article on „Gesichtsempfindungen (i.e., visual sensations)“

for the 11th edition of Müller-Pouillet’s Lehrbuch der Physik, which provided an excellent account of the status of physiological optics including especially color theory  Schrödinger had contributed pioneering papers himself on this topic before 4.

II.3 Erwin Fues and wave mechanics

Fues had been introduced into atomic theory by Sommerfeld. Since the early 1920s he submitted regularly papers on spectra of many-electron atoms:

especially he tried out various potentials to improve their description in terms of the old Bohr-Sommerfeld model. Upon having seen Schrödinger’s first

publication on wave mechanics, he applied it instantly to calculate the energy states and line intensities of diatomic molecules and he became an expert in modern atomic theory. During his Wroclaw period he produced only few scientific papers, often connected with the experimental research of Schaefer and his associates. However, he also published two valuable articles as a supplementary Volume to the well-known Wien-Harms Handbuch der Experimentalphysik : one on diffraction experiments with matter waves, the other providing a very pedagogical introduction to quantum mechanics from the wave-mechanical point of view 5.

In working out the latter he received the help of Fritz Bopp (1909-1987),

who showed up in his institute – he used to tell later: „Ich ging zu Fuß zu Fues

nach Breslau (I went on foot to Fues in Brelau)“ – already before obtaining his

Ph.D. in 1937 at the University of Göttingen (he knew Fues from his Göttingen

lectures). Later Bopp became assistant of Fues (1936-1941) and finally at the

university (1941). Since he was also a collaborator in the secret German

uranium project of World War II, he announced his first lecture course in

Wroclaw only in summer 1943. By that time Fues was about to leave for

Vienna; since he was not replaced, Bopp was the last representative of theoretical physics at the German University Breslau. After Word War II he succeeded (in 1950) Arnold Sommerfeld on the theoretical chhair at the University of Munich.

In conclusion we can say that in Wroclaw the tradition of excellent teaching in physics, established earlier by Lummer and Pringsheim, was certainly

continued by their successors Schaefer, Reiche and Fues and their associates, who included Ludwig Bergmann, Fritz Bopp, Frank Matossi, Ferencz Jüttner and Walter Steubing. A further Breslau speciality, also going back to the times of Lummer and Pringsheim was continued by their heirs: teaching and research on problems of physical optics and its application to technology.

III. Dispersion Electrons – A Path to Quanteum Mechanics

The story of the greatest Wroclaw contribution to quantum physics involves the close cooperation of three scientists: Rudolf Ladenburg, Fritz Reiche and Willy Thomas.

Rudolf Ladenburg was born on 6 June 1882 in Kiel, the son of the chemistry professor Albert Ladenburg (who in 1889 joined the University of Breslau) and his wife Margarethe, daughter of the famous Berlin botanist Nathanael Pringsheim (originating from Silesia). He studied physics at the universities of Heidelberg (1900-1901), Breslau (1901) and Munich (1902- 1906). On obtaining his doctorate with Wilhelm Conrad Röntgen in 1906, he spent a year at Cambridge University before joining Lummer in Wroclaw as an assistant. In 1909 he became Privatdozent, in 1921 he was promoted to an extraordinary professorship – during the World War I years he served first with the cavalry at the front and later by directing a department of the Artillerie- Prüfungskomission (Artillery Test Commission in Berlin). In 1924 Fritz Haber invited him back to Berlin to lead the physics department at his Kaiser Wilhelm Institute in Berlin. Finally Ladenburg went in 1931 to the USA, succeeding Karl T.Compton as professor of physics at Princeton University. He died on 3 April 1952 in Princeton. A pioneer of studying the magneto-optical phenomena and the dispersion of light by atoms, he turned in America to work on neutron physics, and after World War II he published papers on supersonic phenomena.

