- 7.1 Van Vleck’s Two Books and the Quantum Revolution
- 7.2 Van Vleck’s Early Life and Career
- 7.3 The NRC Bulletin
- 7.4 New Research and the Move to Wisconsin
- 7.5 The Theory of Electric and Magnetic Susceptibilities
- 7.6 Kuhn Losses Revisited
- Abbreviations and Archives
Kuhn Losses Regained: Van Vleck from Spectra to
7.1.1 Van Vleck’s Trajectory from Spectra to Susceptibilities, 1926–1932
“The chemist is apt to conceive of the physicist as some one who is so entranced in spectral lines that he closes his eyes to other phenomena.” This observation was made by the American theoretical physicist John H. Van Vleck (1899–1980) in an article on the new quantum mechanics in Chemical Reviews (Van Vleck 1928b, 493). Only a few years earlier, Van Vleck himself would have fit this characterization of a physicist to a tee. Between 1923 and 1926, as a young assistant professor in Minneapolis, he spent much of his time writing a book-length Bulletin for the National Research Council (NRC) on the old quantum theory (Van Vleck 1926b). As its title, Quantum Principles and Line Spectra, suggests, this book deals almost exclusively with spectroscopy. Only after a seemingly jarring change of focus in his research, a switch to the theory of electric and magnetic susceptibilities in gases, did he come to consider his previous focus myopic. In 1927–28, now a full professor in Minnesota, he published a three-part paper on susceptibilities in Physical Review (Van Vleck 1927a; 1927b; 1928a). This became the basis for a second book, The Theory of Electric and Magnetic Susceptibilities (Van Vleck 1932b), which he started to write shortly after he moved to Madison, Wisconsin, in the fall of 1928.
By the time he wrote his article in Chemical Reviews, Van Vleck had come to recognize that a strong argument against the old and in favor of the new quantum theory could be found in the theory of susceptibilities, a subject of marginal interest during the reign of the old quantum theory. As he wrote in the first sentence of the preface of his 1932 book:
The new quantum mechanics is perhaps most noted for its triumphs in the field of spectroscopy, but its less heralded successes in the theory of electric and magnetic susceptibilities must be regarded as one of its great achievements. (Van Vleck 1932b, vii)
What especially struck Van Vleck was that, to a large extent, the new quantum mechanics made sense of susceptibilities not by offering new results, but by reinstating classical expressions that the old quantum theory had replaced with erroneous ones. Both in his articles of the late 1920s and in his 1932 book, Van Vleck put great emphasis on this point.
His favorite example was the value of what he labeled
, a constant in the so-called
The constant also comes into play if we want to determine the dipole moment of a polar molecule such as HCl. Given a gas of these molecules, one can calculate using a measurement of the dielectric constant: the greater the value of , the smaller the value of . Because of the instability of the value of , Van Vleck (1928b) pointed out that, “[t]he electrical moment of the HCl molecule […] has had quite a history” (494).
Fig. (7.1) shows the table with which Van Vleck illustrated this checkered history. The result for whole quanta was found by
These observations, including the table, are reprised in his book on susceptibilities (Van Vleck 1932b, 107). In fact, these fluctuations in the values of and so impressed Van Vleck that the first two columns of this table can still be found in his 1977 Nobel lecture (Van Vleck 1992b, 356).
Van Vleck’s 1932 book on susceptibilities was much more successful than his Bulletin on the old quantum theory, which was released just after the quantum revolution of 1925–26. The Bulletin, as its author liked to say with characteristic self-deprecation, “in a sense was obsolete by the time it was off the press” (Van Vleck 1971, 6, our emphasis). The italicized qualification is important. In the late 1920s and early 1930s, physicists could profitably use the Bulletin despite the quantum revolution. The 1932 book, however, became a classic in the field it helped spawn. Interestingly, given that it grew out of work on susceptibilities in gases, the field in question is solid-state physics. In a biographical memoir about Van Vleck for the National Academy of Sciences (NAS), condensed-matter icon
In this paper we follow Van Vleck’s trajectory from his 1926 Bulletin on spectra to his 1932 book on susceptibilities. Both books, as we will see, loosely qualify as textbooks. As such, they provide valuable insights about the way pedagogical texts written in the midst (the 1926 Bulletin) or the aftermath (the 1932 book) of a scientific revolution reflect such dramatic upheavals.
7.1.2 Kuhn Losses, Textbooks, and Scientific Revolutions
The old quantum theory’s trouble with susceptibilities, masked by its success with spectra, is a good example of what is known in the history and philosophy of science literature as a Kuhn loss. Roughly, a
Incidentally, both Thomas S. Kuhn and
I showed that the factor one-third [in the
Langevin-Debye formula for susceptibilities] got restored in quantum mechanics, whereas in the old quantum theory, it had all kinds of horrible oscillations […] you got some wonderful nonsense, whereas it made sense with the new quantum mechanics. I think that was one of the strong arguments for quantum mechanics. One always thinks of its effect and successes in connection with spectroscopy, but I remember Niels Bohr saying that one of the great arguments for quantum mechanics was its success in these non-spectroscopic things such as magnetic and electric susceptibilities.1
To the best of our knowledge, Kuhn never used the “wonderful nonsense” Van Vleck is referring to here as an example of a Kuhn loss. Still, one can ask whether the example bears out Kuhn’s general claims about
7.1.3 Kuhn Losses
The concept of a
In the transition from classical theory to the old quantum theory, gains in spectroscopy apparently outweighed losses in the theory of susceptibilities just as, at least until the early 1920s, they outweighed losses in dispersion theory. The former Kuhn loss was only regained in the new quantum theory,3 while the latter was recovered in the dispersion theory of
Strictly speaking, of course, when we talk about
Although they are both Kuhn losses of the old quantum theory, the one in susceptibility theory is of a different kind than the one in dispersion theory. In the case of dispersion, there was clear experimental evidence all along for the key feature of the classical theory that was lost in the old quantum theory and recovered in the
The key feature in the case of dispersion is that anomalous dispersion—the phenomenon that in certain frequency ranges the index of refraction gets smaller rather than larger with increasing frequency4—occurs around the absorption frequencies of the dispersive medium. This is in accordance with the classical dispersion theories of
[T]he positions of maximal dispersion and absorption do not lie at the position of the emission lines of hydrogen but at the position of the mechanical frequencies of the model […] the conclusion seems unavoidable to us that the foundations of the Debye-Davysson [sic] theory are incorrect. (Epstein 1922, 107–108; emphasis in the original; quoted and discussed by Duncan and Janssen 2007, 580–581)
By contrast, it was only after the new quantum theory had restored the classical value
Three years later, the experimentalist
7.1.4 Textbooks and Kuhn Losses
Kuhn (1996, chap. 11) famously identified textbooks as the main culprit in rendering the disruption of normal science by scientific revolutions invisible. Textbooks, he argued, by their very nature must present science as a cumulative enterprise. This means that
address themselves to an already articulated body of problems, data, and theory, most often to the particular set of paradigms6 to which the scientific community is committed at the time they are written […] [B]eing pedagogic vehicles for the perpetuation of normal science […] [they] have to be rewritten in the aftermath of each scientific revolution, and, once rewritten, they inevitably disguise not only the role but the very existence of the revolutions that produced them […] [thereby] truncating the scientist’s sense of his discipline’s history. (Kuhn 1996, 136–137)
When he wrote this passage,
