## On the Quantum Theory of Line-Spectra, I-III.

Copenhagen: Bianco Lunos, 1918-22.

First edition, very rare presentation offprints (with ‘Separate Copy’ printed on the front wrappers), **inscribed by Bohr to Kasimir Fajans**, and from the collection of The Garden Ltd., of this fundamental work in which Bohr first gave a clear formulation of, and fully utilized, his ‘correspondence principle’. Besides his ‘On the constitution of atoms and molecules’ (1913), which inaugurated the application of quantum theory to atomic structure, this is by many considered to be Bohr’s greatest contribution to physics. “[T]here was rarely in the history of physics a comprehensive theory which owed so much to one principle as quantum mechanics owed to Bohr’s correspondence principle” (Jammer, p. 118). “Bohr’s treatise … is unified by the fundamental assumption of an analogy between the quantum theory and classical electrodynamics, a point of view which a couple of years later came to be called the ‘correspondence principle’ … [T]his principle … was destined to serve as the principal guide to the progress of the quantum theory in the early twenties, and which later was built into the quantum mechanics” (Nielsen, p. 4). Bohr’s underlying idea was that the quantum theory must satisfy in the limiting cases, e.g., when frequencies *v* tend to zero or quantum numbers *n* tend to infinity, that its predictions approximate those of classical physics. When studying different quantum systems one can thus utilize already established facts from what classical physics predicts in that particular system, and then work backwards to arrive at new quantum theoretic rules for the system. *On the Quantum Theory of Line-Spectra* “was to have been divided into four parts. Part I: *On the general theory* was finished in November 1917, and was ready from the printer on April 27, 1918. Part II: *On the hydrogen spectrum* appeared in print on December 30, 1918. Part III: *On the elements of higher atomic number* was greatly delayed because of the rapid development of the field, by Bohr and others, until it was finally printed in November 1922 together with an Appendix dealing with later work and serving to bring the treatise to a natural conclusion. Part IV, in which the general theory was to have been applied to the constitution of atoms and molecules, was never completed. The treatise was published as a Memoir of the Royal Danish Academy of Sciences and Letters. This medium of publication, prompted by the size of the treatise, and the unsettled conditions existing in Europe at the time, undoubtedly contributed to the slowness with which Bohr’s novel way of treating the problems of quantum theory became known to most physicists” (Nielsen). ABPC/RBH list only the Plotnick copy of these presentation offprints (Christie’s New York, 4 October 2002, lot 22, $6573).

*Provenance*: Kasimir Fajans (1887-1975) (presentation inscription by Bohr, in German, on front wrapper of Part I). Fajans was a “Polish-American physical chemist who discovered the radioactive displacement law simultaneously with Frederick Soddy of Great Britain. According to this law, when a radioactive atom decays by emitting an alpha particle, the atomic number of the resulting atom is two fewer than that of the parent atom. When a beta particle is emitted, the atomic number is one greater. After study at the universities of Leipzig, Heidelberg, Zürich, and Manchester, Fajans served on the faculty of the Technical Academy at Karlsruhe in Germany from 1911 to 1917. In 1913, in collaboration with Otto Göhring, he discovered uranium X2, which is now called protactinium-234m. In 1917 he joined the Institute of Physical Chemistry, Münich, where he rose from associate professor to director. From 1936 to 1957, when he retired, Fajans was a professor at the University of Michigan, Ann Arbor. He became a naturalized citizen of the United States in 1942” (Britannica); Haven O’More (1929-2008) (gilt leather book label of The Garden Ltd.). Born Richard Haven Moore in Austin, Texas, Haven O’More formed a magnificent library which he called ‘The Garden Ltd.’; its books were noted particularly for their fine condition. The collapse of a business partnership forced the sale of much of the collection at Sotheby’s on 9/10 November, 1989. As Nicolas Barker states in his Foreword to the auction catalogue, “The decision to choose the best authors and the best works, in the best available copies, is aptly demonstrated.”

In February 1913 Bohr heard about Johann Balmer’s remarkable discovery of a formula for the frequencies of the lines of the hydrogen spectrum which depended on two integers. As soon as Bohr saw this formula, he immediately recognized that it gave him the missing clue to the correct way to introduce Planck’s quantum of action into the description of atomic systems. The rest of the academic year was spent reconstructing the whole theory upon the new foundation and expounding it in a large treatise, which was published as three papers in the *Philosophical Magazine* entitled ‘On the Constitution of Atoms and Molecules’. “Bohr’s three-part paper postulated the existence of stationary states of an atomic system whose behavior could be described using classical mechanics, while the transition of the system from one stationary state to another would represent a non-classical process accompanied by emission or absorption of one quantum of homogeneous radiation, the frequency of which was related to its energy by Planck’s equation” (Norman).

