![Contents
List of Plates xiv
8 Introduction to Volume 2 1
8.1 Overview 1
8.2 Quantum theory in the early 1920s: deficiencies and discoveries
(exclusion principle and spin) 3
8.3 Atomic structure à la Bohr, X-ray spectra, and the discovery of
the exclusion principle 4
8.3.1 Important clues from X-ray spectroscopy 8
8.3.2 Electron arrangements and the emergence of the exclusion
principle 10
8.3.3 The discovery of electron spin 15
8.4 The dispersion of light: a gateway to a new mechanics 19
8.4.1 The Lorentz–Drude theory of dispersion 20
8.4.2 Dispersion theory and the Bohr model 22
8.4.3 Final steps to a correct quantum dispersion formula 26
8.4.4 A generalized dispersion formula for inelastic light
scattering—the Kramers–Heisenberg paper 33
8.5 The genesis of matrix mechanics 37
8.5.1 Intensities, and another look at the hydrogen atom 37
8.5.2 The Umdeutung paper 41
8.5.3 The new mechanics receives an algebraic framing—Born
and Jordan’s Two-Man-Paper 44
8.5.4 Dirac and the formal connection between classical
and quantum mechanics 46
8.5.5 The Three-Man-Paper [Dreimännerarbeit]—completion of
the formalism of matrix mechanics 48
8.6 The genesis of wave mechanics 53
8.6.1 The mechanical-optical route to quantum mechanics 53
8.6.2 Schrödinger’s wave mechanics 60
8.7 The new theory repairs and extends the old 68
8.8 Statistical aspects of the new quantum formalisms 71
8.9 The Como and Solvay conferences, 1927 80
8.10 Von Neumann puts quantum mechanics in Hilbert space 86](https://image.slidesharecdn.com/107306992-250320100712-f57b53ad/85/Ebook-Constructing-Quantum-Mechanics-by-Anthony-Duncan-and-Michel-Janssen-14-320.jpg)
![2 Introduction to Volume 2
turn to the Umdeutung [reinterpretation] paper with which Heisenberg (1925c) laid the
foundations of matrix mechanics (Chapter 11). We then examine its elaboration in the
Two-Man-Paper of Born and Jordan (1925b) and the Three-Man-Paper [Dreimänner-
arbeit] of Born, Heisenberg, and Jordan (1926), both coming out of Göttingen, as well as
the paper in which Dirac (1925), working in Cambridge, introduced the closely related
q-number theory (Chapter 12).
Wave mechanics, unlike matrix mechanics, grew out of work done by the pio-
neers of quantum theory before Bohr’s 1913 trilogy (see Chapters 2 and 3). These
pioneers—most importantly Planck, Einstein, and Ehrenfest—were not concerned with
the structure of individual atoms, but rather with the statistics of large collections of
simple systems modeled as harmonic oscillators. Instead of spectroscopy, they studied
black-body radiation and the specific heat of solids at low temperatures—two areas where
the equipartition theorem of the kinetic theory of gases of Maxwell and Boltzmann noto-
riously broke down. The year before the arrival of matrix mechanics, Indian physicist
Satyendra Nath Bose (1924) proposed a new derivation of the Planck law for black-body
radiation, using a new statistics for light quanta, which Einstein (1924, 1925a, 1925b)
subsequently transferred to atoms in his quantum theory of the ideal gas. Einstein found
that the density fluctuations in such a gas are given by a formula like the one he had found
earlier for fluctuations in black-body radiation (1909a, 1909b), consisting of a wave and
a particle term. This supported a suggestion made in the dissertation by the French
physicist de Broglie (1924) to extend the wave-particle duality Einstein had proposed
for radiation to matter. Drawing on this work by De Broglie, Bose, and Einstein (Chapter
13), and on the much older optical-mechanical analogy due to William Rowan Hamil-
ton (1834, 1835) (Section 13.3), Schrödinger (1926c, 1926d, 1926f, 1926h), working
in Zurich rather than in one of the centers of the old quantum theory, developed wave
mechanics in a celebrated four-part article (Chapter 14).
In Part IV, consisting of just two chapters (Chapters 16–17), we cover the comple-
tion of the formalism of quantum mechanics and its probabilistic interpretation. By the
middle of 1926, three authors, independently of one another, had already established
that matrix and wave mechanics are intimately related (see Section 14.5). Schrödinger
(1926e) and Eckart (1926) gave their “equivalence proofs” in published papers, and
Pauli (1979, Doc. 131) gave his in private correspondence. In December 1926, Dirac
(1927a) and Jordan (1927b) published, again independently of one another, what has
come to be known as the Dirac–Jordan statistical transformation theory (Section 16.2).
Their formalism unified all four versions of quantum theory that had meanwhile been
proposed: matrix mechanics, wave mechanics, Dirac’s q-number theory, and the oper-
ator calculus of Born and Wiener (1926, see Section 14.5). As the term “statistical”
suggests, Dirac and Jordan generalized the probabilistic interpretation of Schrödinger’s
wave function proposed by Born (1926a, 1926b) in papers on collision processes, and
by Pauli (1979, Doc. 143), true to form, in private correspondence (Section 16.1). As
the term “transformation theory” suggests, canonical transformations familiar to Jordan
and Dirac from the celestial mechanics used in the old quantum theory (see Appendix
A) played a central role in the Dirac–Jordan formalism. These mathematical techniques,
however, proved ill-suited to the task. This was clearly recognized by the young Hun-
garian mathematician Janos von Neumann, who came to work with Hilbert in Göttingen](https://image.slidesharecdn.com/107306992-250320100712-f57b53ad/85/Ebook-Constructing-Quantum-Mechanics-by-Anthony-Duncan-and-Michel-Janssen-23-320.jpg)
![6 Introduction to Volume 2
terms of the optical series of the elements (especially the alkali metals, such as sodium)
using the methods of the old quantum theory (see Section 7.1), as well as a recapitulation
of the applications of correspondence-principle ideas, for example, in the interpretation
of spectra in the Stark effect (see Section 6.3). In the concluding section, Bohr apologizes
for the lack of attention specifically to questions of atomic structure, and points forward
to the need for a more sophisticated approach to the problem of the configuration of
electronic orbits:
In the foregoing I have intentionally not delved more closely into the question of the
structure of atoms and molecules, although this subject is clearly intimately connected with
the theory of the origin of spectra which we have considered . . . On the occasion of the
first treatment of this matter [Bohr’s trilogy of 1913], the author also attempted to sketch
the foundations of a theory for the structure of the atoms of the elements [Part II] and for
the chemical combinations of molecules [Part III] . . . I would like to take the opportu-
nity here, however, to state that from the standpoint of more recent developments of the
quantum theory . . . many of the detailed special assumptions made in the [earlier] the-
ory must be altered, on the basis of disagreement of the theory with experimental results
which have been raised on numerous fronts. In particular it does not seem possible to
justify the orienting assumption introduced at that time [1913], that in the ground state
electrons move in geometrically particularly simple orbits, such as “electron rings.” Con-
siderations of the stability of atoms and molecules with regard to external influences, and
on the possibility of construction of the atom by successive acquisition of single electrons,
require us to insist, first, that the electron configurations to be considered are mechani-
cally stable, and moreover possess a certain stability with regard to the requirements of
the usual mechanics, and second, that the configurations to be used are so constituted
that they may be arrived at by transitions from other stationary states of the atom. These
requirements are in general not fulfilled by such simple configurations as electron rings, and
force us to look around for more complicated form of motion (Bohr 1920, pp. 468–469, our
emphasis).
In the three years following this work, Bohr delved more closely into the problem of
atomic structure, specifically, in determining the mutual orientation of the orbits fol-
lowed by electrons in multi-electronic atoms, and relating the proposed arrangement of
atomic orbits to the observed chemical properties and optical spectra of the elements,
with particular attention to the recurring periodicity of these properties with atomic
number embodied in the periodic table. There would be two short Nature papers (Bohr
1921a, 1921c) as well as a series of longer papers, often elaborated versions of lectures
Bohr presented at various venues—and often to audiences who were not atomic theorists,
and could not therefore follow the more technical aspects of Bohr’s program—such as
Bohr (1921d), a greatly expanded version of a lecture in Copenhagen in October 1921,
and Bohr (1922e), based on his legendary Wolfskehl lectures in Göttingen in June 1922,
which became known as the “Bohr festival” (Bohr Festspiele) (Mehra and Rechenberg
1982a, sec. III.4, pp. 332–358).
The general nexus of ideas proposed in these papers has become known as “Bohr’s
second atomic theory” (see Kragh 1979, 2012, Ch. 7). In these works, Bohr attempted
to develop and apply the ideas visible in embryo form in the quote given above. First, an](https://image.slidesharecdn.com/107306992-250320100712-f57b53ad/85/Ebook-Constructing-Quantum-Mechanics-by-Anthony-Duncan-and-Michel-Janssen-27-320.jpg)
![8 Introduction to Volume 2
be reached (by the third electron) from higher orbits by a transition compatible with
correspondence principle ideas:
Such a transition process, of which one must assume (as a more detailed examination
of the possibilities of motion shows [this is never given]) that it would lead to an atomic
state in which the third electron would appear as a equivalent participant in the col-
lective motion of the three electrons, would in fact be of a quite different type as the
transitions between stationary states connected with the emission spectrum of lithium,
and in contrast to these would display no correspondence to the harmonic components
of the motion of the atom (Bohr 1922b, p. 35).
Sentences of this sort abound in the sequence of articles and lectures mentioned above
whenever the physico-chemical properties of an element required the addition of an
electron in a new, less-bound orbit to the preceding ones. Appeals to the correspon-
dence principle were therefore of critical importance in Bohr’s attempts to understand
the periodicity of these properties in terms of a shell structure (more precisely, a group-
ing into subclasses of similar orbits) of the electrons surrounding larger atoms. Inasmuch
as Bohr’s final (pre-exclusion principle) specification of electronic shell structures, in
a paper co-authored with Dirk Coster (Bohr and Coster 1923, p. 344), still has an
incorrect assignment of quantum numbers (as, for example, in the filled third shell for
elements from copper onward, containing six each of s, p, and d electrons, instead of 2,
6, and 10, respectively), we know that these arguments must have been faulty, even if
they existed in a more explicit mathematical form than is apparent from the published
work. In Chapter 9 we discuss Bohr’s arguments in more detail,3
and demonstrate that
atomic systems subject to the correspondence principle, but lacking spin and exclusion
properties, simply do not behave as Bohr asserted.
8.3.1 Important clues from X-ray spectroscopy
In our discussion in Chapter 7 of the various attempts to deal with the appearance of
“extra” lines—hence, extra quantum states—in the study of the complex optical spec-
tra that occurred especially in larger atoms, we saw that the new levels were at first
absorbed into the theory by the purely phenomenological ruse of introducing a new
quantum number, Sommerfeld’s “inner quantum number”. A dynamical explanation
for this new degree of freedom would eventually (in the few years remaining to the
old quantum theory) come to be widely accepted: namely, that the splitting of Bohr–
Sommerfeld levels associated with this new quantum number was due to the freedom
to alter (discretely) the relative orientation of the orbital angular momentum of an outer
valence electron (the quantum state of which was under consideration) to the orienta-
tion of the combined angular momentum of the atomic core (i.e., all the inner electrons).
A further intriguing phenomenological observation would soon prove to play a critical
3
See also the web resource, Bohr’s Second Theory and Atomic Aufbau before Spin and the Exclusion Principle.](https://image.slidesharecdn.com/107306992-250320100712-f57b53ad/85/Ebook-Constructing-Quantum-Mechanics-by-Anthony-Duncan-and-Michel-Janssen-29-320.jpg)
![10 Introduction to Volume 2
Let us first consider the simplest case, the “L” X-ray terms corresponding to inner
electrons with principal quantum number N = 2, in atoms of large Z (> 50, say).
Pairs of X-ray terms diverging for large Z from each other as the cube of the atomic
number were indeed well established by the work of Gustav Hertz in early 1920, and
were dubbed “regular doublets” by Sommerfeld. Then, there were the “extra” terms,
which maintained a more or less constant (i.e., Z-independent) difference (rather than
increasing as the cube of atomic number) in Moseley ordinate from one of Sommerfeld’s
“regular” terms! In fact, this situation is the one we should have anticipated from the
fact that electron orbits of differing angular momentum would penetrate the innermost
region of the atom (and be exposed to the full nuclear charge) to differing degrees,
and hence, would have different screening constants. Two such terms would then have
Moseley terms exhibiting a constant difference in the leading (i.e., Balmer) energy square
root:
p
(Z − σ1)2 −
p
(Z − σ2)2 ∝ σ1 − σ2 (hence, independent of Z ). This would be
much larger than, and should therefore always dominate, the Sommerfeld fine-structure
shifts (with their extra factor of the square of α). The problem was that one found both
the “regular” doublets, with a splitting that looked just like relativistic fine structure, and
“irregular” doublets with a constant Moseley splitting—which was perfectly reasonable
on the basis of the expected difference in nuclear charge screening for orbits of different
shape (see Figure 9.2), but seemed to lack the expected relativistic corrections for orbits
differing in angular momentum.