The two monographs by Van Vleck examined in this paper would seem to qualify as (graduate) textbooks even though under a strict and narrow definition of the genre they might not. Most of their actual readers may have been research scientists but they were written with the needs of students in mind and both books saw some classroom use. Student notes for a two-semester course on quantum mechanics that Van Vleck offered in Wisconsin in 1930–31 show that, despite the quantum revolution that had supposedly made it obsolete four years earlier, Van Vleck was still using his NRC Bulletin as the main reference for almost two-thirds of the first semester.7 It is unclear whether Van Vleck himself ever used his 1932 book on susceptibilities in his classes. However, one of his colleagues at Wisconsin,
So one can reasonably ask how well Van Vleck’s two books fit with Kuhn’s seductive picture of how the regrouping of a scientific community in response to a scientific revolution is reflected in the textbooks it produces. It will be helpful to separate two aspects of this picture: how textbooks delineate and orient further work in their (sub-)disciplines, and how, in doing so, they inevitably distort the prehistory of these (sub-)disciplines and paper over
Van Vleck’s NRC Bulletin confirms several of his former student’s generalizations about textbooks. The Bulletin is organized around the correspondence principle as a strategy for tackling problems mostly in atomic spectroscopy. Van Vleck thus took the approach he,
Those engaged in work that was marginalized in this way predictably took exception. In a review of the Bulletin, one such colleague,
Selection of, arrangement of, and space devoted to the offerings is heavily influenced by subjective viewpoints and cannot win every reader’s approval everywhere. Instead of the presumably available option of letting all fundamental connections emerge systematically, the author has preferred to put up front what is felt to be the internally most unified part of the quantum theory as it has developed so far, followed by more or less isolated applications to specific problems. (Smekal 1927, 63)
The way in which correspondence-principle techniques take center stage in Van Vleck’s book provides a nice example of how textbooks transmit what
I mean, initially, the concrete problem solutions that students encounter from the start of their scientific education, whether in laboratories, on examinations or at the ends of chapters in science texts. To these shared examples should, however, be added at least some of the technical problem-solutions found in the periodical literature that scientists encounter during their post-educational research careers and that also show them by example how their job is to be done. (Kuhn 1996, 187)
Van Vleck’s Bulletin presented such “technical problem-solutions found in the periodical literature” in a more didactic text that should help its readers become active contributors to this literature themselves.
Confirming another article of Kuhnian doctrine, the problem with susceptibilities, a
Whereas the Bulletin passes over the Kuhn loss in the theory of susceptibilities in silence, the Kuhn loss in dispersion theory in that same transition is flagged prominently. It is easy to understand why. By the time Van Vleck wrote his Bulletin,
In his 1932 book, as we will see in secs. (7.5.2–7.5.5), Van Vleck made even more elaborate use of the recovered Kuhn loss in susceptibility theory to promote his new quantum-mechanical treatment of susceptibilities. He devoted a whole chapter of the book to the problems of the old quantum theory in this area. Of course, the
Like the Bulletin, the 1932 book provided its readers with all the tools they needed to become researchers in the field it so masterfully mapped out for them. Had the correspondence-principle approach to atomic physics been moribund by the time the Bulletin saw print, the approach to electric and magnetic susceptibilities championed in the 1932 book would prove to be remarkably fruitful.
7.1.5 Continuity and Discontinuity in Scientific Revolutions
A couple of
The 1932 book provides the clearest example of this. As mentioned above, Van Vleck devoted an entire chapter to the old quantum theory, putting the problems it ran into with susceptibilities on full display. Yet he conveniently neglected to mention that there had been no clear empirical evidence exposing these problems.
That said, there were many elements in older theories that helped rather than hindered Van Vleck in achieving these objectives. As a result, much of the continuity that can be discerned in the discussions of classical theory and quantum theory in the NRC Bulletin is not, as Kuhn would have it, an artifact of how history is inevitably rewritten in textbooks, but actually matches the historical record tolerably well. Despite its misleading treatment of the experimental state of affairs in the early 1920s, the same can be said about the 1932 book. The final two clauses of the passage from Structure quoted above (“inevitably disguise […]” and “truncating […]”) are clearly too strong.
On the Kuhnian picture of scientific revolutions as paradigm shifts akin to Gestalt switches, it is hard to understand how a post-revolutionary textbook could make the prehistory of its subject matter look more or less continuous and thereby perfectly suitable to its pedagogical objectives without seriously disguising, distorting, and truncating that prehistory. An important part of the explanation, at least in the case of these two books by Van Vleck, is the continuity of mathematical techniques through the conceptual upheavals that mark the transition from classical theory to the old quantum theory, and finally to modern quantum mechanics.
In his recent book, Crafting the Quantum, on the Sommerfeld school in theoretical physics, Suman Seth (2010) makes a similar point. He reconciles the continuous and the discontinuous aspects of the development of quantum theory in the 1920s by emphasizing, as we do, the continuity of mathematical techniques. Scientific revolutions, he writes, “are revolutions of conceptual foundations, not of puzzle-solving techniques. Most simply: Science sees revolutions of principles, not of problems” (Seth 2010, 268). To illustrate his point, Seth quotes Arnold Sommerfeld, who wrote in 1929: “The new development does not signify a revolution, but a joyful advancement of what was already in existence, with many fundamental clarifications and sharpenings” (ibid., 266).
Given the radical conceptual changes involved in the transition from classical physics to quantum physics, it is important to keep in mind that there was at the same time great continuity of mathematical structure in this transition. Both the old quantum theory and matrix mechanics, for instance, retain, in a sense, the laws of classical physics. The old quantum theory just put some additional constraints on the motions allowed by Newtonian mechanics. The basic idea of matrix mechanics, as reflected in the term Umdeutung (reinterpretation) in the title of the paper with which
In the old quantum theory, techniques from perturbation theory in celestial mechanics were used to analyze electron orbits in atoms classically as a prelude to the translation of the results into quantum formulas under the guidance of the correspondence principle (Duncan and Janssen 2007, 592–593, 627–637). This is the procedure that led
One way to highlight the continuity of Van Vleck’s trajectory from spectra to susceptibilities is to note that the derivation of the
The remarkable continuity of mathematical structures and techniques in the transitions from classical theory to the old quantum theory, and then to modern quantum mechanics makes it perfectly understandable that Van Vleck could still use his 1926 Bulletin in his courses on quantum mechanics in the early 1930s. It also explains how Van Vleck could make such rapid progress once he hit upon the problem of susceptibilities not long after he completed the Bulletin and mastered matrix mechanics.
Kuhn had a tendency to see only discontinuity in paradigm shifts. This intense focus on discontinuity is what lies behind his fascination with
Whether one sees continuity or discontinuity in the transition from classical physics to quantum physics depends, to a large extent, on one’s perspective. The historian trying to follow the events as they unfolded on the ground, will probably mainly see continuities. The historian who takes a bird’s eye view and compares the landscapes before and after the transition will most likely be struck first and foremost by discontinuities. A final twist in our story about the recovered Kuhn loss in Van Vleck’s 1932 book nicely illustrates this difference in perspective.