Like previously in Berlin, Fritz Reiche accompanied and supported Ladenburg’s research in Wroclaw, where he also taught and supervised the student Willy Thomas. We do not know much about Thomas life, except that he got his Ph.D. in 1924 with a remarkable thesis, upon which he published in 1925 two weighty articles, which we shall discuss below. Finally, we learn from the preface of Clemens Schaefer to the 3rd edition of Vol. I of his books on

theoretical physics, signed in August 1928 (where he referred to Thomas for many discussions and improvements), that:

“I sadly regret that I can only send my thanks to this highly gifted young

scholar in his grave. He died of an operation, from which he had hoped to recover completly from his sufferings.“

Since Thomas did not publish after 1925, we may assume that the illness mentioned had prevented him to pursue actively further research.

III.1 Ladenburg’s dispersion electrons (1907-1928)

When he came in 1907 to the University of Breslau, Ladenburg began a program of detailed investigations to test the available (classical) description of light by atoms. Thus he concluded, in a joint paper of 1908 with his friend Stanislaw Loria, from measuring the anomalous dispersion of hydrogen a very small value for the number of “dispersion electrons“, in contrast to Planck’s earlier

prediction of 1902 6. In 1914 he suggested a different method to determine the number of dispersion electrons and found, in the case of light scattering from sodium atoms, a far more resonable result, i.e., 40 dispersion electrons per 100 Na atoms 7. During his military service in Berlin he learned to know

thoroughly quantum-theoretical considerations from the masters Planck and Einstein and also from his friend Reiche, and he applied their theoretical ideas in his post-war research. The most influential publication became the article, which he submitted in February 1921 to Zeitschrift für Physik and entitled (in English translation): “The quantum-theoretical interpretation of the number of dispersion electrons)“ 8.

In particular, Ladenburg used Einstein’s 1916 quantum theory of emission and absorption of radiation in order to obtain a new expression for the number of electrons contributing the dispersion process. This expression involved only quantum-theoretical concepts, i.e.,

N = N

( g

/g

) m c

.  8

e





a

, (1)

where the a

denote Einstein’s emission coefficients, g

and g

the statistical weights of the atomic states i and k, N

the number of electrons in the state i, and



the transition frequency ( m and c the mass and velocity constant of light). In the detailed review paper with Reiche of 1923, entitled (in English translation)

“Absorption, scattering and dispersion in Bohr’s atomic theory“, the authors arrived at a relation between the dispersion-theoretical coupling, i.e. the so- called oscillator strength f

, and the Einstein coefficients a

, notably,

f

= a

.  , (2) with  denoting the damping constant known from classical dispersion theory

9. In 1924 Hendrik Kramers in Kopenhagen used the Ladenburg-Reiche

results and proposed a new equation for the quantum-theoretical amplitude P of

light (having the electric vector E), in which he replaced the classical expression

P = E  f

e

/m  4

 (

)

-

 

(3a) by the more complex quantum-mechanical expression

P = E  f

e

/m 4  (

)

- 

 

 E  f

e

/m  4 (

)

- 



(3)

10. This equation extended the one considered by Ladenburg and Reiche earlier by adding the second sum involving “negative oscillator strengths -f

“ (due to Einstein’s stimulated emission of light), and it also represented the right relation according to Niels Bohr’s “correspondence principle“, because passes over in the proper limit of large quantum unmbers into the classical relation (3a).

One should add two points here. First, it took Ladenburg and his later collaborators in Berlin, including Agathe Carst and Hans Kopfermann, several years to establish experimentally the existence of the negative oscillator term (1928) 11. Second, Kramers and Werner Heisenberg derived – in a paper published in early 1925 – an extension of Kramers’ dispersion formula (3) for the case of the incoherent scattering of radiation by atoms: they obtained different sums of more complicated terms for the various special cases of the scattering of light by atoms, in which products of transition amplitudes to and from intermediate states occurred; finally a scattered frequencies different from incident one (by the addition and subtraction of atomic frequencies) resulted

12. The the Kramers-Heisenberg formula of 1925 then predicted, e.g. the so- called Raman effect, found 1928 by Venkata Chandrasekhara Raman and K. S.

Krishnan in India.