As mentioned earlier, an important reason for the increasing interest in X-ray studies
at this point was the hope (already expressed by Sommerfeld in the first edition of Atom-
bau und Spektrallinien; cf. Volume 1, p. 277) that the nuclear Coulomb attraction on an
inner electron would so dominate the relatively small inter-electron repulsive interac-
tions (which could be ignored for the outer electrons) that the motion of these electrons
would be very close to that expected on the basis of the Bohr–Sommerfeld formalism. In
particular, the peculiar complex spectra effects in optical spectra (i.e., in the energies of
the outer electron orbits) presumed to arise from an interaction of the valence electron
orbit with a magnetic field produced by the “core” (i.e., the nucleus plus inner electrons)
should no longer be relevant, as the innermost electrons were the core, and the screening
effects exerted on each other could apparently be accounted for very accurately simply
by a small shift in the effective nuclear charge, Z → Zeff, as indicated above.
Nevertheless, the triplet of L (N = 2) terms, quintuplet of M (N = 3) terms, etc.,
clearly indicated that an additional degree of freedom was in play for these inner electrons
as well, giving rise to “extra” stationary-state energy levels. And the Z-dependence of
the extra splittings, as we have just seen, was inexplicable in terms of simple screening
plus relativistic effects.
8.3.2 Electron arrangements and the emergence
of the exclusion principle
These matters came to a head in an influential paper written in late 1922 by Bohr and
Coster (1923) entitled “Röntgen [X-ray] spectra and the periodic system of the ele-
ments.” Coster had worked on X-ray spectroscopy at Lund University with Manne](https://image.slidesharecdn.com/107306992-250320100712-f57b53ad/85/Ebook-Constructing-Quantum-Mechanics-by-Anthony-Duncan-and-Michel-Janssen-31-320.jpg)
![Atomic structure à la Bohr, X-ray spectra, and the discovery of the exclusion principle 11
Siegbahn’s group in the preceding two years before coming to visit the Niels Bohr Insti-
tute in Copenhagen in the summer of 1922. In fact, his dissertation, completed under the
direction of Paul Ehrenfest at the University of Leyden earlier that year, had been pre-
cisely on the subject of “Röntgen spectra and the atomic theory of Bohr” (Coster 1922).
Coster had at his fingertips all the latest results for X-ray term levels in a wide range of
elements, which now represented a greatly expanded cornucopia of phenomenological
information, compared to the relatively sparse results of Hertz just two years earlier. Bohr
and Coster undertook an extensive review of the data and gave a detailed discussion of
the various screening effects to be expected in the motion of an inner electron. This
motion, while dominated by the very strong nuclear Coulomb field (we are thinking of
atoms with a fairly large atomic number here), would also be affected occasionally by the
penetration of an outer electron with a very elliptical orbit (i.e., low angular momentum)
into the inner region, and more frequently by the presence of the electrostatic repulsion
of the electrons of the same or smaller principal quantum number as the electron in
question. Bohr and Coster asserted that these effects could be subsumed into screening
shifts (such as the factor σ shifting the nuclear charge Z discussed above). Unfortu-
nately, the appearance of extra “anomalous” (in the Bohr–Coster terminology) energy
levels leading to the apparently incompatible simultaneous appearance of “screening”
and “relativistic” doublets could not be reconciled with the fundamental tenet of the
Aufbau program, which assumed that the addition to or subtraction from the atom of
a single electron would not appreciably affect the orbits (or energies) of the remaining
energies. Instead, as we discuss in more detail in Chapter 9, one was forced to assume
that the removal of an inner electron resulted in an essential (wesentliche) change of
the behavior of the other electrons in the same shell (or “group”, in the Bohr–Coster
language). In fact, the authors state
Such a conception is necessitated by the theory if it is assumed that the completion of an
electron group is essentially determined by the interaction of electrons within the various
subgroups [of the shell] (Bohr and Coster 1923, p. 365).
The mysterious “interaction between subshells” referred to here can (with hindsight!)
be regarded as a foreshadowing of the constraints that would, by the end of the next year,
be recognized to arise from the exclusion principle, where the completion of the electron
subshells within a shell (of given principal quantum number) is indeed determined by
the previous occupancy of these subshells. But in the absence of any understanding of
either the additional physical spin degree of freedom of the electrons or the completely
non-classical constraints imposed on the electron states by exclusion, Bohr and Coster
had to admit that “it is still quite unclear how a more detailed execution of this idea
could provide an explanation for the separate appearance of screening- and relativity-
doublets” (p. 365).
A few months after the publication of Bohr and Coster’s paper, a major step toward
the exclusion principle appeared in an article entitled “The distribution of electrons
among atomic levels,” submitted to the Philosophical Magazine by Edmund Stoner](https://image.slidesharecdn.com/107306992-250320100712-f57b53ad/85/Ebook-Constructing-Quantum-Mechanics-by-Anthony-Duncan-and-Michel-Janssen-32-320.jpg)
![16 Introduction to Volume 2
attributed to each valence electron separately, and not to a single putative atomic core.
Kronig later claimed that “it occurred to me immediately that [the fourth quantum num-
ber] might be considered as an intrinsic angular momentum of the electron” (Kronig
1960, p. 20). Kronig did not leave the matter there, but immediately set out to reinter-
pret the puzzling relativistic doublets of X-ray spectroscopy model-theoretically with this
new idea. We remind the reader that the existence of pairs of inner electron states which
appeared to be split in energy by an amount given by the Sommerfeld fine-structure for-
mula from 1916—in particular, with a splitting proportional to the fourth power of the
effective nuclear charge Zeff—which had initially seemed such a great success of the old
quantum theory, had come to be seen as a serious difficulty once further energy levels
were discovered with an energy splitting apparently purely due to screening (and not
relativistic) effects. The critical problem was the absence of obvious screening effects
in the relativistic doublets, which (Landé had argued) meant that these doublets corre-
sponded to pairs of electron states with the same spatial orbits (thus, azimuthal quantum
number k = k1) and so the same screening, but different orientation of the electron
orbital momentum to the core angular momentum. In other words, per Landé, these
splittings would be analogous to the alkali metal doublet splittings which Heisenberg
had calculated in the ancient days of late 1921 with his core model (cf. Section 7.1.2).
The problem was that, in the core model, the doublet splittings, when computed with an
effective nuclear charge Zeff appropriate for an inner electron in a large atom, inevitably
emerged with a dependence proportional to the cube, rather than fourth power, of Zeff,
which was the observed behavior.9
Kronig realized that with the transference of the fourth quantum number to an intrin-
sic spin of the electron (rather than the atomic core) engaged in the transition (optical
or X-ray) of interest, the energy shifts due to the core–valence magnetic interaction
would acquire the additional factor of effective nuclear charge, Zeff, needed to convert
the Heisenberg core–model Z3
eff dependence to the Z4
eff form observed in the relativistic
doublets. The reason is quite simple. In the Heisenberg core model, doublet splittings
arose because of the varying orientation of the magnetic moment associated with the
atomic core with respect to the magnetic field generated by the circling charge −e of
the valence electron. In the new picture, which appears for the first time in Kronig’s
calculations of January 1925, the energy modifications to the Balmer–Bohr term levels
arise because the magnetic moment due to the spin of the circulating electron assumes
different orientations with respect to the magnetic field (in the frame of the moving
electron) arising from the nuclear charge, which is Zeffe: this magnetic field is therefore
Zeff larger than that felt by the core in the earlier model, where the magnetic field at
the core was that generated by a single outer valence electron. The result obtained by
Kronig was about a factor of 2 larger than the empirically observed splitting (on which
more in Chapter 9), but the recovery of the correct Z-dependence was certainly a most
encouraging development. Or so it would seem.
9
Note that an energy correction of order Z4
to the leading order Z2
Bohr–Balmer energy will lead to a
correction of order Z3
in the Moseley ordinate, which reflects the square root of the total binding energy.](https://image.slidesharecdn.com/107306992-250320100712-f57b53ad/85/Ebook-Constructing-Quantum-Mechanics-by-Anthony-Duncan-and-Michel-Janssen-37-320.jpg)
![18 Introduction to Volume 2
paper (less than two pages) to Die Naturwissenschaften in which they proposed a “dif-
ferent approach” to the problem of the fourth quantum number R previously used in
the theory of the anomalous Zeeman effect to indicate the angular momentum of the
atomic core, the magnetic interaction of which with both the valence electron and the
external magnetic field was a crucial part of the phenomenology of the Zeeman effect,
in particular, in the heuristic “derivation” of the formula for the Landé g-factor (cf.
Sections 7.3.2–7.3.3). After describing the ascription by Pauli of a two-valued fourth
quantum number to the valence electron itself (i.e., its removal from the atomic core),
Goudsmit and Uhlenbeck suggest that, while the other three quantum numbers (prin-
cipal, azimuthal, and inner) retained their previous interpretation, the quantum number
R was to be ascribed directly “to the rotation of the electron itself” (Uhlenbeck and
Goudsmit 1925, p. 954). There are no explicit calculations presented to support this
hypothesis, but the authors do recognize in closing that the viability of this hypothesis
rests crucially on its ability, on the basis of the “variable orientations of R to the orbital
plane of the electron [hence, the direction of the orbital angular momentum vector K],”
to yield a correct explanation of the relativity doublets.
Goudsmit and Uhlenbeck’s paper had not had an entirely smooth trip from concep-
tion to publication, as Uhlenbeck had asked Lorentz to assess the dynamical consistency
of an intrinsic electron spin, which, as the acknowledged master of classical electron
theory, Lorentz was clearly in position to do. When Lorentz got back to Uhlenbeck, it
was with the bad news that any interpretation of a spin angular momentum of an elec-
tron based on a model of a rotating charge distribution ran into conflicts with special
relativity, as the surface of the electron sphere would be moving faster than the speed
of light. Objections of this sort had also probably disturbed Pauli much earlier, when
he dismissed Kronig’s suggestion of a mechanical interpretation of his fourth quantum
number. Fortunately for Goudsmit and Uhlenbeck, Ehrenfest, who was responsible for
sending the paper in to the journal, was far more willing to allow his students to entertain
“dangerous” ideas, and their short note, establishing for posterity their priority in the
discovery of electron spin, proceeded to Die Naturwissenschaften unimpeded and was
duly published on November 20.
The day after the paper appeared in print, Heisenberg wrote to Goudsmit enquiring
about the factor of 2 disagreement in the spin–orbit coupling energy for inner electrons—
the empirical splittings seemed systematically about twice as large as one would expect
from a direct calculation of the coupling energy of a magnetic electron moving in the
nuclear field. Everything else about this result (which, of course, goes back to Kronig)
was encouraging, in particular the crucial correct Z4
dependence that had been such
a mystery in the atomic core models. The same issues were raised in a separate letter
to Pauli sent the same day, in which Heisenberg asks for Pauli’s opinion on the whole
idea of a spinning electron. Heisenberg leaves no doubt that, for him, apart from the
annoying extra factor of 2, the spin idea “eliminates all difficulties with a magic stroke”
(Pauli 1979, Doc. 107). The stage was certainly set for a widespread acceptance of the
idea of electron spin in the months to come. By the end of the year, Bohr, on the basis of
conversations with Einstein explaining the relativistic origin of the spin–orbit coupling,
had also accepted the idea of an intrinsic electron angular momentum. As we shall see](https://image.slidesharecdn.com/107306992-250320100712-f57b53ad/85/Ebook-Constructing-Quantum-Mechanics-by-Anthony-Duncan-and-Michel-Janssen-39-320.jpg)
![20 Introduction to Volume 2
solid.14
Newton’s experiments with prisms in the 1660s had established the essential
facts underlying optical dispersion: a circular “pencil” of white light could be decom-
posed after passage through a prism into an elliptical spot displaying the colors of the
spectrum; if a particular color (say, blue) was selected by passing the light through a
small hole at the appropriate spot, the light remained blue in subsequent refractions
through other prisms; and the light at the blue end of the spectrum was more bent (more
refrangible, in Newton’s terminology) than the light at the red end. In terms of the index
of refraction n, which had already been introduced in quantitative accounts of refrac-
tion by Snell and Descartes, the existence of a variable refrangibility indicated that the
index of refraction, which was a quantitative expression of the extent to which a light
ray underwent bending in passage from air into a transparent medium, was a number
greater than one that increased monotonically as one went from the red to the blue end
of the spectrum in typical refractive media (such as prism glass).
With the advent of the wave theory 150 years later, and the ascription of color to the
varying wavelengths λ (inversely related to frequency ν by λ = c/ν, where c is the speed
of light), the task of physical optics became clear. Any complete theory of the interaction
of light with a material medium through which it passes must be capable of explaining
quantitatively the functional dependence of the index of refraction on frequency, n(ν).
By the close of the nineteenth century, Maxwell’s electromagnetic theory of light was
the dominant paradigm, and the interaction of light with matter was presumed to be
determined by the effect of the oscillating electric field generated by a light wave as it
passes over the constituent charged “ions” of opposite and, in bulk, cancelling signs in
an electrically neutral medium. The point of view generally accepted for the underlying
process can be seen in the following excerpt from the second edition of Drude’s textbook
on optics, in the first section of Chapter 5, “The Dispersion of Bodies:”
One arrives at a theory which adequately represents the observed phenomena by intro-
ducing the assumption that the smallest parts . . . of a body possess the possibility of
characteristic vibrations. These [vibrations] are excited to a greater or lesser sense to the
extent that their [natural] period lies nearer or further from that of the light vibrations
impinging from outside. Such vibrations, stimulated by a light wave (i.e., an oscillating
electrical force) are immediately understandable if we generalize the conception (made
necessary by electrolysis) that each molecule of a body is composed out of positive or
negative atoms, the so-called ions. These ions are not in general identical with those
obtained in electrolysis and they are to be referred to by a different word, for example
electrons (Drude 1906, pp. 362–363).