Van Vleck covered the troublesome recent history of its subject matter in chap. V, “Susceptibilities in the old quantum theory contrasted with the new.” This chapter, as we will show in more detail in secs. (7.5.2–7.5.5), allows us to see important elements of continuity in the transition from the old to the new quantum theory. Toward the end of his life, Van Vleck began revising his 1932 classic with the idea of publishing a new edition (Fellows 1985, 258, 262–263, 266).9 Wanting to add a chapter on modern developments without changing the total number of chapters, he intended to cut chap. V, on the grounds that by then it only had historical value.10 Even in 1932 he began the chapter apologizing to his readers that “it may seem like unburying the dead to devote a chapter to the old quantum theory” (Van Vleck 1932b, 105). Note also the one reservation Anderson (1987, 509) expressed about the book in his NAS memoir: “It is marked—perhaps even slightly marred, as a modern text for physicists poorly trained in classical mechanics—by careful discussion of the ways in which quantum mechanics, the old quantum theory, and classical physics differ.” As it happened, the new edition of the book never saw the light of day, but if it had, it would have been a confirming instance of an amended version of Kuhn’s thesis, namely that, going through multiple editions, textbooks eventually suppress or at least sanitize the history of their subject matter and paper over
7.1.6 Van Vleck as Teacher
Although it will be clear from the preceding subsections that our main focus in this paper is not on Van Vleck’s books as pedagogical tools, it seems appropriate to devote a short subsection to Van Vleck as a teacher.
A good place to start is to compare testimony by
By the 1940s […] his teaching style had become unique, and is remembered with fondness by everyone I spoke to. Most of the material was written in his inimitable scrawl on the board […] Especially in group theory [taught from (Wigner 1931) in the original German], his intuitive feeling for the subject often bewildered us as he scribbled […] in an offhand shorthand to demonstrate what we thought were exceedingly abstruse points. (Anderson 1987, 524)
Anderson’s assessment is actually consistent with Kuhn’s, even though the latter evidently did not share his fellow student’s enthusiasm for the unique style of their advisor: “One of the courses that I then took was group theory with Van Vleck. And I found that somewhat confusing […] Van Vleck was not a terribly good teacher” (Baltas et.al. 2000, 272).
Van Vleck’s teaching style must have been less idiosyncratic in his earlier years. As
John Van Vleck was my professor at Wisconsin. The first year I was there he gave a course in quantum mechanics. No one wanted to take a degree that year. Everyone put it off because it was useless—there weren’t any jobs. The next year Van had the same bunch of students, so he gave us advanced quantum mechanics. The year after that he gave us advanced quantum mechanics II. Van was extremely good, a good teacher and an outstanding physicist.11 (Serber 1992, xxiv)
Anderson offered the following explanation for Van Vleck’s effectiveness as a teacher:
In all of his classes […] he used two basic techniques of the genuinely good teacher. First, he presented a set of carefully chosen problems […] Second, he supplied a “crib” for examination study, which we always thought was practically cheating, saying precisely what could be asked on the exam. It was only after the fact that you realized that it contained every significant idea of the course. (Anderson 1987, 524–525)
7.2 Van Vleck between two fans at 1300 Sterling Hall, University of Wisconsin–Madison, ca. 1930 (picture courtesy of John Comstock).
Even before the Great Depression, students sometimes took Van Vleck’s quantum course more than once.
In his first year at Madison, 1928–29, Van Vleck immediately started supervising two postdocs,
Physics 212, “Quantum mechanics and atomic structure,” was the only lecture course Van Vleck offered during his first few years in Wisconsin (ibid., 230). It was not until 1931–33, the period described by
7.1.7 Structure of Our Paper
The balance of this paper is organized as follows. In sec. (7.2), we sketch Van Vleck’s early life against the backdrop of theoretical physics coming of age and maturing in the United States. Our main focus is on his years in Minneapolis leading up to the writing of his NRC Bulletin (1923–26). Throughout the paper, but especially in the more biographical secs. (7.2) and (7.4), we make heavy use of the superb dissertation on Van Vleck by Fred Fellows (1985). In sec. (7.3), we turn to the Bulletin itself (Van Vleck 1926b). In sec. (7.3.1), we recount how what had originally been conceived as a review article of average length eventually ballooned into a 300-page book. In sec. (7.3.2) we give an almost entirely qualitative discussion of its contents, focusing on the derivation of
7.2 Van Vleck’s Early Life and Career
John Hasbrouck Van Vleck (1899–1980) was born in Middletown, Connecticut, to Edward Burr Van Vleck and Hester Laurence Van Vleck (née Raymond). In 1906 the family moved to Madison, Wisconsin, where his father was appointed professor of mathematics.13 He had been named after his grandfather, John Monroe Van Vleck, but his mother, not fond of her father-in-law, called him Hasbrouck (Fellows 1985, 6–8). To his colleagues, he would always be Van. A nephew of Van’s wife, Abigail June Pearson (1900–1989), recalls that a telegram from Japan congratulating Van Vleck on winning the Nobel prize was addressed to “Professor Van” (John Comstock, private communication).
In 1916 Van Vleck began his undergraduate studies at the University of Wisconsin, where he eventually majored in physics. In the fall of 1920, he enrolled at Harvard as a graduate student in physics.14 He took
The following year, Van Vleck accepted a position as an instructor in Harvard’s physics department. This demanding job left him with little time for his own work. Most of his time was spent preparing for lectures and lab sessions (Fellows 1985, 49). In the midst of this daily grind, the job offer that arrived from the University of Minnesota in early 1923 must have looked especially attractive. As Van Vleck (1992a, 351) would reflect later, it was an “unusual move” for such an institution at that time—indicative, one may add, of the American physics community’s growing recognition of the importance of quantum theory—to offer him an assistant professorship “with purely graduate courses to teach.”
At first, Van Vleck was hesitant to accept the position.17 He and
In October 1924, after a preliminary report in the Journal of the Optical Society of America (Van Vleck 1924a), a two-part paper appeared in Physical Review in which Van Vleck (1924b; 1924c) used correspondence-principle techniques to analyze the interaction between matter and radiation in the old quantum theory. Its centerpiece was Van Vleck’s own correspondence principle for absorption, but the paper also contains a detailed derivation of the
After his 1924 paper, however, Van Vleck did not push this line of research any further. He had meanwhile been ‘invited’ to produce the volume to which we now turn our attention. Its completion would occupy nearly all of his available research time for the next two years.