“As for the principle by which the possible stationary states are selected, Bohr was still very far from a general formulation; indeed, he was keenly aware of the necessity of extending the investigation to configurations other than the simple ones to which he had restricted himself. The search for sufficiently general quantum conditions defining the stationary states of atomic systems was going to be a major problem in the following period of development of the theory …

“The first of the main problems requiring consideration was the generalization of the quantum conditions defining the stationary states. Bohr … started from the premise that slow deformations of a system would not change its quantal state, and developed it into a ‘principle of mechanical transformability,’ which proved quite efficient within a limited scope. The idea was to transform one type of motion continuously into another by slow variation of some parameter; if the determination of the stationary states had been accomplished for one of the two motions, one could derive stationary states, by such a transformation, for the other. To this end, one could take advantage of the existence of dynamical quantities, the ‘adiabatic invariants,’ which have the property of remaining unchanged under slow mechanical transformations … In their general from, the quantum conditions stated that a certain set of adiabatic invariants should be integral multiples of Planck’s constant …

“The theory of multiply periodic systems offered the possibility of a more rational treatment of the question which Bohr had tackled in his very first reflections on the nuclear atomic model: the gradual building up of the periodicities in the atomic structures revealed by Mendeleev’s periodic table. The starting point was the consideration of the individual stationary orbits of each single electron in the electrostatic field of the nucleus, ‘screened’ by the average field of the other electrons; the residual interaction of the electrons could then be treated by the perturbation methods originally developed for use by astronomers …

“The confrontation of the theory with the relevant spectroscopic evidence led to partial success: the main features of the empirical term sequences were well reproduced by the theory, and the spectroscopic quantum numbers on which these features depended accordingly acquired a simple mechanical interpretation … but the fine structure of the term sequences presented a complexity for which the atomic model offered no mechanical counterpart.

“In spite of this imperfection, the model could be expected to give reliable guidance at least in the investigation of the broader outlines of atomic structures. The primitive ring configurations of Bohr’s previous attempt [in ‘On the constitution of atoms and molecules’] were now replaced by groupings of individual electron orbits in ‘shells’ specified by definite sets of quantum numbers, according to rules that were inferred from the spectroscopic data. This conception of the shell structure of atomic systems did not merely account for the main classification of the stationary states; its scope could be extended to include the interpretation of the empirical rules established by the spectroscopists for the intensities of the quantal transitions between these states. This was a much more difficult problem than that of the formulation of quantum conditions for the stationary states; the complete breakdown of classical electrodynamics, reflected in Bohr’s quantum postulates, seemed at first to remove the very foundation on which a comprehensive theory of atomic radiation could rest. It was in taking up this challenge that Bohr was led to one of his most powerful conceptions: the idea of a general correspondence between the classical and the quantal descriptions of the atomic phenomena.

“Bohr seized upon the only link between the emission of light in a quantal transition and the classical process of radiation: the requirement that the classical description should be valid in the limiting case of transitions between states with very large quantum numbers. If the atom were treated as a multiply periodic system, its states of motion could be represented as superpositions of harmonic oscillations of specified frequencies and their integral multiples, each occurring with a definite amplitude … in the limit of large quantum numbers, then, the classical amplitudes could be used directly to calculate the intensities of the quantal transitions. Bohr boldly postulated that such a correspondence should persist, at least approximately, even for transitions between states of small quantum numbers; in other words, the amplitudes of the harmonics of the classical motion should in all cases give an estimate of the corresponding quantal amplitudes …

“By 1918 Bohr had visualized, at least in outline, the whole theory of atomic phenomena … he at once started writing up a synthetic exposition of his arguments and of all the evidence upon which they could have any bearing … he could hardly keep pace with the growth of the subject; the paper he had in mind at the beginning developed into a four-part treatise, ‘On the Theory of Line Spectra,’ publication of which dragged on over four years without being completed; the first three parts appeared between 1918 and 1922, and the fourth, unfortunately, was never published” (DSB).

“In the Introduction to *Quantum Theory of Line-Spectra* (p. 4), the general tendency of the entire work is indicated by the statement: ‘it will be shown that it seems possible to throw some light on the outstanding difficulties [of quantum theory] by trying to trace the analogy between the quantum theory and the ordinary theory of radiation as closely as possible.’ In Part I, pp. 15-16, occurs a statement of the correspondence principle for a system of one degree of freedom … In the next section of Part I the principle is extended to conditionally periodic systems [i.e., multiply-periodic systems] (pp. 31-32) …

“Part I contains a detailed discussion of all the general postulates and principles applied to the interpretation of line spectra, including Bohr’s postulate of the existence of stationary states and his frequency relation. From Einstein’s derivation of Planck’s formula, Bohr adopts the concept of the probability of transitions between the stationary states … he recognizes the fundamental importance of Ehrenfest’s ‘adiabatic hypothesis’, which he terms the principle of ‘mechanical transformability’ of stationary states. He realizes the importance of this principle for the definition of energy, and he discusses in detail the modification required in the use of the principle when degeneracy occurs …