8.4.1 The Lorentz–Drude theory of dispersion
The idea that physical optics had become a study of the interaction of light with harmon-
ically bound “mobile ions” within the atoms or molecules of a material substance had
already been used by Lorentz a decade before the second edition of Drude’s textbook
14
See the dissertation of Marta Jordi Taltavull (2017) for a history of dispersion in the period of interest to
us (1870–1925).](https://image.slidesharecdn.com/107306992-250320100712-f57b53ad/85/Ebook-Constructing-Quantum-Mechanics-by-Anthony-Duncan-and-Michel-Janssen-41-320.jpg)
![26 Introduction to Volume 2
in general be brought into agreement with the actual number of radiating charged par-
ticles in the optical medium of interest, introduced a concept that would play a very
important role in the 1924/1925 period, and which would lead to a final, and concep-
tually satisfactory, solution to the quantum dispersion problem for atomic models of
the Bohr type. Namely, it was assumed that optical dispersion phenomena could be
understood by treating the radiating charged particles in atoms as “substitute oscilla-
tors” (Ersatzoszillatoren). In other words, as far as phenomena of absorption, emission,
or coherent scattering (dispersion) were concerned, atomic electrons circulating in mul-
tiperiodic, quasi-Keplerian stationary state orbits, for some reason, acted as though they
were actually charged simple harmonic oscillators, with each Bohr transition r → s with
emitted frequency νrs associated with a corresponding substitute oscillator with a natu-
ral frequency of oscillation equal to νrs. The peculiar values of the Sellmeier numerator
factors (now connected to the Einstein spontaneous emission coefficient As
r) could now
be adjusted on the assumption that the substitute oscillators were to be treated as having
non-standard values for the electron charge e and/or mass m. The idea of such oscilla-
tors as the responsible entities for the emission of radiation from atoms was not in fact
entirely new: in a response to a question from Langevin in the question session follow-
ing his talk at the 1921 Solvay conference (“Notes on the theory of electrons”), H. A.
Lorentz had speculated
one could for example imagine that besides the atoms of Bohr a [radiating] gas contains
true “vibrators” [oscillators] which could provisionally store the energy lost by the atom
in passing from one stationary state to another; it would be necessary in this case for
the quantity of [stored] energy be exactly the quantum corresponding to the proper
frequency of the vibrator [i.e., hν for an oscillator of natural frequency ν] . . . Perhaps
also the atom temporarily changes itself into a vibrator (Solvay 1923, p. 24).
8.4.3 Final steps to a correct quantum dispersion formula
The derivation of correct quantum theoretical expressions for the interaction of bound
atomic electrons with the electromagnetic field would be accomplished, more or less
independently, by two physicists working in different continents and in very different
scientific environments. The first, Kramers, Bohr’s trusted assistant in Copenhagen, we
have already met on several occasions in connection with the Stark effect and the Bohr–
Kemble model of helium (see Sections 6.3 and 7.4). The other physicist, the young
American theorist John Van Vleck, we encountered in our account of the final (failing)
attempts of the old quantum theory to deal with the binding energy of two electrons
in the helium atom (see Section 7.4). Indeed, the extra technical equipment essential
in their attack on the helium atom, a complete facility with canonical perturbation the-
ory in classical mechanics, was the essential ingredient that allowed both Kramers and
Van Vleck to go beyond the work of Ladenburg and Reiche, and arrive at formulas for
emission, absorption, and (in the case of Kramers, and later, Heisenberg) both coher-
ent (dispersion) and incoherent (Raman) scattering, formulas that survived completely
intact the transition to modern quantum mechanics. Of course, the formulas in question](https://image.slidesharecdn.com/107306992-250320100712-f57b53ad/85/Ebook-Constructing-Quantum-Mechanics-by-Anthony-Duncan-and-Michel-Janssen-47-320.jpg)
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- 8. Constructing Quantum Mechanics Volume Two The Arch: 1923–1927 Anthony Duncan and Michel Janssen
- 9. Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Anthony Duncan and Michel Janssen 2023 The moral rights of the authors have been asserted All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2023930769 ISBN 978–0–19–888390–6 DOI: 10.1093/oso/9780198883906.001.0001 Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work.
- 10. Dedicated to the memory of Philip M. Stehle (1919 –2013) and Roger H. Stuewer (1934 –2022)
- 11. Preface This is the second of two volumes on the genesis of quantum mechanics in the first quarter of the twentieth century. For some general comments on our goals and approach in these volumes, see the preface of Volume 1. As we did for Parts I and II in Volume 1 (see Chapter 1), we begin Volume 2 with an overview, as non-technical as possible, of the contents of Parts III and IV (see Chapter 8). As in Volume 1, we wrote many sections of Volume 2 with the idea that they be read in tandem with the original papers discussed in them. The papers most important for the development of both matrix and wave mechanics (covered in Part III) are avail- able in English translations. The anthology edited by B. L. van der Waerden (1968) presents translations of the papers documenting the development of matrix mechan- ics (see Chapters 10–12). Schrödinger (1982) conveniently collects translations of the papers in which he introduced wave mechanics (see Chapter 14). Partial translations of De Broglie’s (1924) dissertation and five key papers by Schrödinger can be found in an anthology on wave mechanics edited by Günther (or Gunter) Ludwig (1968). We refer to the papers included in van der Waerden (1968) and Schrödinger (1982) by using the page numbers of these volumes. Unfortunately, several of the papers in which the theory was developed further in 1926–1927 (covered in Part IV) have not yet been translated. Our discussions of the original papers serve three purposes. First, they should enable a reader with a background in physics and mathematics comparable to that of an advanced physics undergraduate or beginning graduate student to follow the arguments and derivations in these papers.1 Secondly, we explain how these papers are related to other papers upon which they build or to which they respond. Finally, we place these papers in the broader context of the debates over the developing quantum theory, in published papers, at public talks and conferences, and in private correspondence. For this third task we relied heavily on editions of correspondence (as well as papers and manuscripts) of several prominent protagonists in our story. Especially important for Volume 2 were Niels Bohr’s Collected Works (Bohr 1972–2008, especially Vols. 4 and 6), The Collected Papers of Albert Einstein (Einstein 1987–2021, Vols. 12–15), and the late Karl von Meyenn’s edition of the correspondence of Wolfgang Pauli (in particular Vol. 1, Pauli 1979). The microfilms of the Archive for History of Quantum Physics (referred to as AHQP microfilm) are our source for unpublished correspondence. A finding aid prepared by Thomas S. Kuhn, John L. Heilbron, Paul Forman, and Lini Allen (1967) gives the reel and frame numbers for all letters included in this collection. 1 To this end, we added several more web resources (cf. the preface of Volume 1): on Bohr’s atomic theory of the early 1920s, on Thomas precession, on intensities of spectral lines in the old and the new quantum theory, on time-dependent perturbation theory in the Three-Man-Paper of Born, Heisenberg, and Jordan, on extremal principles, and on (Schrödinger’s original handling of) the radial Schrödinger equation.
- 12. Preface vii For the third task we also made heavy use of the massive six-volume history of quantum mechanics by Jagdish Mehra and Helmut Rechenberg (1982a, 1982b, 1982c, 1982d, 1987, 2000–2001). While Mehra and Rechenberg offer little help to those trying to understand the detailed arguments and derivations, their work remains invaluable in placing those arguments and derivations in a broader biographical and institutional con- text. Later reminiscences, although they need to be handled with care, were useful as well (see, e.g., Fierz and Weisskopf 1960; Rozental 1967; Heisenberg 1971). So were several biographies (both those written for a scholarly and those written for a general audience), such as David Cassidy (1991, 2009) on Werner Heisenberg, Max Dresden (1987) on Hans Kramers, Michael Eckert (2013) on Arnold Sommerfeld, Graham Farmelo (2009) and Helge Kragh (1990) on Paul Dirac, Nancy Greenspan (2005) on Max Born, Wal- ter Moore (1989) on Erwin Schrödinger, Abraham Pais (1982a, 1991) on Einstein and Bohr, and, more recently, Ananyo Bhattacharya (2022) on John von Neumann. We were able to find on Wikipedia the dates and other biographical information for all but the most obscure characters. We have also tried, as much as we could, to consider the sprawling secondary literature on the genesis of quantum mechanics. Especially important for Volume 2 were Guido Bacciagaluppi and Antony Valentini’s (2009) edition of the proceed- ings of 1927 Solvay conference (with a detailed historical introduction) and, as for Volume 1, Olivier Darrigol’s (1992) From c-Numbers to q-Numbers. Other important secondary sources were an edited volume (Badino and Navarro 2013), a pair of disser- tations (Jordi Taltavull 2017; Jähnert 2016, since published as a book: Jähnert 2019) and a pair of papers (Joas and Lehner 2009; Blum et al. 2017) coming out of the quantum project in the context of which we started writing our book. For more infor- mation about this project, which ran from 2006 to 2013 and had its headquarters at the Max Planck Institute for History of Science in Berlin, see the preface to Vol- ume 1 and the foreword by Alexander Blum, Christoph Lehner, and Jürgen Renn to Eckert (2020). We also made use of our own earlier work. For dispersion theory and its role in the run-up to Heisenberg’s (1925c) “reinterpretation” (Umdeutung) paper (see Chapter 10), we used Duncan and Janssen (2007). For Jordan’s derivation of Einstein’s fluc- tuation formula in the Three-Man-Paper (Dreimännerarbeit) of Born, Heisenberg, and Jordan (1926) (see Section 12.3.5), we used Duncan and Janssen (2008). For our dis- cussion of “Kuhn losses” (see Section 15.2), we used Midwinter and Janssen (2013). For the Stark effect in wave mechanics (see Section 15.3.2), we used Duncan and Janssen (2014, 2015). For the statistical transformation theory of Jordan (1927b) and Dirac (1927a) and von Neumann’s (1927a, 1927b) Hilbert space formalism (see Chapters 16 and 17), we used Duncan and Janssen (2013). Finally, for the arch-and-scaffold metaphor (see Chapter 18), we used Janssen (2019). For permission to reuse this mate- rial, we are grateful to Elsevier, publisher of Studies in History and Philosophy of Modern Physics (for Duncan and Janssen 2008, 2014); Springer, publisher of Archive for History of Exact Sciences (for Duncan and Janssen 2007); the Royal Danish Academy of Sciences and Letters (for Duncan and Janssen 2015); Edition Open Access (for Midwinter and Janssen 2013); and University of Minnesota Press (for Janssen 2019).
- 13. viii Preface In addition to those friends and colleagues we already acknowledged in the preface of Volume 1, we thank Rudrajit (Rudi) Banerjee, Alex Blum, Victor Boantza, Cindy Cat- tell, Patrick Cooper, Paul Crowell, Mike Cuffaro, Rafael Fernandez, Michael Janas, Jim Kakalios, Alex Kamenev, Karl-Henning Rehren, and Kurt Schönhammer for helpful discussion. A significant part of this volume was written in Barcelona, where we enjoyed the company of and discussions with Enric Pérez. We thank Olival Freire Jr. for inviting us to contribute a brief synopsis of an important strand in our account of the genesis of quantum theory (i.e., the early history of quantization conditions) to The Oxford Hand- book of the History of Quantum Interpretations (see Duncan and Janssen 2022, in Freire 2022). Olivier Darrigol once again deserves our special thanks for his detailed comments on drafts of several chapters of this volume. We are grateful to Laurent Taudin for draw- ing the “quantum cathedral” in Figure 18.1. One of us (MJ) thanks the Alexander von Humboldt Foundation for a Research Grant that provided generous financial support for work on this book. We thank the Emilio Segrè Visual Archives of the American Institute of Physics (AIP), the Bibliothèque nationale de France, the Solvay Institutes, Brussels, the Pauli Archive at CERN, the Niels Bohr Archive in Copenhagen, the MIT Museum, the Amer- ican Philosophical Society, and the Mathematisches Forschungsinstitut Oberwolfach for permission to use images in their collections for the plates in this volume. We thank Dominique Bogaerts, Urte Brauckmann, Joe DiLullo, Jennifer Hinneburg, Samantha Holland, Allison Rein, Rob Sunderland, Jens Vigen, and Ariel Weinberg for their help in obtaining these permissions. It has once again been a pleasure to work with Sonke Adlung, our editor at Oxford University Press. We thank Michele Marietta for her careful copy-editing of the manuscript and Mark Ajin Millet for overseeing the production of this volume. We dedicate this volume to the memory of Philip M. Stehle (1919–2013) and Roger H. Stuewer (1934–2022). Phil Stehle is responsible for one of us (AD) developing a deeper appreciation for and an interest in contributing to the history of physics. Phil Stehle is probably best known as one of the authors of a textbook on classical mechanics that is still in print (Corben and Stehle 1994, first published in 1950) but he also wrote an excellent textbook on quantum mechanics (Stehle 1966). He even wrote a book on the same topic as ours: Order, Chaos, Order. The Transition from Classical to Quantum Physics (Stehle 1994). Roger Stuewer, in addition to his many other contributions to the history of physics, created the position of a historian of physics embedded in the School of Physics and Astronomy at the University of Minnesota currently held by one of us. Roger Stuewer is probably best known for his authoritative publications on the history of nuclear physics, synthesized in his last monograph, The Age of Innocence: Nuclear Physics between the First and Second World Wars (Stuewer 2018). This is a topic beyond the scope of our book. But he also wrote the definitive account of the discovery of the Compton effect (Stuewer 1975), a classic in the history of physics on which our treatment of this topic is based (see Section 10.4). We were also able to consult many books in his impres- sive and extensive library, which he donated to the School of Physics and Astronomy at the University of Minnesota. Our book is written in the spirit and in fond memory of Phil Stehle and Roger Stuewer.