7.3 The NRC Bulletin
7.3.1 Writing the Bulletin
Later in life, when interviewed by
I was already writing some chapters on that on rainy days in Switzerland in 1924. I would say I started writing that perhaps beginning in the spring of 1924, and finished it in late 1925. I worked on it very hard that summer […] I was sort of a “rara avis” at that time. I was a young theoretical physicist presumably with a little more energy than commitments than the older people interested in these subjects, so they asked me if I’d write this thing. I think it was by invitation rather than by my suggestion.18
The invitation had come from
The committees organized the Bulletins of the NRC, which existed to present “contributions from the National Research Council […] for which hitherto no appropriate agencies of publication [had] existed” (Swann et.al. 1922, 173–174). This sounds rather vague and overly inclusive, and on reading the motley assortment of topics covered by the Bulletins through 1922, one finds that it was rather vague and overly inclusive. The Bulletins served to disseminate whatever information the myriad committees deemed important. A brief list of topics covered by these publications includes “The national importance of scientific and industrial research,” “North American forest research,” “The quantum theory,” “Intellectual and educational status of the medical profession as represented in the United States Army,” and “The scale of the universe” (ibid.). The Bulletins tended to be short, averaging about 75 pages. Several were even shorter, coming in under 50 pages. The longest at the time Van Vleck was invited to write one on line spectra was a 172-page book, Electrodynamics of Moving Media (Swann et.al. 1922). It had been written by four authors, including
Given the Bulletin’s publication history, Van Vleck was not making an unreasonable commitment when he accepted
As we saw, Van Vleck later recalled having begun his Bulletin in the spring of 1924, but he must have started much earlier than that. In March 1924, Foote returned a draft to Van Vleck along with extensive comments. “This has been read very carefully by
Like you I “am wondering” when my paper for the Research Council will ever be ready. I am sorry to be progressing so slowly but I hope you realize that I am devoting to this report practically all of my time not occupied with teaching duties. I still hope to have the manuscript ready by Christmas except for finishing touches.22
Van Vleck would blow the Christmas deadline as well. It was not until August that he submitted a new draft:
I hope the bulletin will be satisfactory, as with the exception of one three-month period it has taken all my available research time for two years.
You wrote me that the bulletin should be “fairly complete.” My only fear is that it may be too much so. I made sure to include references to practically all the important theoretical papers touching on the subjects covered in the various chapters. Four new chapters have been included since an early draft of the manuscript was sent to you a year ago […]
You will note that I have used a new title “Quantum Principles and Line-Spectra” as this is much briefer and perhaps more a-propos than “The Fundamental Concepts of the Quantum Theory of Line-Spectra.”23
It is worth noting the change in title. The old quantum theory was strongly focused on the phenomena of line spectra. Van Vleck’s new title conveys at once this focus even as he had significantly broadened the scope of his project.
It is clear that however the project began, and whatever Van Vleck’s initial expectations, in the end the Bulletin was intended by its author as a comprehensive and up-to-date review of quantum theory. This makes it useful not only as a review of the old quantum theory, but also as a window into Van Vleck’s own perception and understanding of the field.
Despite some critical notes,25 the Bulletin was “on the whole, well-received” (Fellows 1985, 88). Van Vleck must have read
This excellent bulletin will prove extremely useful to all who are interested in atomic physics […] [T]he fundamental theorems of Hamiltonian dynamics and perturbation methods of quantization are treated in a very readable fashion […] The chapter on the quantization of neutral helium is authoritative […] The author’s treatment of the “correspondence principle” is refreshingly clear […] The whole book is surprisingly up-to-date. Even the theory of spinning electrons and matrix dynamics are touched upon. It is to be hoped that this report will run through many revised editions as quantum theory progresses, for it fills a real need. (Ruark 1926)
Many readers will not agree with the author’s conclusion that
“Kramers’s dispersion theory […] furnishes by far the most satisfactory theory of dispersion” [Van Vleck 1926b, 156–157] […] the reviewer believes that a final solution cannot be achieved until we have a much more thorough knowledge of the dispersion curves of monatomic gases and vapors. (Ruark 1926)
Subsequent developments would prove that Van Vleck’s confidence in the Kramers dispersion formula was well-placed. It carried over completely intact to the new quantum mechanics (Duncan and Janssen 2007, 655).
7.3.2 The Bulletin and the Correspondence Principle
The central element in Van Vleck’s presentation of the old quantum theory in his NRC Bulletin is the correspondence principle. It forms the basis of 11 out of a total of 13 chapters.27 As it says in the preface,
Bohr’s correspondence principle is used as a focal point for much of the discussion in Chapters I–X. In order to avoid introducing too much mathematical analysis into the discussion of the physical principles underlying the quantum theory, the proofs of certain theorems are deferred to Chapter XI, in which the dynamical technique useful in the quantum theory is summarized. (Van Vleck 1926b, 3)
By the early 1920s, the correspondence principle had become a sophisticated scheme used by several researchers for connecting formulas in classical mechanics to formulas in the old quantum theory. The most important result of this approach was the
As a concrete example of the use of the correspondence principle in the old quantum theory in the early 1920s, we sketch Van Vleck’s derivation of the Kramers dispersion formula.28 This formula and what Van Vleck (1926b, 162) called the “correspondence principle for dispersion” are presented in a section of only two and a half pages in chap. X of the NRC Bulletin (ibid., sec. 51, 162–164). The reason that Van Vleck could be so brief at this point is that the various ingredients needed for the derivation of the formula are all introduced elsewhere in the book, especially in chap. XI on mathematical techniques. At 50 pages, this is by far the longest chapter of the Bulletin.
Consider some (multiply-)periodic system—anything from a charged simple harmonic oscillator to an electron orbiting a nucleus—struck by an electromagnetic wave of a frequency
not too close to that system’s characteristic frequency
To obtain the
As with all such derivations in the old quantum theory, the part involving classical mechanics called for advanced techniques borrowed from celestial mechanics. As we mentioned in sec. (7.2), Van Vleck had thoroughly mastered these techniques as a graduate student at Harvard. Decades later, when the Dutch Academy of Sciences awarded him its prestigious Lorentz medal, Van Vleck related an anecdote in his acceptance speech that demonstrates his early mastery of this material:
In 1924 I was an assistant professor at the University of Minnesota. On an American trip,
[Paul] Ehrenfest gave a lecture there […] [He] said he would like to hear a colloquium by a member of the staff. I was selected to give a talk on my “Correspondence Principle for Absorption” [Van Vleck 1924a, 1924b, 1924c] […] I remember Ehrenfest being surprised at my being so young a man. The lengthy formulas for perturbed orbits in my publication on the three-body problem of the helium atom [Van Vleck 1922] had given him the image of a venerable astronomer making calculations in celestial mechanics. (Van Vleck 1974, 9; quoted by Duncan and Janssen 2007, 627)
Van Vleck put his expertise in classical mechanics to good use. Using canonical perturbation theory in action-angle variables, he derived an expression in classical mechanics for the dipole moment of a charged multiply-periodic system hit by an electromagnetic wave of small amplitude, which could then be translated into a quantum-theoretical expression.
In general coordinates and their conjugate momenta (where , with the number of degrees of freedom), Hamilton’s equations are:
is the Hamiltonian and dots indicate time derivatives. Given the Hamiltonian of some multiply-periodic system, one can often find special coordinates,
, called action-angle variables, such that the Hamiltonian in the new coordinates only depends on the new momenta, the action variables
, and not on the new coordinates, the angle variables
. In that case,
The first of these equations shows what makes the use of action-angle variables so attractive in celestial mechanics. It makes it possible to extract the characteristic periods of the system from the Hamiltonian without having to know the details of the orbit.