III. 2 The Thomas-Reiche-Kuhn sume rule and Heisenberg’s quantum condition (1925)

In March 1924 Willy Thomas submitted an excerpt from his Ph.D. thesis in the paper on an “Approximate calculation of orbits and transition probabilities of series electrons in sodium atoms“ to Zeitschrift für Physik 13. Since the quantum laws for many-electron atoms were not yet known, the author

suggested to compare the Fourier coefficients calculated for the electron orbits

due to the Bohr-Sommerfeld model (by the way, he also used Fues’ potential for

the sodium atom!) with the transition amplitudes derived from the dispersion

measurements of Ladenburg and collaborators (including Rudolf Minkowski, a

nephew of Hermann Minkowski). Thomas concluded that the two values thus

obtained agreed at least in order of magnitude. In the following year he went on

with his dispersion-theoretical research, and on 18 June 1925 he signed a letter

to the weekly German journal Die Naturwissenschaften, which was published in

the issue of 7 July under the title: „On the number of dispersion electrons

associated with a stationary state“ 14. In this short note he suggested, for a non-degenerate periodic atomic sytsem containing one electron, the relation  f

  f

= degree of periodicity. (4) Thomas expressed this relation in the classical theory „anschaulich (visually“

like this: If one multiplies the sum of the coupling strengths for absorption minus the the sum of the coupling strengths for emission by a factor, one obtains the number of dispersion electrons.

At about the same time as he submitted the letter to Naturwissenschaften, Werner Heisenberg achieved in Helgoland his break-through to creating

quantum mechanics by what he called “the quantum-theoretical interpretation of kinematic and mechanical relations“ – so also the translated title of his

pioneering paper emerging from it (and submitted in July 1925) 15. In particular, he proposed now a re-interpretation of the (semi-classical) Bohr- Sommerfeld quantum condition, i.e., the phase integral  p dq = n h. That is, he rewrote it for a periodic motion (frequency 

) of a particle, having mass m , the action variable J = nh and the Fourier coefficients a

, as

1 = 2 m   d/dn ( 

a



), (5a) with the -sum extending over all integers from minus infinity to plus infinity.

Thus classical expression he translated according to the Born-Heisenberg discretization rule of 1923 as

h = 4 m    a(n,n)

(n,n)  + a(n, n-)

(n,n-)  , (5)

where the -sum now extended from 0, 1, 2 to infinity. Heisenberg noticed explicitly in a footnote that relation (5) was supported empirically by Thomas’

spectroscopic sum rule for the oscillator strengths, Eq.(4), or a similar one derived by the Swiss research fellow Werner Kuhn (whom he had met in spring 1925 in Copenhagen) 16. Soon afterwards Max Born and Pascual Jordan in Göttingen would express Heisenberg’s new quantization rule in their matrix scheme, derived from Heisenberg’s pioneering paper: namely, as the

“commutation rule“ for the quantum mechanical momentum p and position q,

p q – q p = h/2 i , (6)

which constitutes the fundamental formula in the new quantum mechanics 17.

Let us add here the observation that Heisenberg’s keen procedure in deriving

his quantum-mechanical relations in May and June 1925 was the same which

also Willy Thomas sketched in his letter to Naturwissenschaften as the path to

derive his Eq.(4). That is, both referred to a “suitable re-interpretation of the

classical quantities as quantum-theoretical ones and replacing (some) differential quotients (in classical dynamical equations) as difference quotients“. Evidently, Thomas sum rule represented an special case of Heisenberg’s Eq.(5), or the Born-Jordan commutation relation (6). In August 1925 then Fritz Reiche and Willy Thomas submitted a detailed paper to Zeitschrift für Physik, in which they generalized the sum rule (4) to degenerate atomic systems, e.g., having two equivalent electrons 18; therefore it is often referred to as the “Thomas- Reiche-Kuhn sum rule“. Reiche and Thomas also established the connection between the total number of dispersion electrons associated with a given atomic energy state to the degree of periodicity s of the atomic system considered: this number is equal to s/3. In any case, Thomas’ Eq.(4) represents the first example of a sum rule in atomic theory. In the 1950s and 1960s such sum rules for

various quantities would play an important role in microphysics, especially to obtain relations for the properties of strongly interacting elementary particles.