- 14. Contents List of Plates xiv 8 Introduction to Volume 2 1 8.1 Overview 1 8.2 Quantum theory in the early 1920s: deficiencies and discoveries (exclusion principle and spin) 3 8.3 Atomic structure à la Bohr, X-ray spectra, and the discovery of the exclusion principle 4 8.3.1 Important clues from X-ray spectroscopy 8 8.3.2 Electron arrangements and the emergence of the exclusion principle 10 8.3.3 The discovery of electron spin 15 8.4 The dispersion of light: a gateway to a new mechanics 19 8.4.1 The Lorentz–Drude theory of dispersion 20 8.4.2 Dispersion theory and the Bohr model 22 8.4.3 Final steps to a correct quantum dispersion formula 26 8.4.4 A generalized dispersion formula for inelastic light scattering—the Kramers–Heisenberg paper 33 8.5 The genesis of matrix mechanics 37 8.5.1 Intensities, and another look at the hydrogen atom 37 8.5.2 The Umdeutung paper 41 8.5.3 The new mechanics receives an algebraic framing—Born and Jordan’s Two-Man-Paper 44 8.5.4 Dirac and the formal connection between classical and quantum mechanics 46 8.5.5 The Three-Man-Paper [Dreimännerarbeit]—completion of the formalism of matrix mechanics 48 8.6 The genesis of wave mechanics 53 8.6.1 The mechanical-optical route to quantum mechanics 53 8.6.2 Schrödinger’s wave mechanics 60 8.7 The new theory repairs and extends the old 68 8.8 Statistical aspects of the new quantum formalisms 71 8.9 The Como and Solvay conferences, 1927 80 8.10 Von Neumann puts quantum mechanics in Hilbert space 86
- 15. x Contents Part III Transition to the New Quantum Theory 9 The Exclusion Principle and Electron Spin 97 9.1 The road to the exclusion principle 98 9.1.1 Bohr’s second atomic theory 98 9.1.2 Clues from X-ray spectra 110 9.1.3 The filling of electron shells and the emergence of the exclusion principle 117 9.2 The discovery of electron spin 126 10 Dispersion Theory in the Old Quantum Theory 135 10.1 Classical theories of dispersion 135 10.1.1 Damped oscillations of a charged particle 139 10.1.2 Forced oscillations of a charged particle 143 10.1.3 The transmission of light: dispersion and absorption 146 10.1.4 The Faraday effect 149 10.1.5 The empirical situation up to ca. 1920 151 10.2 Optical dispersion and the Bohr atom 153 10.2.1 The Sommerfeld–Debye theory 153 10.2.2 Dispersion theory in Breslau: Ladenburg and Reiche 155 10.3 The correspondence principle in radiation and dispersion theory: Van Vleck and Kramers 164 10.3.1 Van Vleck and the correspondence principle for emission and absorption of light 167 10.3.2 Dispersion in a classical general multiply periodic system 170 10.3.3 The Kramers dispersion formula 179 10.4 Intermezzo: the BKS theory and the Compton effect 185 10.5 The Kramers–Heisenberg paper and the Thomas–Reiche–Kuhn sum rule: on the verge of Umdeutung 197 11 Heisenberg’s Umdeutung Paper 209 11.1 Heisenberg in Copenhagen 209 11.2 A return to the hydrogen atom 216 11.3 From Fourier components to transition amplitudes 221 11.4 A new quantization condition 224 11.5 Heisenberg’s Umdeutung paper: a new kinematics 232 11.6 Heisenberg’s Umdeutung paper: a new mechanics 239 12 The Consolidation of Matrix Mechanics: Born – Jordan, Dirac and the Three-Man-Paper 255 12.1 The “Two-Man-Paper” of Born and Jordan 255 12.2 The new theory derived differently: Dirac’s formulation of quantum mechanics 276
- 16. Contents xi 12.3 The “Three-Man-Paper” of Born, Heisenberg, and Jordan 293 12.3.1 First chapter: systems of a single degree of freedom 298 12.3.2 Second chapter: foundations of the theory of systems of arbitrarily many degrees of freedom 313 12.3.3 Third chapter: connection with the theory of eigenvalues of Hermitian forms 320 12.3.4 Third chapter (cont’d): continuous spectra 329 12.3.5 Fourth chapter: physical applications of the theory 337 13 De Broglie’s Matter Waves and Einstein’s Quantum Theory of the Ideal Gas 349 13.1 De Broglie and the introduction of wave–particle duality 349 13.2 Wave interpretation of a particle in uniform motion 350 13.3 Classical extremal principles in optics and mechanics 353 13.4 De Broglie’s mechanics of waves 362 13.5 The new statistics of Bose and Einstein and Einstein’s quantum theory of the ideal gas 371 14 Schrödinger and Wave Mechanics 384 14.1 Schrödinger: early work in quantum theory 384 14.2 Schrödinger and gas theory 388 14.3 The first (relativistic) wave equation 392 14.4 Four papers on non-relativistic wave mechanics 405 14.4.1 Quantization as an eigenvalue problem. Part I 406 14.4.2 Quantization as an eigenvalue problem. Part II 416 14.4.3 Quantization as an eigenvalue problem. Part III 437 14.4.4 Quantization as an eigenvalue problem. Part IV 441 14.5 The “equivalence” paper 453 14.6 Reception of wave mechanics 471 15 Successes and Failures of the Old Quantum Theory Revisited 478 15.1 Fine structure 1925–1927 479 15.2 Intermezzo: Kuhn losses suffered and recovered 502 15.3 External field problems 1925–1927 507 15.3.1 The anomalous Zeeman effect: matrix-mechanical treatment 508 15.3.2 The Stark effect: wave-mechanical treatment 515 15.4 The problem of helium 530 15.4.1 Heisenberg and the helium spectrum: degeneracy, resonance, and the exchange force 532 15.4.2 Perturbative attacks on the multi-electron problem 545 15.4.3 The helium ground state: perturbation theory gives way to variational methods 550
- 17. xii Contents Part IV The Formalism of Quantum Mechanics and Its Statistical Interpretation 16 Statistical Interpretation of Matrix and Wave Mechanics 559 16.1 Evolution of probability concepts from the old to the new quantum theory 559 16.2 The statistical transformation theory of Jordan and Dirac 570 16.2.1 Jordan’s and Dirac’s versions of the statistical transformation theory 571 16.2.2 Jordan’s “New foundation (Neue Begründung) . . . ” I 580 16.2.3 Hilbert, von Neumann, and Nordheim on Jordan’s “New foundation (Neue Begründung) . . . ” I 599 16.2.4 Jordan’s “New foundation (Neue Begründung) . . . ” II 603 16.3 Heisenberg’s uncertainty relations 613 16.4 Discussions of the new quantum theory in Como and Brussels 1927 627 17 Von Neumann’s Hilbert Space Formalism 642 17.1 “Mathematical foundation . . . ” 647 17.2 “Probability-theoretic construction . . . ” 658 17.3 From canonical transformations to transformations in Hilbert space 667 18 Conclusion: Arch and Scaffold 671 18.1 Continuity and discontinuity in the quantum revolution 671 18.2 Continuity and discontinuity in two early quantum textbooks 673 18.3 The inadequacy of Kuhn’s model of a scientific revolution 678 18.4 Evolution of species and evolution of theories 680 18.5 The role of constraints in the quantum revolution 682 18.6 Limitations of the arch-and-scaffold metaphor 683 18.7 Substitution and generalization 689 Appendix C. The Mathematics of Quantum Mechanics 695 C.1 Matrix algebra 696 C.2 Vector spaces (finite dimensional) 698 C.3 Inner-product spaces (finite dimensional) 701 C.4 A historical digression: integral equations and quadratic forms 710 C.5 Infinite-dimensional spaces 714 C.5.1 Topology: open and closed sets, limits, continuous functions, compact sets 716 C.5.2 The first Hilbert space: l2 719 C.5.3 Function spaces: L2 726 C.5.4 The axiomatization of Hilbert space 732
- 18. Contents xiii C.5.5 A new notation: Dirac’s bras and kets 734 C.5.6 Operators in Hilbert space: von Neumann’s spectral theory 736 Bibliography 753 Index 784
- 19. List of Plates 1 George Uhlenbeck, Hans Kramers, Samuel Goudsmit. AIP Emilio Segrè Visual Archives, Goudsmit Collection. 2 Copenhagen, 1926. Back row left to right: David Dennison, Ralph Kronig, Bidhu Bhusan Ray. Front row left to right: Yoshio Nishina and Werner Kuhn. Niels Bohr Archive, Copenhagen. 3 Llewellyn Hilleth Thomas. AIP Emilio Segrè Visual Archives. 4 Outdoors at a cabin in Pontresina, Switzerland. Front row left to right: Otto Hahn, Max Born, Rudolf Ladenburg, Robert Pohl, Fritz Reiche. Back row left to right: Bertha Reiche, Auguste Pohl, Mrs. Pummerer, Else Ladenburg, Edith Hahn, and Hedwig Born. AIP Emilio Segrè Visual Archives, Gift of H. Reiche, Mrs. Ewald Mayer, and B. P. Winnewisser. 5 Paul Ehrenfest and Pascual Jordan, 1926. AIP Emilio Segrè Visual Archives, Segrè Collection. 6 Werner Heisenberg. AIP Emilio Segrè Visual Archives. Gift of Jost Lemmerich. 7 Norbert Wiener and Max Born in 1925. Photograph by George H. Davis, Jr. MIT Museum. 8 Paul Dirac. AIP Emilio Segrè Visual Archives. Gift of Mrs. Zemansky. 9 Satyendra Nath Bose. Falguni Sarkar. Courtesy AIP Emilio Segrè Visual Archives. 10 Louis de Broglie. Bibliothèque nationale de France. 11 Erwin Schrödinger. Ruth Braunitzer. Courtesy of Christian Joas. 12 Arthur Holly Compton, Werner Heisenberg, George Monk, Paul Dirac, Carl Eckart, Henry Gale, Robert Mulliken, Friedrich Hund, Frank C. Hoyt. Chicago 1929. AIP Emilio Segrè Visual Archives. Gift of David C. Cassidy. 13 John H. Van Vleck. AIP Emilio Segrè Visual Archives, Francis Simon Collection. 14 John C. Slater. MIT museum. 15 Max Born and Wolfgang Pauli. Courtesy of the Pauli Archive, CERN. 16 Born’s tombstone at the Stadtfriedhof in Göttingen. Picture by Michel Janssen (with thanks to Kurt Schönhammer). 17 Schrödinger’s tombstone, Alpbach, Austria. Picture by Therese Kakalios. 18 Enrico Fermi, Werner Heisenberg, and Wolfgang Pauli at the Como Conference, 1927. Photograph by Franco Rasetti. Courtesy of AIP Emilio Segrè Visual Archives, Segrè Collection. 19 Fifth Solvay Conference, 1927. Photograph by Benjamin Couprie. Courtesy of the Solvay Institutes, Brussels.
- 20. List of Plates xv 20 Ernst Hellinger. Archives of the Mathematisches Forschungsinstitut Oberwolfach. 21 David Hilbert. Photograph by A. Schmidt, Göttingen, courtesy AIP Emilio Segrè Visual Archives, Landé Collection. 22 John von Neumann. Stanislaw Ulam Papers. American Philosophical Society.