Action-angle variables played a central role in the old quantum theory. They are used to formulate the
For orbits with high values for all quantum numbers, there is only a small difference between the values of the Hamiltonian for and for (with the values of all ’s with fixed). The differential quotients in the first equation in eq. (7.2) can then be approximated by difference quotients:
The two values of the Hamiltonian in the numerator give the energies and of two orbits, close to each other, with high values for all quantum numbers (all, except for the -th one, equal for the two orbits). Eq. (7.4) is thus of the form:
In the limit of high quantum numbers, this equation for the orbital frequency
of the electron—and thereby, according to classical electrodynamics, the frequency of the radiation emitted because of the electron’s acceleration in that orbit—coincides with
for the frequency of the radiation emitted when an electron jumps from an initial orbit (quantum number ) to a final orbit (quantum number ). This asymptotic connection between this classical formula for the orbital frequencies and Bohr’s quantum formula for the radiation frequencies is what Van Vleck (1926b, sec. 11, 23–24) called “the correspondence theorem for frequencies.”
Such asymptotic connections can be used in two ways, either to check that a given quantum formula reduces to its classical counterpart in the limit of high quantum numbers, or to make an educated guess on the basis of the classical formula assumed to be valid for high quantum numbers as to what its quantum-theoretical counterpart, valid for all quantum numbers, might be. While
To derive a formula for the classical dipole moment from which its counterpart in the old quantum theory can be constructed (or against which it can be checked), one treats the electric field of the electromagnetic wave striking the periodic system under consideration as a small perturbation of the system in the absence of such disturbances. The full Hamiltonian is then written as the sum of an unperturbed part and a small perturbation (where ‘int’ stands for ‘interaction’). Using action-angle variables in such perturbative calculations, one can derive the formula for the classical dipole moment without having to know anything about the dynamics of the unperturbed system other than that it is solvable in these variables.30
The classical formula Van Vleck eventually arrived at for the dipole moment of a multiply-periodic system has the form of a derivative with respect to the action variables
of an expression involving squares of the amplitudes of the Fourier components and the characteristic frequencies
of the motion of the unperturbed system. The correspondence principle, as it was understood by
1Replace the characteristic frequencies , the orbital frequencies of the motion in the unperturbed multiply-periodic systems under consideration, by the frequencies of the radiation emitted in the transition from the -th to the -th orbit.
2Replace squares of the amplitudes of the Fourier components of this motion by transition probabilities given by the
coefficients for spontaneous emission in the quantum theory of radiation proposed by
3Replace the derivatives with respect to the action variables by difference quotients as in eq. (7.4). This last substitution is often attributed to Born but it was almost certainly discovered independently by Born, Kramers, and Van Vleck (Duncan and Janssen 2007, 637–638, 668).
Although this construction guarantees that the quantum formula merges with the classical formula for high quantum numbers, it still took a leap of faith to assert that the quantum formula would continue to hold all the way down to small quantum numbers. In the case of the Kramers dispersion formula, however, there were other considerations, besides this correspondence-principle argument for it, that inspired confidence in the result.
As mentioned in sec. (7.1.3), the Kramers dispersion formula amounted to the recovery of a
The recovery of this
The correspondence-principle translation scheme outlined above was central to the research in the early 1920s of both Van Vleck (1924b; 1924c) and Born (1924). In fact, their approaches were so similar that the two men had a testy correspondence about the proper assignment of credit for various results and insights (Duncan and Janssen 2007, 569–571, 638–639). Moreover, both Born (1925) and Van Vleck (1926b) wrote a book on the old quantum theory in which they organized the material covered around the correspondence principle as they had come to understand and use it in their research.33
Van Vleck apparently told his former student
[T]he work is deliberately conceived as an attempt […] to ascertain the limit within which the present principles of atomic and quantum theory are valid and […] to explore the ways by which we may hope to proceed […] [T]o make this program clear in the title, I have called the present book “Vol. I;” the second volume is to contain a closer approximation to the “final” atomic mechanics […] The second volume may, in consequence, remain for many years unwritten. In the meantime let its virtual existence serve to make clear the aim and spirit of this book. (Born 1925, v)
By the time the English translation of
[I]t seems to me that the time is [sic] not yet arrived when the new mechanics can be built up on its own foundations, without any connection with classical theory […] Further, I can state with a certain satisfaction that there is practically nothing in the book which I wish to withdraw. The difficulties are always openly acknowledged […] Lastly, I believe that this book itself has contributed in some small measure to the promotion of the new theories, particularly those parts which have been worked out here in Göttingen.36 (Born 1927, xi)
Quantum mechanics continued to develop rapidly in the late 1920s (Duncan and Janssen 2013). Only three years after the English translation of his 1924 book, the sequel Born had promised in the preface to the original German edition appeared. The book, co-authored with his former student
[t]his book is the continuation of the “Lectures on atomic mechanics” published in 1925; it is the “second volume” that was announced in the preface, of which “the virtual existence should serve to make clear the aim and spirit of this book.” The hope that the veil that was still hanging over the real structure of the laws of the atom would soon be parted has been realized in a surprisingly fast and thorough fashion. (Born and Jordan 1930, v)
The authors then warned their readers that they had made a conscious effort to see how much could be done with “elementary, i.e., predominantly algebraic means” (ibid., vi). In other words, elementary quantum mechanics, for Born and Jordan, was essentially matrix mechanics. They relegated wave-theoretical methods to a future book they promised to write “as soon as time and energy permit” (ibid).
In his review of Elementary Quantum Mechanics in Die Naturwissenschaften,
Contrary to Born, Van Vleck only seems to have realized how serious the problems facing the old quantum theory were after its demise. Talking to
In one of the sections of chap. VIII, sec. 35, entitled “Standard Quantum Conditions and Correspondence Theorem for Frequencies Remain Valid Even if Classical Mechanics Break [sic] Down,” he wrote:
[T]o escape from the difficulties thus encountered [in the preceding section] it appeared necessary to assume that the classical mechanics do [sic] not govern the motions of the electrons in the stationary states of atoms with more than one electron. It might seem that this bold proposal would invalidate the considerable degree of success already sometimes attained in complicated atoms […] Such successful applications, however, need not be forfeited if only we assume that the Bohr frequency condition and the standard quantum conditions retain their validity, even though the motions quantized by the latter are not in accord with ordinary dynamics in atoms with more than one electron. (Van Vleck 1926b, 108, our emphasis)
As bold as Van Vleck may have thought his proposal was, by the time his Bulletin was in print,
By the time he wrote the article about the new quantum theory in the Chemical Reviews from which we quoted at the beginning of this paper, Van Vleck had certainly understood that the transition from the old to the new quantum theory required much more radical steps than the ones he had contemplated in his NRC Bulletin. As he explained to his colleagues in chemistry,
one cannot use a meter stick to measure the diameter of an atom, or an alarm clock to record when an electron is at the perihelion of its orbit. Consequently we must not be surprised […] that models cannot be constructed with the same kind of mechanics as Henry Ford uses in designing an automobile. (Van Vleck 1928b, 468, quoted and discussed by Duncan and Janssen 2007, 666)
In the years following the Bulletin’s publication, Van Vleck’s perceptions of the old quantum theory would change a great deal. Specifically, he would come to see its shortcomings through the lens of his subsequent work on susceptibilities and his own accomplishments in this area as providing powerful arguments against the old and in favor of the new quantum theory.