In the development of quantum mechanics, from the previous „old quantum theory“ the dispersion-theretical approach, inaugurated by Rudolf Ladenburg played a prominent role 19. Thus Bartel Leendert Van der Waerden in his Sources of Quantum Mechanics 20 consitently started out with discussing this approach and reprinted, as the first paper, a translation of Ladenburg’s

pioneering paper 8. This evaluation tells enough about the importance of the work of Ladenburg, Reiche and Thomas in the history of physics. It certainly was a highlight in the development of modern atomic theory, and at the same time an outstanding contribution of the physicists in the German period of Breslau Univerity .

References

1 M. Born: My Life - Recollections of a Nobel Laureate. Taylor & Francis and Charles Scribners’s Sons, London and New York 1978, pp.123-124.

2 C. Schaefer and F.Matossi: Das ultrarote Spektrum. Springer, Berlin 1930.

3 F. Reiche: Die Quantentheorie in ihrer Entwicklung. Julius Springer, Berlin 1921; English translation: The Quantum Theory. Methuen, London 1922.

4 E. Schrödinger. Gesichtsempfindungen. In O. Lummer, ed.: Müller-

Pouillet’s Lehrbuch der Physik,11th edition, Vol. II/1. Fr. Vieweg & Sohn, Braunschweig 1926, pp. 456-561.

5 E. Fues: Beugungsversuche mit Materiewellen; Einführung in die Quantenmechanik. In W. Wiens and F. Harms, eds.: Handbuch der Experimentalphysik, Suppl. Volume. AVG, Leipzig 1935.

6 R. Ladenburg and S. Loria. Über die Dispersion des leuchtenden Wasserstoffs. Physikalische Zs. 9, 875-879 (1908).

7 R. Ladenburg: Über die Zahl der an der Emission von Spektrallinien

beteiligten Atome. Verhandlungen d. Deutsch. Physik. Ges. (2) 16, 765-779

(1914).

8 R. Ladenburg: Die quantentheoretische Deutung der Zahl der Dispersionselektronen. Z. Physik 4, 451-468 (1921).

9 R. Ladenburg and F. Reiche: Absorption, Zerstreuung und Dispersion in the Bohrschen Atomtheorie. Naturwissenschaften 11, 584-598 (1923).

10 H. Kramers: The law of dispersion in Bohr’s theory of spectra. Nature 113, 673-674 (1924).

11 A. Carst and R. Ladenburg: Untersuchungen über die anomale Dispersion des Wasserstoffs. Z. Physik 48, 192-204 ( 1928).

12 H. Kramers and W. Heisenberg: Über die Streuung der Strahlung durch Atome. Z. Physik 31, 681-708 (1925).

13 W. Thomas: Näherungsweise Berechnung der Bahnen und

Übergangswahrscheinlichkeiten des Serienelektrons von Natriumatomen.

Z. Physik 24, 169-196 (1924).

14 W. Thomas: Über die Zahl der Dispersionselektronen, die einem

stationären Zustande zugeordnet sind. Naturwissenschaften 13, 627 (1925).

15 W. Heisenberg: Über die quantentheoretische Umdeutung kinematischer und mechanischer Beziehungen. Z. Physik 33, 879-893 (1925).

16 W. Kuhn: Über die Gesamtstärke der von einem Zustande ausgehenden Absorptionslinien. Z. Physik 33, 408-412 819259.

17 M. Born and P. Jordan: Zur Quantenmechanik. Z. Physik 34, 858-888 (1925).

18 F. Reiche and W. Thomas: Über die Zahl der Dispersionselektronen, die einem stationären Zustand zugeordnet sind. Z. Physik 34, 510-525 (1925).

19 J. Mehra and H. Rechenberg: The Historical development of Quantum Theory. Springer Verlag, New York, Heidelberg, etc., 1982-2001, especially Vol. 1(1982), 2(1982) and 6/1 (2000).

20 B. L. Van der Waerden: Sources of Quantum Mechanics, North Holland, Amsterdam 1967.