- 22. 8 Introduction to Volume 2 8.1 Overview This is the second of two volumes on the genesis of quantum mechanics in the first quarter of the twentieth century. It covers the period 1923–1927 and tracks how what we now call the old quantum theory gave way to modern quantum mechanics. As in Volume 1 (see Chapter 1),1 we begin with a detailed overview, as non-technical as possible, of the contents of Volume 2 (Chapter 8). Except for this introductory chapter and a short conclusion (Chapter 18), the chapters in this volume, like the ones in Volume 1, are divided into two parts, Part III and Part IV. Mirroring the appendix on the mathematics of the old quantum theory in Volume 1 (Appendix A), this volume has an appendix on the mathematics for the new quantum theory (Appendix C). In Part III, consisting of seven chapters (Chapters 9–15), we cover the transition from the old to two distinct forms of the new quantum theory. In Chapter 9, we start with Bohr’s (unsuccessful) attempts in the early 1920s to account for the peri- odicity of the table of elements, and the introduction of two ingredients, the first of which was indispensable for this task: Pauli’s (1925b) exclusion principle and Uhlenbeck and Goudsmit’s (1925) electron spin (after Pauli had talked Kronig out of publishing the idea). The next five chapters are devoted to the emergence of matrix mechanics (Chapters 10–12) and wave mechanics (Chapters 13–14). In Chapter 15, the final chapter of Part III, we revisit the successes and failures of the old quantum theory (see Chapters 6–7) and show how the new theory took care of the failures and put the successes on a more solid basis. Matrix mechanics and wave mechanics resulted from different strands in the devel- opments covered in Volume 1 (cf. Darrigol 2009). Matrix mechanics grew out of the atomic theory introduced by Bohr in his path-breaking 1913 trilogy (see Chapter 4) and elaborated by Sommerfeld and others, mostly working in the three main centers of the old quantum theory, Copenhagen, Munich, and Göttingen (see Chapters 5–7). After examining the roots of matrix mechanics in dispersion theory (Chapter 10), we 1 In all further references to (sections, equations, and figures in) Parts I and II, Chapters 1–7, and Appen- dices A and B, we will no longer note explicitly that these are references to material in Volume 1 of our book. Constructing Quantum Mechanics. Anthony Duncan and Michel Janssen, Oxford University Press. © Anthony Duncan and Michel Janssen (2023). DOI: 10.1093/oso/9780198883906.003.0008
- 23. 2 Introduction to Volume 2 turn to the Umdeutung [reinterpretation] paper with which Heisenberg (1925c) laid the foundations of matrix mechanics (Chapter 11). We then examine its elaboration in the Two-Man-Paper of Born and Jordan (1925b) and the Three-Man-Paper [Dreimänner- arbeit] of Born, Heisenberg, and Jordan (1926), both coming out of Göttingen, as well as the paper in which Dirac (1925), working in Cambridge, introduced the closely related q-number theory (Chapter 12). Wave mechanics, unlike matrix mechanics, grew out of work done by the pio- neers of quantum theory before Bohr’s 1913 trilogy (see Chapters 2 and 3). These pioneers—most importantly Planck, Einstein, and Ehrenfest—were not concerned with the structure of individual atoms, but rather with the statistics of large collections of simple systems modeled as harmonic oscillators. Instead of spectroscopy, they studied black-body radiation and the specific heat of solids at low temperatures—two areas where the equipartition theorem of the kinetic theory of gases of Maxwell and Boltzmann noto- riously broke down. The year before the arrival of matrix mechanics, Indian physicist Satyendra Nath Bose (1924) proposed a new derivation of the Planck law for black-body radiation, using a new statistics for light quanta, which Einstein (1924, 1925a, 1925b) subsequently transferred to atoms in his quantum theory of the ideal gas. Einstein found that the density fluctuations in such a gas are given by a formula like the one he had found earlier for fluctuations in black-body radiation (1909a, 1909b), consisting of a wave and a particle term. This supported a suggestion made in the dissertation by the French physicist de Broglie (1924) to extend the wave-particle duality Einstein had proposed for radiation to matter. Drawing on this work by De Broglie, Bose, and Einstein (Chapter 13), and on the much older optical-mechanical analogy due to William Rowan Hamil- ton (1834, 1835) (Section 13.3), Schrödinger (1926c, 1926d, 1926f, 1926h), working in Zurich rather than in one of the centers of the old quantum theory, developed wave mechanics in a celebrated four-part article (Chapter 14). In Part IV, consisting of just two chapters (Chapters 16–17), we cover the comple- tion of the formalism of quantum mechanics and its probabilistic interpretation. By the middle of 1926, three authors, independently of one another, had already established that matrix and wave mechanics are intimately related (see Section 14.5). Schrödinger (1926e) and Eckart (1926) gave their “equivalence proofs” in published papers, and Pauli (1979, Doc. 131) gave his in private correspondence. In December 1926, Dirac (1927a) and Jordan (1927b) published, again independently of one another, what has come to be known as the Dirac–Jordan statistical transformation theory (Section 16.2). Their formalism unified all four versions of quantum theory that had meanwhile been proposed: matrix mechanics, wave mechanics, Dirac’s q-number theory, and the oper- ator calculus of Born and Wiener (1926, see Section 14.5). As the term “statistical” suggests, Dirac and Jordan generalized the probabilistic interpretation of Schrödinger’s wave function proposed by Born (1926a, 1926b) in papers on collision processes, and by Pauli (1979, Doc. 143), true to form, in private correspondence (Section 16.1). As the term “transformation theory” suggests, canonical transformations familiar to Jordan and Dirac from the celestial mechanics used in the old quantum theory (see Appendix A) played a central role in the Dirac–Jordan formalism. These mathematical techniques, however, proved ill-suited to the task. This was clearly recognized by the young Hun- garian mathematician Janos von Neumann, who came to work with Hilbert in Göttingen
- 24. Quantum theory in the early 1920s: deficiencies and discoveries (exclusion principle and spin) 3 in 1927. Building on recent work by Hilbert and other mathematicians on what is now known as functional analysis (Appendix C), von Neumann (1927a) introduced the mod- ern Hilbert space formalism in a paper presented to the Göttingen Academy in May 1927 and used it to formulate quantum mechanics in a way that steered clear of such mathematical monstrosities as the Dirac delta function, unavoidable in Dirac and Jor- dan’s approach. Prompted in part by Heisenberg’s (1927b) uncertainty paper (Section 16.3), von Neumann published two more papers later that year (von Neumann 1927b, 1927c). In the second installment of this 1927 trilogy, drawing on von Mises’s ideas about probability (soon to be published in book form:von Mises 1928), von Neumann (1927b) reformulated the statistical interpretation of quantum mechanics in terms of his Hilbert space formalism. This paper only appeared after the Como and Solvay confer- ences of September and October 1927, which mark the beginning of the contentious and ongoing debate over the interpretation of quantum mechanics, a topic beyond the scope of our book. We therefore decided to cover Heisenberg’s (1927b) uncertainty principle and Bohr’s (1928b) complementarity principle (first presented at Como and Solvay) in Chapter 16, and to end with von Neumann’s Hilbert-space formalism in Chapter 17. In the short concluding Chapter 18, we elaborate and reflect on the arch-and-scaffold metaphor we used for the subtitles of Volumes 1 and 2 ( Janssen 2019). As will become clear in the course of this volume, the “quantum revolution” of the mid-1920s, while a major turning point in the history of physics, does not follow the pattern of a “paradigm shift” familiar from Thomas S. Kuhn’s (1962) classic The Structure of Scientific Revo- lutions, in which the old paradigm crumbles under the weight of an accumulation of anomalies and a new paradigm is erected on its ruins. The old quantum theory had run into serious difficulties by the early 1920s, but the way out of the resulting Kuhnian crisis was not, as Kuhn (1970, pp. 256–257) himself clearly recognized, to trash the old theory and start from scratch. A more apt image, our analysis suggests, is that the new quantum theory was built as an arch on the scaffold provided by the old quantum theory and, in turn, the classical mechanics and electrodynamics upon which it was built. Much of this scaffolding was discarded once the arch could support itself, making the discontinuity in its construction look, in hindsight, much more pronounced than it actually was. 8.2 Quantum theory in the early 1920s: deficiencies and discoveries (exclusion principle and spin) By the middle of 1923, as we saw in Chapter 7, numerous serious deficiencies had been exposed in the formal framework of the old quantum theory, as developed in the after- math of Sommerfeld’s groundbreaking papers of 1916, in which the Bohr–Sommerfeld quantization rules were introduced and applied with remarkable success to several cen- tral problems of atomic physics. These deficiencies can broadly be divided into two categories. First, there were problems that seemed at least amenable to the accepted framework of the old quantum theory—formulation of a mechanical problem using the Hamilton–Jacobi technique, given that the problem allowed treatment in terms of a mul- tiply periodic motion, followed by imposition of the quantization rules to the associated
- 25. 4 Introduction to Volume 2 action variables, but in which the execution of the formalism simply led to an incor- rect result—one in clear disagreement with the experimental data. The quintessential example of this type was the failure of the old theory to obtain the correct binding energy of the ground state of helium, despite the discovery of a perfectly “reasonable” multiply periodic coordinated motion for the two electrons consistent with classical mechanics. The second set of failures—more numerous than the first—comprised those problems in atomic physics in which no single consistent treatment based on mechanical principles and the Bohr–Sommerfeld methodology could be found for the phenomena in question. The cluster of problems associated with the interpretation of the complex structure of optical spectral multiplets, and the explanation of the complicated rules that had arisen from the study of anomalous Zeeman multiplets belong to this second category. Other problems in this category crop up in the theory of optical dispersion and in the theory of the Stark effect. In all these cases, the root of the problem was the association of energy levels with definite classical orbits of charged (but non-magnetic) electrons in miniature solar systems. The early 1920s saw an enormous increase in the interest in, and attention paid to, two more topics, which were eventually found to elude consistent treatment with the methods of the old quantum theory—the periodic nature of the chemical and optical properties of the elements, presumably to be explained by some shell-like arrangement of the electrons in larger atoms, and the appearance of unexplained “superfluous” lines in X-ray spectra, with spectral splittings with an inexplicable (on old quantum theory grounds) dependence on the atomic number Z. This cluster of problems would turn out to be deeply connected with the two final major discoveries of the quantum theory prior to the advent of modern quantum mechanics: the exclusion principle for atomic electron states proposed in December 1924 by Wolfgang Pauli, and the existence of an intrinsic electron spin angular momentum, first proposed by Ralph Kronig in January 1925, and independently rediscovered in November of the same year by George Uhlen- beck and Samuel Goudsmit. Ignorance of these two basic facts, quite apart from the (in hindsight) deep kinematical and dynamical defects of the old theory which would be cured by matrix and wave mechanics, explains a large part of the failure to arrive at a consistent treatment of complex spectra or the anomalous Zeeman effect within the framework of the old quantum theory. The discovery of the exclusion principle in late 1924, and shortly after, of electron spin, is our focus in Chapter 9. It is useful here, how- ever, to begin our broad overview of the construction of modern quantum mechanics by describing the dramatic progress made in these two areas in the years leading up to the extraordinary developments of 1925. 8.3 Atomic structure à la Bohr, X-ray spectra, and the discovery of the exclusion principle As we saw in Chapter 4, Bohr’s earliest efforts in attempting to insert quantum principles into an understanding of atomic structure (post the Rutherford nuclear atom) were sum- marized in the memorandum of 1912 that he sent to Rutherford. Here, we can see him
- 26. Atomic structure à la Bohr, X-ray spectra, and the discovery of the exclusion principle 5 wrestling already with the problem of the configuration of electron orbits in an atom with more than one electron. These rudimentary considerations were much amplified and deepened in the second paper of his 1913 trilogy, entitled “Systems containing only a single nucleus” (but in general, more than one electron—see Section 4.5). The basic conclusions at this early stage of the old quantum theory were necessarily tentative, and indeed, only tenable given the rudimentary quantitative evidence available at the time concerning the detailed energetics of the quantized states of atoms. For example, even for the simplest multi-electron atom, helium, the total binding energy of the atom was only very roughly known. Bohr’s earliest attempts in this direction, which we may term here “Bohr’s first atomic theory”, can be summarized in the following statements, as regards the configuration of electron orbits: 1. All the electrons circulate in a single plane in (roughly) concentric circular orbits. This is, of course, analogous to the solar system, in which the major planets orbit the sun in roughly (to within 6◦ ) the same (“ecliptic”) plane. 2. Every electron in the ground state of the atom has a quantized value of h/2π for its orbital angular momentum. 3. For a given nuclear charge, there is a maximum number of electrons that can be accommodated in each of the concentric rings containing electrons. This number was determined by a mechanical stability condition with regard to small perturbations of the orbit perpendicular to the “ecliptic” plane of the atom. Even in the absence of detailed quantitative information on electron binding energies, there was clear evidence from chemistry and optical spectroscopy of a “periodic” char- acter in the properties of the elements, which had already been codified in Mendeleev’s periodic table. Bohr’s (1913b) stability arguments were an early, but (as Bohr himself recognized) far from successful, attempt to come to terms with this periodicity. By the early 1920s, it was clear that none of these three principles could possibly be valid. For numerous reasons, including, but not limited to, the diamagnetism of the rare gases, the crystal structure of various elements, optical dispersion measurements, and (perhaps most critically) the extraordinary dual feature of the helium spectrum, which divided into two disconnected sets of lines separately connecting the terms of “ortho- helium” and “parahelium”, the picture of electrons circulating in coplanar orbits, as originally imagined by Bohr in 1913, had become completely untenable. The definitive abandonment of this picture in the case of helium came with the introduction of the non- coplanar Bohr–Kemble model of the ground state of helium (see Section 7.3), which was itself a natural descendant of the Sommerfeld–Landé model of helium, developed in the process of wrestling with the peculiarities of the helium spectrum. Remarks at the end of a long paper entitled “On the series spectra of the elements” (Bohr 1920), submitted to Zeitschrift für Physik in the Fall of 1920, offer a glimpse of the methodology Bohr was to adopt in the years 1920–1923 in the study of atomic structure. Most of this paper is devoted to a description of the understanding which had been achieved of the remarkable efficacy of Rydberg-type formulas (see Appendix B) for the