7.4 New Research and the Move to Wisconsin
Only after the Bulletin was sent to press was Van Vleck able to confront matrix mechanics. By late March of 1926, he had no doubt caught up with current developments, in part through his own reading and in part through direct contact with Born, who lectured in Madison that month (Fellows 1985, 102). In January of 1926,
In April 1926, Van Vleck read a paper submitted to the Physical Review by
About a month after Van Vleck read Pauling’s paper, a paper by
I remember in particular [Tate] showing me an article by Dennison written in Copenhagen [while on an International Education Board (IEB) fellowship] which had the matrix elements for the symmetrical top. I realized this was just what was needed to compute the dielectric constant of a simple diatomic molecule. I requested Dennison’s permission to use them in advance of their appearing in print, and remember his wiring me permission to do so. I found that they made the factor in the
Debye formula […] for the susceptibility reacquire the classical value , replacing the nonsensical values yielded by the old quantum theory.42 (Van Vleck 1971, 8)
Van Vleck’s calculation was analogous to
I must confess that that rather burned me up because I felt it was quite a significant achievement in quantum theory. When I mentioned it to
Bohr he said “you should have got me to endorse it, it would have gone through quicker” [see also Van Vleck (1968, 1235)]. As it was, I think [Lucy] Mensing and Pauli beat me to it on being the first to publish that factor one-third. It was essentially a triple tie, though [Ralph de Laer] Kronig had it too, all three of us.43,44
Van Vleck (1971) later called it a “quadruple tie” (7), adding a paper by
While crossing the Atlantic in June 1926, Van Vleck finished another calculation in quantum mechanics only to discover upon reaching Copenhagen that he had been scooped by
When Van Vleck returned to the United States, he found that quantum theorists were in high demand and that the publication of his NRC Bulletin had earned him a reputation as one of the few in the United States who had a grasp of the theory. He had also found time that summer to write a short report on the new quantum mechanics for the Progress Committee of the Optical Society of America (Van Vleck 1928c). Leonard R. Ingersoll at the University of Wisconsin called it “the only readable synopsis of the present situation in this difficult subject” (Fellows 1985, 162).
As Van Vleck’s fame increased, he found himself wooed more and more doggedly by other universities. From the fall of 1926 through the spring of 1928, he declined offers from the University of Chicago, Princeton, and the Mellon Institute. Many of these he rejected out of a sense of loyalty to the University of Minnesota, which had been so generous to him. The department continued to recognize Van Vleck’s value, following up with raises and promotions. In June 1926 he had become an associate professor, and only a year later he became a full professor. By the summer of 1927, having married Abigail June Pearson, a native Minnesotan, he had established family ties to the state as well. It took an offer from his alma mater to win him over, and even then he vacillated for over a year before accepting a position at the University of Wisconsin (Fellows 1985, 169–175). He arrived at Madison in time for the fall semester of 1928.
Over the same period, Van Vleck had been busy pursuing the line of inquiry that would secure him fame as an expert in magnetism. He published a three-part paper that advanced a general theory of susceptibilities (Van Vleck 1927a; 1927b; 1928a). This trilogy would form the basis for The Theory of Electric and Magnetic Susceptibilities (Van Vleck 1932b).
7.3 Van Vleck receiving the National Medal of Science in 1966 from President Lyndon B. Johnson with Lady Bird Johnson looking on (picture courtesy of John Comstock).
Before turning to that volume in the next section, we wrap up this section with some brief comments about Van Vleck’s career after he left the Midwest. In early 1934, Van Vleck was offered an associate professorship at Harvard to replace
During World War II, Van Vleck was the head of the theory group at Harvard’s Radio Research Laboratory, thinking about ways to jam enemy radar, and a consultant to MIT’s much bigger Radiation Laboratory (Anderson 1987, 514).
From 1945 to 1949 he was chair of Harvard’s physics department (ibid., 519). In 1951, he succeeded
Even though Van Vleck spent the better part of his career at Harvard, he always retained a soft spot for Minnesota and Wisconsin. Together with
7.5 The Theory of Electric and Magnetic Susceptibilities
7.5.1 Writing the 1932 Book
In 1928 Van Vleck had been thinking about writing his own book on quantum mechanics, but he became interested that fall when
The delays were of a different nature than the trials and tribulations that had prevented a slightly younger Van Vleck from publishing his completed “article” in the NRC Bulletin. This time, he made his own original research a higher priority. He also accepted several invitations to give talks in Iowa, Minneapolis, and New York. This, and supervising the research of his graduate students and postdocs, took up most of his time during the 1928–29 school-year. He did manage to squeeze in one chapter, however. “I have actually, mirab[i]le dictu, completed one chapter of my book,” he wrote to
After spending the summer on research, he devoted all of his free time in the fall to the book and completed another chapter. The following spring, 1930, he negotiated a sabbatical leave in which he received half of his salary from Wisconsin, and made up the rest with a Guggenheim fellowship. He and Abigail went to Europe, making stops in England, Holland, and Germany. Finally, Van Vleck went to Switzerland while Abigail joined his parents for a tour of Italy. Unfortunately, when Van Vleck arrived at the Eidgenössische Technische Hochschule (ETH) in Zurich, he discovered that Pauli and other faculty were away on lengthy spring vacations (ibid, 240–241). Van Vleck turned this to his advantage:
The janitor at the ETH, fortunately, was very friendly and arranged for me to have the use of the library. I lived comfortably at the Hotel Waldhaus Dolder, and with a portable typewriter and no distractions by colloquia, social life or sight-seeing, I probably wrote more pages of my ‘Theory of Electric and Magnetic Susceptibilities’ in my first month at Zurich than in any other comparable time interval. (Van Vleck 1968, 1236, quoted by Fellows 1985, 242)
In June 1930, Van Vleck received an invitation to the
After receiving permission from Wisconsin, he extended his trip into the fall, finally returning in October with the book almost complete.
Reviewers immediately recognized its importance.50 Even
7.5.2 The 1932 Book and Spectroscopic Stability
Van Vleck’s The Theory of Electric and Magnetic Susceptibilities is remarkable both for the wide range of concepts it covers and techniques it assembles, and for the amount of discussion devoted to the historical development of the theories under consideration. Even though the main focus of the book is on gases, it ended up, as we mentioned in the introduction, setting “a standard and a style for American solid-state physics” (Anderson 1987, 524). As Van Vleck explained in the preface:
At the outset I intended to include only gaseous media, but the number of paramagnetic gases is so very limited that any treatment of magnetism not applicable to solids would be rather unfruitful. (Van Vleck 1932b, vii)
In the book, Van Vleck clearly demonstrated how his general
The book can be roughly divided into two parts, separated by an interstitial aside concerning the defects and demise of the old quantum theory. Chaps. I–IV constitute the first part. Here Van Vleck surveyed the classical theories of electric and magnetic susceptibilities. In addition to marshaling resources that will be drawn from in later chapters, Van Vleck carefully examined the failings of the classical theories, motivating the quantum-mechanical approach that is developed in the book’s second half.