“In Part II he applies the general theory to a detailed discussion of the spectra of the hydrogen atom when the relativistic corrections and the effects of external electric and magnetic fields are taken into consideration. He makes extensive use of perturbation theory … by studying the perturbed motions and applying the correspondence principle, it is possible to determine the polarizations and estimate the intensities of the components into which a hydrogen line is split. In particular, it is possible in some cases to predict that certain transitions cannot occur, while others can occur only with a change of one unit in the appropriate quantum number …

“A manuscript of Part III of *Quantum Theory of Line-Spectra*, which was to deal with the spectra of elements of higher atomic number, was already in existence in 1918 when Part I appeared. However, after the appearance of Part II, Bohr felt that a revision was needed because of difficulties that had meanwhile come to light both through his own work and the theoretical and experimental work of others. This revision was, however, never completed. On May 24, 1922, Kramers wrote to Sommerfeld about the difficulties that Bohr was having with Parts III and IV. On June 1, Sommerfeld replied that in his opinion Bohr should not publish Part IV, but that he might publish Part III in the original form as an epilogue to Parts I and II, giving references to recent work in an Appendix. Bohr accepted this advice. He finished the Appendix in September 1922, and Part III was published on November 30, 1922.

“In the main body of the printed version of Part III, Bohr describes the general appearance of the series spectra of the higher elements and mentions the conclusion, drawn by him already in 1913, that these spectra arise from transitions of an electron, the so-called series electron, which is moving at a distance from the nucleus that is large compared with the distances of the other electrons. He mentions Sommerfeld’s formal interpretation of these spectra by assuming that the series electron moves approximately in a central field. Different series correspond to different values of the quantum number that determines the angular momentum of the series electron, and this quantum number can change only by ±1. He refers to his work with Kramers on the helium atom and discusses briefly possible structures of lithium and beryllium. Finally, he mentions problems presented by the Stark and Zeeman effects.

“The Appendix consists of four notes, one for each section of the original Part III, and it covers briefly the major progress made in the understanding of series spectra since 1918 … He points out that the models of He, Li and Be contemplated in the original Part III are untenable, and emphasizes in particular the conclusion … that, in the ground state of the alkali and other atoms, the series electron has a principal quantum number greater than one, and that its orbit may penetrate into the region of the inner electrons … He mentions that the complex structure cannot be explained by a central motion of the series electron and that its description requires a third quantum number. Finally, he discusses the advances made, especially by Landé, in explaining the anomalous Zeeman effect, but concludes that this effect is still not well understood” (Nielsen, pp. 5-9).

“Bohr’s theory of the periodic system of the elements, based essentially on the analysis of the evidence of the spectra, renewed the science of chemistry by putting at the chemists’ disposal rational spectroscopic methods much more refined than the traditional ones. This was dramatically illustrated in 1922, by the identification, at Bohr’s institute, of the element with atomic number 72. This discovery was made by Dirk Coster and Hevesy, under the direct guidance of Bohr’s theoretical predictions of the properties of this element; they gave it the name ‘hafnium,’ from the Latinized name of Copenhagen. The conclusive results were obtained just in time to be announced by Bohr in the address he delivered when he received the Nobel Prize in physics for that year” (DSB).

“Einstein was impressed by the advances Bohr had made. ‘His intuition is much to admire,’ he said to Sommerfeld. Much later, in his autobiographical notes written in 1946, Einstein characterized Bohr’s atomic theory as a ‘miracle’ and an expression of ‘the highest form of musicality in the sphere of thought.’ It was, he said, due to Bohr’s ‘unique instinct and tact’ that he had been able ‘to discover the major laws of the spectral lines and of the electron-shells of the atoms together with their significance for chemistry’” (Kragh, p. 203).

‘On the Quantum Theory of Line-Spectra, Part I-III’ was published in *Det Kongelige Danske Videnskabernes Selskabs **Skrifter**, Ottende Rakke, Naturvidenskabelig og Matematisk Afdeling*, 1918-23, pp. 1-118. The offprints precede the journal, which cannot have appeared until after November 1923, as its Index states ‘Fortegmelse over Selskabets Medlemmer November 1923’ (‘List of Publications November 1923’). These author’s presentation offprints are all rare, but the third part is particularly so. Commercially available separates of the three parts were issued by the same publisher, and with the same dates, as the offprints, but these are much less rare (these commercial separates are sometimes incorrectly referred to as offprints).

Jammer, *The Conceptual Development of Quantum Mechanics* (New York: McGraw Hill, 1966). Kragh, *Niels Bohr and the Quantum Atom* (Oxford: Clarendon Press, 2012). Nielsen (ed.), *Niels Bohr Collected Works, Vol. 3. The Correspondence Principle* (*1918-1923*) (Amsterdam: North-Holland, 1976).

Three author’s presentation offprints. 4to (268 x 214 mm), pp. [3], 4-36; [1], 38-100; [1], 102-118. Original printed wrappers (corners of front wrapper of Part I chipped, small repair to inner margin of front wrapper of Part II). Housed in a cloth drop-back box with paper label on spine and gilt leather book-label of The Garden Ltd. Very good copies.

Item #4920

**
Price:
$25,000.00
**