- 27. 6 Introduction to Volume 2 terms of the optical series of the elements (especially the alkali metals, such as sodium) using the methods of the old quantum theory (see Section 7.1), as well as a recapitulation of the applications of correspondence-principle ideas, for example, in the interpretation of spectra in the Stark effect (see Section 6.3). In the concluding section, Bohr apologizes for the lack of attention specifically to questions of atomic structure, and points forward to the need for a more sophisticated approach to the problem of the configuration of electronic orbits: In the foregoing I have intentionally not delved more closely into the question of the structure of atoms and molecules, although this subject is clearly intimately connected with the theory of the origin of spectra which we have considered . . . On the occasion of the first treatment of this matter [Bohr’s trilogy of 1913], the author also attempted to sketch the foundations of a theory for the structure of the atoms of the elements [Part II] and for the chemical combinations of molecules [Part III] . . . I would like to take the opportu- nity here, however, to state that from the standpoint of more recent developments of the quantum theory . . . many of the detailed special assumptions made in the [earlier] the- ory must be altered, on the basis of disagreement of the theory with experimental results which have been raised on numerous fronts. In particular it does not seem possible to justify the orienting assumption introduced at that time [1913], that in the ground state electrons move in geometrically particularly simple orbits, such as “electron rings.” Con- siderations of the stability of atoms and molecules with regard to external influences, and on the possibility of construction of the atom by successive acquisition of single electrons, require us to insist, first, that the electron configurations to be considered are mechani- cally stable, and moreover possess a certain stability with regard to the requirements of the usual mechanics, and second, that the configurations to be used are so constituted that they may be arrived at by transitions from other stationary states of the atom. These requirements are in general not fulfilled by such simple configurations as electron rings, and force us to look around for more complicated form of motion (Bohr 1920, pp. 468–469, our emphasis). In the three years following this work, Bohr delved more closely into the problem of atomic structure, specifically, in determining the mutual orientation of the orbits fol- lowed by electrons in multi-electronic atoms, and relating the proposed arrangement of atomic orbits to the observed chemical properties and optical spectra of the elements, with particular attention to the recurring periodicity of these properties with atomic number embodied in the periodic table. There would be two short Nature papers (Bohr 1921a, 1921c) as well as a series of longer papers, often elaborated versions of lectures Bohr presented at various venues—and often to audiences who were not atomic theorists, and could not therefore follow the more technical aspects of Bohr’s program—such as Bohr (1921d), a greatly expanded version of a lecture in Copenhagen in October 1921, and Bohr (1922e), based on his legendary Wolfskehl lectures in Göttingen in June 1922, which became known as the “Bohr festival” (Bohr Festspiele) (Mehra and Rechenberg 1982a, sec. III.4, pp. 332–358). The general nexus of ideas proposed in these papers has become known as “Bohr’s second atomic theory” (see Kragh 1979, 2012, Ch. 7). In these works, Bohr attempted to develop and apply the ideas visible in embryo form in the quote given above. First, an
- 28. Atomic structure à la Bohr, X-ray spectra, and the discovery of the exclusion principle 7 electrically neutral multi-electronic atom with an atomic nucleus of charge +Ze would be built up in stages by first putting a single electron into the lowest energy quantized orbit available for the bare nucleus, and then a second electron would be added in the lowest quantized energy state compatible with the execution of a multiply periodic motion2 by both electrons. A third electron would then be added in the same way, allowed to settle down into an orbit which necessarily had to possess a high degree of “cooperation” with the motions of the preceding two electrons—and so on, until a full complement of Z electrons had been added to the atom, rendering it electrically neutral. The multiply periodic character of the motion at each stage would allow for the appli- cation of the Bohr–Sommerfeld quantization principles, so that the orbit of each electron could be characterized by assigning the two quantum numbers familiar from the old theory: the principal quantum number n, the main determinant of the orbital size and energy, and the azimuthal (orbital angular momentum) quantum number k, so that one could speak of an electron in the state nk (with k = 1, 2, . . ., n). Two guiding principles were supposed to be applied in the execution of this Auf- bau (“build-up”) process: (a) the quantum numbers of the previously added electrons would not be altered by the addition of subsequent electrons, although of course the detailed shape and size of the inner orbits would be presumed to change as new elec- trons were added, in conformance with the classical equations of motion; and (b) the final orbit reached by the latest electron added must correspond to an allowed transition as prescribed in the correspondence principle. Here, one should recall that the extended correspondence principle developed by Bohr and Kramers in the late teen years asso- ciated the probability of a quantum transition in which electromagnetic radiation was absorbed (or emitted) with the Fourier components of the motion of the absorbing (or emitting) electron. This association was quantitatively exact in the limit of large quan- tum numbers, but assumed to hold roughly for small quantum numbers. In particular, a transition disallowed for some symmetry reason (giving zero Fourier components for the corresponding emitted light quantum) was assumed to be disallowed for all (and not just large) quantum numbers, an extension that led to the remarkably successful application of the correspondence principle in deriving selection rules for optical transitions. The guiding principles Bohr attempted to apply to his “second theory of atomic struc- ture” seem fairly straightforward in principle, but turned out to be extremely slippery, and ultimately untenable, in practice, as we examine in detail in Chapter 9. His refer- ences to the correspondence principle as determining the choice of orbits for each newly added electron in the Aufbau procedure were unsupported by explicit calculations and for the most part expressed in vague and obscure language. A typical example is the third electron in lithium, which cannot (because of the exclusion principle, as we now know) occupy 1s (n = 1, k = 1) orbits like the first two, but which (according to Bohr) is prevented from joining the first two electrons in a 1s state because such a state cannot 2 We remind the reader that a multiply periodic motion of N electrons is one in which the time depen- dence of each of the 3N Cartesian coordinates of the system of electrons can be written as a Fourier series combination involving at most 3N distinct frequencies.
- 29. 8 Introduction to Volume 2 be reached (by the third electron) from higher orbits by a transition compatible with correspondence principle ideas: Such a transition process, of which one must assume (as a more detailed examination of the possibilities of motion shows [this is never given]) that it would lead to an atomic state in which the third electron would appear as a equivalent participant in the col- lective motion of the three electrons, would in fact be of a quite different type as the transitions between stationary states connected with the emission spectrum of lithium, and in contrast to these would display no correspondence to the harmonic components of the motion of the atom (Bohr 1922b, p. 35). Sentences of this sort abound in the sequence of articles and lectures mentioned above whenever the physico-chemical properties of an element required the addition of an electron in a new, less-bound orbit to the preceding ones. Appeals to the correspon- dence principle were therefore of critical importance in Bohr’s attempts to understand the periodicity of these properties in terms of a shell structure (more precisely, a group- ing into subclasses of similar orbits) of the electrons surrounding larger atoms. Inasmuch as Bohr’s final (pre-exclusion principle) specification of electronic shell structures, in a paper co-authored with Dirk Coster (Bohr and Coster 1923, p. 344), still has an incorrect assignment of quantum numbers (as, for example, in the filled third shell for elements from copper onward, containing six each of s, p, and d electrons, instead of 2, 6, and 10, respectively), we know that these arguments must have been faulty, even if they existed in a more explicit mathematical form than is apparent from the published work. In Chapter 9 we discuss Bohr’s arguments in more detail,3 and demonstrate that atomic systems subject to the correspondence principle, but lacking spin and exclusion properties, simply do not behave as Bohr asserted. 8.3.1 Important clues from X-ray spectroscopy In our discussion in Chapter 7 of the various attempts to deal with the appearance of “extra” lines—hence, extra quantum states—in the study of the complex optical spec- tra that occurred especially in larger atoms, we saw that the new levels were at first absorbed into the theory by the purely phenomenological ruse of introducing a new quantum number, Sommerfeld’s “inner quantum number”. A dynamical explanation for this new degree of freedom would eventually (in the few years remaining to the old quantum theory) come to be widely accepted: namely, that the splitting of Bohr– Sommerfeld levels associated with this new quantum number was due to the freedom to alter (discretely) the relative orientation of the orbital angular momentum of an outer valence electron (the quantum state of which was under consideration) to the orienta- tion of the combined angular momentum of the atomic core (i.e., all the inner electrons). A further intriguing phenomenological observation would soon prove to play a critical 3 See also the web resource, Bohr’s Second Theory and Atomic Aufbau before Spin and the Exclusion Principle.
- 30. Atomic structure à la Bohr, X-ray spectra, and the discovery of the exclusion principle 9 role—when a weak magnetic field was applied, a valence energy level would split into a number of close levels equal to twice the inner quantum number. By 1920, the increasing accuracy and detail available from X-ray studies—especially, the studies of X-ray absorption edges by Gustav Hertz (the nephew of Heinrich Hertz) at the Physikalisches Institut of the University of Berlin—had shown quite clearly the existence of a similar set of “extra” levels in the atomic energy levels (“terms”) associated with the innermost, and most tightly bound, electrons of larger atoms (all the way up to uranium, with Z = 92). The L levels, corresponding to principal quantum number 2, were found to be associated with a triplet of energy values, which were labeled by Sommerfeld as the L1, L2, and L3 levels. The increasing importance of X-ray studies, more specifically, of focusing attention on the innermost electrons, was precisely that such electrons should most closely approx- imate the behavior of single electrons in a hydrogenic atom (now of large nuclear charge) given the dominance of the nuclear electrostatic field, and the relative weakness of the influence of the other single electrons (each having a much smaller charge than the nucleus). Outer electrons were assumed to surround the inner ones (and nucleus) more or less symmetrically, and therefore, by a well-known theorem of classical electrostatics, exert no (or very little) net Coulomb force on the inner ones. An inner electron (say, an L electron with principal quantum number N = 2) would see almost the full effect of the atomic nucleus of charge +Ze, and the small screening effect of the other inner electrons could be taken into account quite accurately by replacing Z → Zeff = Z − σ, where the screening factor σ was typically of order unity (compared to Zs of interest in the X-ray studies between 50 and 90). The square root of the Bohr binding energy for electrons of a given principal quantum number, which was the quantity typically plotted as the vertical ordinate in X-ray term diagrams post-Moseley,4 was therefore roughly proportional to Z for medium to large Z. Sommerfeld’s calculation of relativistic effects, giving rise to the famous fine structure formula (see Eqs. (6.51)–(6.53)), showed that the varying eccentricity of orbits of dif- fering n (orbital angular momentum) for a fixed principal quantum number5 N would lead to different relativistic corrections to the electron binding energy—corrections of order of magnitude (v/c)2 relative to the leading energy term, where the typical veloc- ity v of an electron circulating in a Bohr orbit around a nucleus of charge Ze satisfies v/c ≃ Zα (with α ≃ 1/137 the Sommerfeld fine structure constant). The presence of such relativistic effects would therefore necessarily induce shifts in the inner electron term energies of order Z4 (compared to the leading Z2 Balmer term), or a divergence of order Z3 between the roughly linear Moseley square-root-of-binding energy ordinates for orbits of differing n value (hence, orbital eccentricities). 4 Moseley’s X-ray data were typically displayed with the atomic number on the vertical axis (ordinate) and the square root of the term energy on the horizontal axis (abscissa). Cf. Figure 6.3. 5 We apologize here for the confusing evolution of conflicting notations for electronic quantum numbers, with N, n (for principal and orbital angular quantum numbers) of the late teens giving way later to n, k (as in our discussion of Bohr’s Aufbau), and subsequently n, l. We have tried to hold to the notation used by the authors being discussed as much as possible.