Chap. V is the interstitial aside, which we will discuss in more detail in sec. (7.5.4). Chap. VI begins the book’s second half, which develops a quantum-theoretical approach to electric and magnetic susceptibilities. Like chap. XI of the NRC Bulletin on mathematical techniques, this chapter on “Quantum-Mechanical Foundations,” is by far the longest of the book. It takes up 59 pages (chap. XI of the Bulletin ran to 50 pages). It is so complete that, as we mentioned in sec. (7.1.4), it was sometimes used by itself as an introductory text in courses on the new theory. Although Van Vleck’s work had largely been in the tradition of matrix mechanics, his general exposition of quantum mechanics, in his book as well as in his lectures (as evidenced by the lecture notes mentioned in note 7), has none of the “Göttingen parochialism” (Duncan and Janssen 2008, 641) of
I have tried to correlate and intermingle the use of wave functions and of matrices, rather than relying exclusively on the one or the other, as is too often done. It is hoped that this chapter may be helpful as a presentation of the perturbation machinery of quantum mechanics, quite irrespective of the magnetic applications.51 (Van Vleck 1932b, viii)
Chaps. VII–XII interrogate and extend Van Vleck’s general
The book does exactly what a good textbook ought to do according to
The book reflects Van Vleck’s own trajectory, from his early work in the old quantum theory to the line of work in the new quantum theory that won him his reputation as one of the pioneering theorists of solid-state physics in the United States (cf. the remark by
is the number of molecules,
is a constant,
is the permanent electric moment of the molecule under consideration,
The idea of induced polarization is an old one […] The suggestion that part of the electric susceptibility might be due to alinement [sic] of permanent moments, resisted by temperature agitation, does not appear to have been made until 1912 by
Debye . A magnetic susceptibility due entirely to the orientation of permanent moments was suggested some time previously, in 1905, by [Paul] Langevin [1905a, 1905b], and the second term of [eq. 7.7] is thus an adaptation to the electric case of Langevin’s magnetic formula. (In the electric case, a formula such as [7.7] is commonly called just the Debye formula, but we use the compound title Langevin-Debye in order to emphasize that the mathematical methods which we use to derive the second term of [eq. 7.7] apply equally well to magnetic or electric dipoles). (Van Vleck 1932b, 30)
It is this temperature-dependent second term that Van Vleck was most interested in. We can write this term as
Both classical theory and quantum mechanics correctly predict that, under very general conditions, . The two theories agree except at very low temperatures, where the classical theory breaks down and where quantum mechanics gives deviations from (Van Vleck 1932b, 185, 197). Other than that, the factor is a remarkably robust prediction of both theories. It is true for a wide range of models (e.g., dumbbell, symmetrical top) and it is independent of the choice of a -axis for the quantization of the -component of the angular momentum in these models. The latter feature is an example of what Van Vleck called “spectroscopic stability.” As he put it in Part I of the trilogy that provided the backbone for his 1932 book:
[T]he high spectroscopic stability characteristic of the new quantum mechanics is the cardinal principle underlying the continued validity of the
Langevin-Debye formula. We shall not attempt a precise definition of the term “spectroscopic stability.”54 It means roughly that the effect of orientation or of degeneracy in general is no greater than in the classical theory, and this usually implies that summing over a discrete succession of quantum-allowed orientations gives the same result as a classical average over a continuous distribution.55 (Van Vleck 1927a, 740)
The old quantum theory gave values for
much greater than
That the susceptibility of a gas of rigid rotators does not depend on the axis of quantization is an example of spectroscopic stability. In his book, Van Vleck devoted considerable space to the “principle” or the “theorem” of spectroscopic stability (Van Vleck 1932b, 111, 139). Before giving a mathematical proof (ibid., sec. 35, 137–143), he explained the situation qualitatively in the chapter on the old quantum theory (ibid., sec. 30, 111–113). After conceding that the term, which he took from
[I]t can for our purposes be considered identical with the idea that the susceptibility is invariant of the type of quantization, or in the special case of spacial quantization, that summing over the various quantized orientations is equivalent, as far as results are concerned, to a classical integration over a random orientation of orbit. It is indeed remarkable that a discrete quantum summation gives exactly the same answers as a continuous integration. This was not at all true in the old quantum theory. (Van Vleck 1932b, 111)
In the three subsections that follow, we present derivations of the formula for the electric susceptibility in gases in classical theory (sec. 7.5.3), the old quantum theory (sec. 7.5.4), and quantum mechanics (sec. 7.5.5). In the quantum theory, old and new, we focus on the special case in which the gas molecules are modeled as rigid rotators. We will see how the robustness of the value
was established, lost, and regained. In secs. (7.5.3) and (7.5.5), we follow Van Vleck (1932b). In sec. (7.5.4), we follow
7.5.3 Susceptibilities in Classical Theory
The susceptibility of a gas, , is a measure of how the gas responds to external fields. We will consider the electric susceptibility in particular. The field, , and polarization, , are assumed to be parallel, and the medium is assumed to be both isotropic and homogenous. Predictions of require one to deal with the motions of the systems used as models for the gas molecules and their constituent atoms: the specific behavior of these systems in response to the external field will determine their electric moments, and in turn, the polarization of the medium.
Consider a small volume of a gas of molecules with permanent dipole moments, such as HCl. When an electric field is applied, say in the
-direction of the coordinate system we are using, the molecules experience a torque that tends to align them with the field. In addition, the charges in each molecule will rearrange themselves in response. If the field is too weak to cause ionization, the charges will settle into equilibrium with the field, creating a temporary induced electric moment. Both of these effects contribute to a molecule’s electric moment
. Following Van Vleck, we largely focus on the first of these effects, which, as mentioned above, is responsible for the temperature-dependent term in the
To find the polarization, , we need to take two averages over the component of these electric moments in the direction of the field , in this case the component. First, we need to average over the period(s) of the motion of the molecule (or in the case of quantum theory, over the stationary state). This is indicated by a single overbar: . Second, we need to average this time-average over a thermal ensemble of a large number of such molecules. This is indicated by a double overbar: . All derivations of expressions for the susceptibility call for this two-step averaging procedure.58
The strength of the polarization is given by:
The electric susceptibility, , is defined as the ratio of the strengths of the polarization and the external field:
When it comes to the derivation of expressions for , the various theories differ only in how and are obtained.
We first go through the calculation in the classical theory, covered elegantly in chap. II of Van Vleck’s book, “Classical Theory of the Langevin-Debye Formula” (Van Vleck 1932b, 27–41). Consider a multiply-periodic system with degrees of freedom, which, in its unperturbed state, is described by the Hamiltonian , and which is subjected to a small perturbation coming from an external electric field in the -direction. The Hamiltonian for the perturbed system can then be written as the sum , where . In this case, the full Hamiltonian is given by:
As in his NRC Bulletin, Van Vleck (1932b, 38) used action-angle variables ) for the unperturbed Hamiltonian , even when dealing with the full Hamiltonian (see also Van Vleck 1927b, 50; cf. our discussion in sec. 7.3.2).
The -component of the polarization of the system, , can be written as a Fourier expansion. For a system with only one degree of freedom the expansion is given by:59
Essentially the same Fourier expansion is the starting point both for the derivation of the
To ensure that in eq. (7.12) is real, the complex amplitudes must satisfy . Eq. (7.12) also gives the expansion for a system with degrees of freedom, if, following Van Vleck, we introduce the abbreviations , , , and (Van Vleck 1932b, 38). Through , the full Hamiltonian, , in eq. (7.11) depends on , so the action-angle variables are not action-angle variables for . The phase space element, however, is invariant under the transformation from action-angle variables for to action-angle variables for , i.e., (ibid., 39).