- 31. 10 Introduction to Volume 2 Let us first consider the simplest case, the “L” X-ray terms corresponding to inner electrons with principal quantum number N = 2, in atoms of large Z (> 50, say). Pairs of X-ray terms diverging for large Z from each other as the cube of the atomic number were indeed well established by the work of Gustav Hertz in early 1920, and were dubbed “regular doublets” by Sommerfeld. Then, there were the “extra” terms, which maintained a more or less constant (i.e., Z-independent) difference (rather than increasing as the cube of atomic number) in Moseley ordinate from one of Sommerfeld’s “regular” terms! In fact, this situation is the one we should have anticipated from the fact that electron orbits of differing angular momentum would penetrate the innermost region of the atom (and be exposed to the full nuclear charge) to differing degrees, and hence, would have different screening constants. Two such terms would then have Moseley terms exhibiting a constant difference in the leading (i.e., Balmer) energy square root: p (Z − σ1)2 − p (Z − σ2)2 ∝ σ1 − σ2 (hence, independent of Z ). This would be much larger than, and should therefore always dominate, the Sommerfeld fine-structure shifts (with their extra factor of the square of α). The problem was that one found both the “regular” doublets, with a splitting that looked just like relativistic fine structure, and “irregular” doublets with a constant Moseley splitting—which was perfectly reasonable on the basis of the expected difference in nuclear charge screening for orbits of different shape (see Figure 9.2), but seemed to lack the expected relativistic corrections for orbits differing in angular momentum. As mentioned earlier, an important reason for the increasing interest in X-ray studies at this point was the hope (already expressed by Sommerfeld in the first edition of Atom- bau und Spektrallinien; cf. Volume 1, p. 277) that the nuclear Coulomb attraction on an inner electron would so dominate the relatively small inter-electron repulsive interac- tions (which could be ignored for the outer electrons) that the motion of these electrons would be very close to that expected on the basis of the Bohr–Sommerfeld formalism. In particular, the peculiar complex spectra effects in optical spectra (i.e., in the energies of the outer electron orbits) presumed to arise from an interaction of the valence electron orbit with a magnetic field produced by the “core” (i.e., the nucleus plus inner electrons) should no longer be relevant, as the innermost electrons were the core, and the screening effects exerted on each other could apparently be accounted for very accurately simply by a small shift in the effective nuclear charge, Z → Zeff, as indicated above. Nevertheless, the triplet of L (N = 2) terms, quintuplet of M (N = 3) terms, etc., clearly indicated that an additional degree of freedom was in play for these inner electrons as well, giving rise to “extra” stationary-state energy levels. And the Z-dependence of the extra splittings, as we have just seen, was inexplicable in terms of simple screening plus relativistic effects. 8.3.2 Electron arrangements and the emergence of the exclusion principle These matters came to a head in an influential paper written in late 1922 by Bohr and Coster (1923) entitled “Röntgen [X-ray] spectra and the periodic system of the ele- ments.” Coster had worked on X-ray spectroscopy at Lund University with Manne
- 32. Atomic structure à la Bohr, X-ray spectra, and the discovery of the exclusion principle 11 Siegbahn’s group in the preceding two years before coming to visit the Niels Bohr Insti- tute in Copenhagen in the summer of 1922. In fact, his dissertation, completed under the direction of Paul Ehrenfest at the University of Leyden earlier that year, had been pre- cisely on the subject of “Röntgen spectra and the atomic theory of Bohr” (Coster 1922). Coster had at his fingertips all the latest results for X-ray term levels in a wide range of elements, which now represented a greatly expanded cornucopia of phenomenological information, compared to the relatively sparse results of Hertz just two years earlier. Bohr and Coster undertook an extensive review of the data and gave a detailed discussion of the various screening effects to be expected in the motion of an inner electron. This motion, while dominated by the very strong nuclear Coulomb field (we are thinking of atoms with a fairly large atomic number here), would also be affected occasionally by the penetration of an outer electron with a very elliptical orbit (i.e., low angular momentum) into the inner region, and more frequently by the presence of the electrostatic repulsion of the electrons of the same or smaller principal quantum number as the electron in question. Bohr and Coster asserted that these effects could be subsumed into screening shifts (such as the factor σ shifting the nuclear charge Z discussed above). Unfortu- nately, the appearance of extra “anomalous” (in the Bohr–Coster terminology) energy levels leading to the apparently incompatible simultaneous appearance of “screening” and “relativistic” doublets could not be reconciled with the fundamental tenet of the Aufbau program, which assumed that the addition to or subtraction from the atom of a single electron would not appreciably affect the orbits (or energies) of the remaining energies. Instead, as we discuss in more detail in Chapter 9, one was forced to assume that the removal of an inner electron resulted in an essential (wesentliche) change of the behavior of the other electrons in the same shell (or “group”, in the Bohr–Coster language). In fact, the authors state Such a conception is necessitated by the theory if it is assumed that the completion of an electron group is essentially determined by the interaction of electrons within the various subgroups [of the shell] (Bohr and Coster 1923, p. 365). The mysterious “interaction between subshells” referred to here can (with hindsight!) be regarded as a foreshadowing of the constraints that would, by the end of the next year, be recognized to arise from the exclusion principle, where the completion of the electron subshells within a shell (of given principal quantum number) is indeed determined by the previous occupancy of these subshells. But in the absence of any understanding of either the additional physical spin degree of freedom of the electrons or the completely non-classical constraints imposed on the electron states by exclusion, Bohr and Coster had to admit that “it is still quite unclear how a more detailed execution of this idea could provide an explanation for the separate appearance of screening- and relativity- doublets” (p. 365). A few months after the publication of Bohr and Coster’s paper, a major step toward the exclusion principle appeared in an article entitled “The distribution of electrons among atomic levels,” submitted to the Philosophical Magazine by Edmund Stoner
- 33. 12 Introduction to Volume 2 (1924), a graduate student working with Rutherford and Fowler at the Cavendish Lab- oratory in Cambridge. In discussing Stoner’s contribution, we should first note yet another shift in terminology in describing electronic configurations. Where Bohr and Coster speak of electron “groups” and “sub-groups”, we find in Stoner the designation of electron “levels” (electrons sharing a given value of the principal quantum number n), which are subdivided into “sub-levels” characterized by a given value of the azimuthal (orbital angular momentum) quantum number k (=1, 2, . . ., n).6 For Stoner the appearance of “extra” terms in X-ray spectra, which seemed very sim- ilar to the extra terms which had been known in optical “complex” spectra for decades, was more than a coincidence. The number of terms appearing in the X-ray data, just as in the optical case, suggested that each stationary state with an azimuthal number k = 2, 3, . . . (thus, the p, d, . . . states) was duplicated, with the two states distinguished by the numerological artifice of an extra “inner” quantum number ( j for Stoner), which could take the two values k − 1 and k. The s states, with k = 1, on the other hand, appeared singly, with no extra “twin.” Now in the optical case, numerous studies of the anomalous Zeeman effect had shown that in a weak magnetic field a stationary state with inner quantum number j splits into exactly 2j evenly spaced energy levels. Stoner suggested that this splitting revealed precisely the number of dynamically possible distin- guishable electron states, not just for the outer light-emitting valence electrons relevant to the optical spectra, but also for the inner electrons occupying already filled levels (or “groups”, or “shells”, . . . ) in larger atoms.7 Stoner refused to commit himself to any specific model-theoretic “explanation” for the existence of these states, except to accept the widely held proposition that the split energy levels revealed in the Zeeman effect cor- responded to a space-quantized variation of the orientation of the orbits. He concludes the fifth chapter of his paper with the summary In brief, then, it is suggested that, corresponding roughly to the definite indication in the optical case, the number of possible states of the atom is equal to 2j; so, for the X- ray sub-levels, 2j gives the number of possible orbits differing in orientation relative to the atom as a whole; and that electrons can enter a sub-level until all the orbits are occupied (Stoner 1924, p. 726, our emphasis). Note that the term “sub-level” here indicates the specification of all three quantum num- bers: principal (n), azimuthal (k), and inner ( j ). This remark is followed immediately in the next section of the paper by the assertion that all 2j electron states (for given n and k) have identical statistical weight, “for there is then one electron in each possible equally probable state” (p. 726). Although Stoner does not put it this way, it is very hard to see how all of this works if one does not also grant that more than one electron is in 6 We are now adopting, in place of Sommerfeld’s use of n for the azimuthal quantum number, the denotation k, which was increasingly used in the last few years of the old quantum theory. The modern notation n for the principal quantum number also becomes more usual at this time. The extra inner quantum number also exhibited a confusing abundance of different notations: for Bohr and Coster it was k2 (with k1 = k the usual azimuthal quantum number), while Stoner, following Landé, uses j for the inner quantum number. 7 It is important to realize that the relatively very small splittings induced by application of a weak magnetic field were not at this stage empirically accessible in the X-ray data.
- 34. Atomic structure à la Bohr, X-ray spectra, and the discovery of the exclusion principle 13 fact excluded from occupying the same orbit, at least in the case of completed groups. However, it must be admitted that the statement of identical statistical weight for the various levels revealed after magnetic splitting cannot be taken to imply that each such level can be occupied at most by a single electron, only that the probability of occupancy on average is the same for the levels (in the weak field limit). A simple example of how Stoner’s configurations differ from Bohr and Coster’s can be seen in the suggested distribution of electrons in the inert gas argon (atomic number Z = 18). In the Bohr scheme (using the nk notation) there were two 11 (1s) states; four 21 (2s) and four 22 (2p) states; and four 31 (3s) and four 32 (3p) states. Stoner’s assignments instead correspond exactly to the modern distribution of electrons: two 1s, two 2s, six 2p (divided into a sub-shell of two j = 1 and four j = 2 electrons), two 3s, and six 3p electrons (again divided into two sub-shells of j = 1, 2). In the final part of his paper, Stoner gives some heuristic arguments to support his numerical assignments of electron orbits. The most convincing of these come from consideration of the intensity of X-ray lines arising from transitions of electrons out of different sub-levels, as in the comparison of L2 → K and L3 → K transitions, and from the relative absorption of X-rays by electrons in various sub-levels, as in the relative absorption of L1, L2, and L3 electrons. Stoner’s paper was read with great enthusiasm by Wolfgang Pauli, who fully approved of the strategy of trying to combine information available from X-ray spectroscopy with the insights obtained from the (by now) multi-decade struggle with the mysteries of the anomalous Zeeman effect. The results of his study of the problem were published in a paper entitled “On the connection between the closing of electron groups in the atom with the complex structure of spectra,” submitted on January 16, 1925 (Pauli 1925b). After a brief review of the difficulties encountered by the old quantum theory in arriving at a satisfying—that is, dynamically consistent model-theoretic—account of the anomalous Zeeman effect for weak magnetic fields, Pauli turns to the most basic problem faced by the quantum theory in explaining the regularities of the periodic table: namely, the appearance of the period lengths 2, 8, 18, 32, . . . (more concisely, the numbers 2n2 for n = 1, 2, 3, 4, . . .), which suggest that the group of electrons with principal quantum number n is unable to accommodate the addition of any further electrons once there are, as in the case of the rare gases, already 2n2 electrons present. Pauli fully accepts Stoner’s interpretation of this counting, reproducing both Bohr’s and Stoner’s proposed electron distributions for the rare gases from helium up to “radium emanation” (i.e., radon) (Pauli 1925b, Tables 1 and 2, respectively). In common with Stoner, Pauli also proposes that critical insights into the filling of atomic electron groups could be obtained from the splittings observed in the optical Zeeman effect, but in Pauli’s case, the important clues are to be found in the special case of large magnetic fields (Paschen–Back regime; cf. Section 7.2.3), where the structure of the splitting simplifies, as the interaction between the magnetic moment due to the orbital angular moment of a valence electron with that of the putative “atomic core” becomes negligible, and the magnetic energy shifts take on the form of a sum of two terms, one proportional to the component of the valence electron angular momentum in the direction (say, z) of the magnetic field, and the other to the z-component of the
- 35. 14 Introduction to Volume 2 core magnetic momentum (enhanced by a factor of 2, reflecting the “double magnetism” of the core). However, by this time in late 1924, Pauli had become convinced, on the basis of var- ious calculations of the effects of relativity on the motion of inner “core” electrons, that the second term had to be associated with some completely non-classical and essentially mysterious “double-valuedness” (Zweideutigkeit) of single electrons in stationary states (Pauli 1925b, p. 768). The simplification of the dynamics of electron states in special circumstances was essential for both Stoner and Pauli: in the case of the former, the negli- gibility of inter-electron interactions for inner electrons dominated by the Coulomb field of a large nuclear charge; for the latter, as a result of the unimportance of the core–orbit (later spin–orbit) interaction once the external magnetic field became very large. Pauli showed that the counting of Stoner—in particular, the division into sub-levels containing a number of electrons given by twice the inner quantum number—could be reproduced by introducing a set of four quantum numbers, which seemed relevant in describing the magnetically split levels of a single valence electron in the strong field (Paschen–Back) Zeeman effect. These were n, the principal quantum number, k1, the azimuthal orbital angular momentum (thus s, p, d, . . . electrons for k1 = 1, 2, 3, . . .), k2, the inner quantum number, which by now had been interpreted by Landé as equal to j + 1/2 (where j was the total angular momentum, valence plus core, of the atom), and finally, a magnetic quantum number m1 = −j, −j + 1, . . ., j − 1, j, which came into play once the levels were magnetically split, as it reflected the space-quantized component of the total angular momentum in the direction of the magnetic field. To the classification of dynamically allowed electron states specified by the set (n, k1, k2, m1), Pauli added an additional—and crucial—stricture not to be found, at least explicitly, in Stoner: the prohibition of double occupancy of any state specified by a unique value of these four quantum numbers. As we shall see, this prohibi- tion, which would soon become known as the “Pauli exclusion principle” (Paulisches Ausschliessungsprinzip, or more simply, the Pauli-Prinzip or Pauli-Verbot), leads to exactly the same distribution of electron shells into sub-levels as Stoner’s scheme. The eight electrons in the L shell of principal quantum number n = 2, for example, would be divided into three sub-levels containing two 2s electrons (with k1 = 1, k2 = 1, and m1 = ±1/2), two 2p electrons (with k1 = 2, k2 = 1, and m1 = ±1/2), and four 2p elec- trons (with k1 = 2, k2 = 2, and m1 = +3/2, +1/2, −1/2, −3/2), respectively. This is clearly different from Bohr’s scheme, which simply apportioned electrons in completed shells equally among the different k1 values (thus, four each of 2s and 2p electrons). Pauli’s article concludes with the remark that, in contradistinction to Bohr’s reliance on correspondence-principle arguments in arriving at proposed electronic arrangements in multi-electron atoms, arguments based on the “principle of invariance of statistical weights of quantum states” (p. 783) provide a much firmer foundation for the assign- ment of the quantum numbers to electrons in closed shells of atoms.8 This principle is, in fact, a direct corollary of the second of the three “guiding principles” of the old quantum 8 Pauli had made just this point in a much more explicit way in a letter to Bohr in early December 1924. See Section 9.1.