Using the standard formula for the canonical ensemble average, we find for (ibid., 38):61
To first order in the field , the Boltzmann factor is given by:
Assuming there is no residual polarization in the absence of an external field (which is true for gases if not always for solids), i.e., for , we have
For we insert its Fourier expansion
Only the terms on the right-hand side will contribute to the integral of over in eq. (7.16). All terms are periodic functions of , which vanish when integrated over a full period of these functions. Hence,
where in the last step we used that
This relation holds both in the classical theory and in quantum mechanics. That it does not hold in the old quantum theory is central, as we will see, to that theory’s failure to reproduce the
denotes the statistical mean square of in the absence of the field , i.e. the average over only the part of the phase space, weighted according to the Boltzmann factor, of the time average value of [in our notation: ] for a molecule having given values of the ’s [recall that short-hand for ]. Now if the applied electric field is the only external field, all spacial orientations will be equally probable when , and the mean squares of the , , and components of moment will be equal [i.e., ]. This will also be true even when there are other external fields (e.g. a magnetic field) besides the given electric field[,] provided, as is usually the case, these other fields do not greatly affect the spacial distribution. We may hence replace by one-third the statistical mean square of the vector momentum of the molecule. (Van Vleck 1932b, 39–40)
In the old quantum theory, as pointed out by
Van Vleck (1932b) called eq. (7.19) “a sort of generalized Langevin-Debye formula” (40). No particular atomic model need be assumed for its derivation. To obtain the familiar Langevin-Debye formula (7.7) with terms corresponding to permanent and induced electric moments, we need to adopt a model for the molecule of the gas similar to that underlying the classical dispersion theory of
where is the -component of the permanent electric dipole moment of the molecule, and where the coefficients are real positive numbers. Similar expressions obtain for the - and -components of .
Since positive and negative displacements will cancel during the averaging process, for (ibid., 40). If we associate a ‘spring constant’ with the linear force binding the i-th charge, then, by the equipartition theorem, we get: . Inserting eq. (7.21) for and similar equations for and for the components of in eq. (7.19) and using the relations for and , we find (ibid., 37):
As desired, the first term gives us the contribution of the permanent moment with a factor of , and the second is of the form , where is independent of temperature.
Unfortunately, the assumption that electrons can be thought of as harmonically-bound charges in the atom had to be discarded as the old quantum theory began to shed light on atomic structure. This is the same development that was responsible for the old quantum theory’s
A model such as we have used, in which the electronic motions are represented by harmonic oscillators, is not compatible with modern knowledge of atomic structure […] Inasmuch as we have deduced a generalized
Langevin-Debye formula for any multiply periodic system, the question naturally arises whether [eq. 7.19] cannot be specialized in a fashion appropriate to a real Rutherford atom instead of to a fictitious system of oscillators mounted on a rigid rotating framework. This, however, is not possible. (Van Vleck 1932b, 41)
The reason Van Vleck gave for this is that, in the Rutherford(-Bohr) atom, the energy of the electron ranges from
7.5.4 Susceptibilities in the Old Quantum Theory
Attempts to derive a formula for susceptibility in the old quantum theory, similar to the one in classical theory given above, ran afoul of some of the old quantum theory’s most striking yet little-known inconsistencies. The old quantum theory was at its best when physicists could be agnostic about the details of the multiply-periodic motion in atoms or molecules (as in the case of the
the old quantum theory replaced the factor [in the
Langevin-Debye formula 7.7] by a constant whose numerical value depended rather chaotically on the type of model employed, whether whole or half quanta were used, whether there was “weak” or “strong” spacial quantization, etc.63 This replacement of by caused an unreasonable discrepancy with the classical theory at high temperatures, and in some instances the constant even had the wrong sign. (Van Vleck 1927a, 728)
The issue of ‘weak’ or ‘strong’ quantization mentioned in this passage has to do with the question of how to quantize the unperturbed motion in the old quantum theory. Consider a rotating molecule. If a strong enough electric field is present, it makes sense to quantize the molecule’s rotation with respect to the direction of the field. But how to quantize in the absence of an external field? In that case, there is no reason to assume a preferred direction in space, and it seems arbitrary to preclude entire classes of rotational states. Yet one had to proceed somehow. Two different kinds of quantization could be assumed (Van Vleck 1932b, 106). In the first, called ‘strong spatial quantization,’ rotation was assumed to be quantized with respect to the field even when there was, as yet, no field. In the other, called ‘weak spatial quantization,’ molecules were assumed to be in some intermediate state between ‘strong quantization’ and a classical distribution of rotational states.64 Van Vleck highlighted this conceptual conundrum:
Spacial quantization cannot be effective unless it has some axis of reference. In the calculation of
Pauli and Pauling […] the direction of the electric field is taken as such an axis […] [I]n the absence of all external fields […] there is no reason for choosing one direction in space rather than another for the axis of spacial quantization. (Van Vleck 1932b, 108)
We need to take a closer look at these calculations by
Implicitly assuming strong spatial quantization,
The equations on the blackboard behind Van Vleck in the picture in fig. (3.1) may serve as a reminder that even this sanitized version (7.26) of the quantum conditions (7.24–7.25) is not how angular momentum is quantized in modern quantum mechanics.67 This modern treatment of angular momentum underlies the calculations of susceptibilities by
To find the susceptibility of a gas of rigid rotators in the old quantum theory,
, the value of
, is the total energy of the molecule (Pauli 1921, 322; Pauling 1926b, 570). Using eq. (7.28),
In the evaluation of
, a distinction needs to be made between two energy regimes (Pauli 1921, 322). In the first, the molecules have energies
much smaller than
, the energy of the interaction between the electric moment and the field. In the second,
is much larger than
. The calculations of
The ratio on the right-hand side corresponds to the time average for the unperturbed system. In the classical theory,68 but not in the old quantum theory, the ensemble average, , of this time average, (both for the unperturbed system) is equal to . This is the same point that Van Vleck (1932b, 39–40) made in one of the passages we quoted in sec. (7.5.3): (see eq. 7.20). It thus follows from the classical counterpart of eq. (7.29) (see note 68) that the ensemble average, , of the time average, (now both for the perturbed system) vanishes.
According to the classical theory, in other words, there is no contribution to the susceptibility at all from molecules in the energy regime
for which the classical counterpart of eq. (7.29) (see note 68) was derived. As
This energy, in turn, can be expressed in terms of a new quantity (Pauli 1921, 326):
The ensemble average of is given by (Pauli 1921, 325):
(Pauli 1921, 326; Pauling 1926b, 571). Evaluating these sums for integer and half-integer quantum numbers, respectively, and multiplying by
As we mentioned in sec. (7.4),
Hence, up to an additive constant, the energy is given by squares of half-integers rather than integers, as
In the old quantum theory, would be given by times the expression on the right-hand side of eq. (7.29) for . In the new quantum theory, is given by
for and , respectively (ibid., 512).73