- 36. Atomic structure à la Bohr, X-ray spectra, and the discovery of the exclusion principle 15 theory, Ehrenfest’s adiabatic principle (cf. Section 5.2). For Pauli, Bohr’s vague asser- tions that electrons could only enter certain correspondence-principle selected orbits in the “build-up” process of constructing an electrically neutral atom by adding electrons to the initially bare nucleus could easily lead to erroneous assignments. By contrast, the adiabatic stability of quantum states, whereby single stationary states revealed by the “dissection” of states induced by application of an initially weak magnetic field would transform continuously as the magnetic field was increased (without alteration of their discrete quantum number assignments) all the way to the strong field Paschen–Back end, led directly, with the addition of an exclusion postulate, given the unique identification of the electron state in terms of four quantum numbers, to the numerology observed in the periodic table. Pauli’s demotion of the correspondence principle in atom-building, to which he gave full credit as an “indispensable aid” (p. 770) in understanding the origin of selection rules for atomic transitions, but which he abandoned in his arguments for the exclusion principle, are connected with a deep allergy to classical-mechanical intuition, which he had developed by late 1924, primarily as a result of the enormous efforts expended on the Zeeman effect over many years by theorists, who were ultimately unable to arrive at a dynamically consistent (according to classical mechanics) description of the motion of an atomic electron in an applied magnetic field using the methods of the old quantum theory. He states quite explicitly that “it is not in fact a requirement of the correspon- dence principle to assign to each electron in a stationary state a uniquely determined orbit in the sense of the usual kinematics” (p. 770) Indeed, the word “orbit” (Bahn) appears only twice in the article, while “state” (Zustand) occurs twenty-six times. Pauli had clearly become convinced that any satisfactory new quantum theory would nec- essarily have to eschew the conventional visualizations of particle motion which had become ingrained over three centuries of dominance of Newtonian mechanics. We will see that, this attitude, while in many respects sanctioned by subsequent developments, would lead to an unfortunate rigidity—even hostility—in his initial reaction to the idea of an intrinsic electron spin. 8.3.3 The discovery of electron spin Few important discoveries in the development of quantum theory can be dated with as much precision as that of electron spin, which we can confidently place a day or two following the arrival of the German physicist Ralph Kronig in Tübingen on January 7, 1925. Kronig at the time was a doctoral student at Columbia University in New York, and was in the middle of an extended tour of several important European loci of research in atomic theory, having already made stops in Cambridge and Leyden (where he worked with Samuel Goudsmit on the Zeeman effect). In Tübingen, Kronig would be hosted primarily by Alfred Landé, who had already made important contributions to the theory of the anomalous Zeeman effect and the helium spectrum (see Sections 7.3.2–7.3.3). Shortly after his arrival in Tübingen, Kronig was handed a letter that Landé had just received from Pauli summarizing his recent progress in interpreting the closure of electron groups in atoms with the use of a fourth two-valued quantum number directly
- 37. 16 Introduction to Volume 2 attributed to each valence electron separately, and not to a single putative atomic core. Kronig later claimed that “it occurred to me immediately that [the fourth quantum num- ber] might be considered as an intrinsic angular momentum of the electron” (Kronig 1960, p. 20). Kronig did not leave the matter there, but immediately set out to reinter- pret the puzzling relativistic doublets of X-ray spectroscopy model-theoretically with this new idea. We remind the reader that the existence of pairs of inner electron states which appeared to be split in energy by an amount given by the Sommerfeld fine-structure for- mula from 1916—in particular, with a splitting proportional to the fourth power of the effective nuclear charge Zeff—which had initially seemed such a great success of the old quantum theory, had come to be seen as a serious difficulty once further energy levels were discovered with an energy splitting apparently purely due to screening (and not relativistic) effects. The critical problem was the absence of obvious screening effects in the relativistic doublets, which (Landé had argued) meant that these doublets corre- sponded to pairs of electron states with the same spatial orbits (thus, azimuthal quantum number k = k1) and so the same screening, but different orientation of the electron orbital momentum to the core angular momentum. In other words, per Landé, these splittings would be analogous to the alkali metal doublet splittings which Heisenberg had calculated in the ancient days of late 1921 with his core model (cf. Section 7.1.2). The problem was that, in the core model, the doublet splittings, when computed with an effective nuclear charge Zeff appropriate for an inner electron in a large atom, inevitably emerged with a dependence proportional to the cube, rather than fourth power, of Zeff, which was the observed behavior.9 Kronig realized that with the transference of the fourth quantum number to an intrin- sic spin of the electron (rather than the atomic core) engaged in the transition (optical or X-ray) of interest, the energy shifts due to the core–valence magnetic interaction would acquire the additional factor of effective nuclear charge, Zeff, needed to convert the Heisenberg core–model Z3 eff dependence to the Z4 eff form observed in the relativistic doublets. The reason is quite simple. In the Heisenberg core model, doublet splittings arose because of the varying orientation of the magnetic moment associated with the atomic core with respect to the magnetic field generated by the circling charge −e of the valence electron. In the new picture, which appears for the first time in Kronig’s calculations of January 1925, the energy modifications to the Balmer–Bohr term levels arise because the magnetic moment due to the spin of the circulating electron assumes different orientations with respect to the magnetic field (in the frame of the moving electron) arising from the nuclear charge, which is Zeffe: this magnetic field is therefore Zeff larger than that felt by the core in the earlier model, where the magnetic field at the core was that generated by a single outer valence electron. The result obtained by Kronig was about a factor of 2 larger than the empirically observed splitting (on which more in Chapter 9), but the recovery of the correct Z-dependence was certainly a most encouraging development. Or so it would seem. 9 Note that an energy correction of order Z4 to the leading order Z2 Bohr–Balmer energy will lead to a correction of order Z3 in the Moseley ordinate, which reflects the square root of the total binding energy.
- 38. Atomic structure à la Bohr, X-ray spectra, and the discovery of the exclusion principle 17 The explanation of the relativistic doublets in terms of the varying orientation of spin and a given orbital angular momenta (k = 2 for the L terms, for example) of the inner electrons in the X-ray spectra also cleared up the difficulty of the screening doublets, which simply corresponded to the energy shifts expected between states of the same principal quantum number but different orbital angular momentum (s versus p states in the L terms, for example).10 As the screening effects appear in the leading Balmer–Bohr energy term, they dominate the energy difference and lead to a roughly constant offset in the Moseley ordinate, as we saw previously. Kronig’s idea seems to have met with a positive reception by his host Landé, who had struggled mightily with the problem of relativistic versus screening doublets in the previous year. Unfortunately, with Pauli’s physical arrival in Tübingen, the whole devel- opment came to a screeching halt. According to Kronig’s later recollections,11 Pauli dismissed out of hand his attempts to associate his fourth quantum number with an explicit mechanical source—the angular momentum of an electron viewed as a ball of charge spinning on its own axis—as “a quite clever insight, but Nature is just not like that” (“ein ganz witziges aperçu, aber so ist die Natur schon nicht”). We admittedly do not know the specific criticisms Pauli leveled at the idea of an intrinsic electron spin angular momentum, but difficulties with reconciling a classically motivated picture of a ball of charge (with various assumed charge distributions) spinning fast enough to generate a spin h/4π (or ℏ/2 in modern notation with ℏ ≡ h/2π) in a relativistically consistent way are not hard to find.12 We shall see that obstacles of this kind emerged once again toward the end of the year when the idea of spin was independently discovered by Uhlenbeck and Goudsmit. In their case, the objections came from the master of classical electron theory, H. A. Lorentz. Kronig took Pauli’s criticism very seriously (as did Landé, who withdrew his previ- ous support for Kronig’s idea) and his calculation showing the agreement (apart from the annoying factor of 2) of the relativistic doublet splitting with a spin–orbit magnetic interaction of the electron was never written up and submitted for publication. However, Kronig does seem to have explained his calculation to Heisenberg in the course of his subsequent visit to the Bohr Institute in Copenhagen.13 Heisenberg’s awareness of the potential importance of the spin idea would play a significant role, as we shall see shortly, in the subsequent emergence, and final acceptance, of electron spin. The definitive appearance of the idea of electron spin would come ten months after Kronig’s ill-fated visit to Tübingen. In mid-October 1925, two graduate students of Paul Ehrenfest in Leyden, Samuel Goudsmit and George Uhlenbeck, submitted a short 10 In modern notation, for the N = 2 L states, the relativistic doublets expose the energy difference of 2p3/2 and 2p1/2 states, which arises solely from the spin–orbit interaction (as they have the same orbital angular momentum, thus the same screening), while the pair 2p1/2, 2s1/2 correspond to different orbits, as they have different orbital angular momentum, and are therefore subject to the leading order screening effect. 11 Interview with Kronig for the AHQP by John Heilbron in 1962. 12 In his AHQP interview (see note 11), Kronig suggests that Pauli’s objection might have concerned the excessively large electromagnetic mass which a spinning electron would necessarily develop if its intrinsic angular momentum was as large as h/4π. This is clearly a speculation on his part. 13 In his AHQP interview (see note 11), Kronig recalled that this happened “on the ferry between Gedser and Warnemünde . . . sometime in the Spring of 1925.”
- 39. 18 Introduction to Volume 2 paper (less than two pages) to Die Naturwissenschaften in which they proposed a “dif- ferent approach” to the problem of the fourth quantum number R previously used in the theory of the anomalous Zeeman effect to indicate the angular momentum of the atomic core, the magnetic interaction of which with both the valence electron and the external magnetic field was a crucial part of the phenomenology of the Zeeman effect, in particular, in the heuristic “derivation” of the formula for the Landé g-factor (cf. Sections 7.3.2–7.3.3). After describing the ascription by Pauli of a two-valued fourth quantum number to the valence electron itself (i.e., its removal from the atomic core), Goudsmit and Uhlenbeck suggest that, while the other three quantum numbers (prin- cipal, azimuthal, and inner) retained their previous interpretation, the quantum number R was to be ascribed directly “to the rotation of the electron itself” (Uhlenbeck and Goudsmit 1925, p. 954). There are no explicit calculations presented to support this hypothesis, but the authors do recognize in closing that the viability of this hypothesis rests crucially on its ability, on the basis of the “variable orientations of R to the orbital plane of the electron [hence, the direction of the orbital angular momentum vector K],” to yield a correct explanation of the relativity doublets. Goudsmit and Uhlenbeck’s paper had not had an entirely smooth trip from concep- tion to publication, as Uhlenbeck had asked Lorentz to assess the dynamical consistency of an intrinsic electron spin, which, as the acknowledged master of classical electron theory, Lorentz was clearly in position to do. When Lorentz got back to Uhlenbeck, it was with the bad news that any interpretation of a spin angular momentum of an elec- tron based on a model of a rotating charge distribution ran into conflicts with special relativity, as the surface of the electron sphere would be moving faster than the speed of light. Objections of this sort had also probably disturbed Pauli much earlier, when he dismissed Kronig’s suggestion of a mechanical interpretation of his fourth quantum number. Fortunately for Goudsmit and Uhlenbeck, Ehrenfest, who was responsible for sending the paper in to the journal, was far more willing to allow his students to entertain “dangerous” ideas, and their short note, establishing for posterity their priority in the discovery of electron spin, proceeded to Die Naturwissenschaften unimpeded and was duly published on November 20. The day after the paper appeared in print, Heisenberg wrote to Goudsmit enquiring about the factor of 2 disagreement in the spin–orbit coupling energy for inner electrons— the empirical splittings seemed systematically about twice as large as one would expect from a direct calculation of the coupling energy of a magnetic electron moving in the nuclear field. Everything else about this result (which, of course, goes back to Kronig) was encouraging, in particular the crucial correct Z4 dependence that had been such a mystery in the atomic core models. The same issues were raised in a separate letter to Pauli sent the same day, in which Heisenberg asks for Pauli’s opinion on the whole idea of a spinning electron. Heisenberg leaves no doubt that, for him, apart from the annoying extra factor of 2, the spin idea “eliminates all difficulties with a magic stroke” (Pauli 1979, Doc. 107). The stage was certainly set for a widespread acceptance of the idea of electron spin in the months to come. By the end of the year, Bohr, on the basis of conversations with Einstein explaining the relativistic origin of the spin–orbit coupling, had also accepted the idea of an intrinsic electron angular momentum. As we shall see
- 40. The dispersion of light: a gateway to a new mechanics 19 in Chapter 9, the whole issue was settled fairly conclusively a few months later, as L. H. Thomas was able to show that in virtue of a rather subtle relativistic precession effect, the spin–orbit energy of a bound electron with an intrinsic magnetic moment in an external nuclear field was indeed reduced by exactly a factor of 2. Pauli never publicly acknowledged the role he had played in robbing Kronig of the credit for one of the most spectacular discoveries in the history of quantum mechanics. To some degree, however, he did make amends in a letter of recommendation for Kronig, which had been solicited by Coster in an attempt to procure a position for Kronig in Groningen. This letter is quoted in a scientific biography of Kronig. Pauli wrote: In my opinion Mr. Kronig is in every respect, on the basis of his scientific achievements, ready for a professorship of theoretical physics . . . All his work is characterized by great originality in the framing of the problem and a particular reliability and thoroughness in its execution. It is especially to be emphasized that, although there is no publication from his hand concerning this, at one point he came very close to the discovery of the spinning electron (Casimir 1996, p. 58). Close, but no cigar, as the saying goes. Before proceeding to a survey of the remarkable evolution that quantum theory was already undergoing at the time of Uhlenbeck and Goudsmit’s work, it is important to acknowledge that, although the introduction of the exclusion principle and electron spin were to prove indispensable in resolving the fail- ures of the old theory (see Chapter 7), these ideas played practically no role in the initial radical transformations of the theory initiated by Heisenberg with his matrix mechanics, or a little later by Schrödinger in his wave mechanics. For Heisenberg, the indispens- able workhorse on which to exercise the tenets of his “reinterpreted kinematics” was the one-dimensional anharmonic oscillator; for Schrödinger, it was the hydrogen atom, sans any fine structure or spin effects. In neither case did electron spin nor the Pauli exclu- sion principle play a role in the initial developments of the new formalisms. They were, however, crucial in the next stage, where the new methods were systematically applied to the resolution of the incorrect results and loose ends of the old quantum theory, most especially in the theory of the anomalous Zeeman effect and of the helium atom (see Chapter 15). 8.4 The dispersion of light: a gateway to a new mechanics The study of visible light emitted from and absorbed by atoms was by far the most important source of empirical data constraining the evolution of atomic models follow- ing Bohr’s introduction of the quantum into atomic theory in 1913. Apart from the study of emission and absorption of light, a third aspect of optical research that would turn out to have a profound impact on the development of modern quantum mechanics was dispersion—technically, the coherent scattering of light by the atoms (or molecules) comprising an extended homogeneous medium, typically a gas, liquid, or transparent
- 41. 20 Introduction to Volume 2 solid.14 Newton’s experiments with prisms in the 1660s had established the essential facts underlying optical dispersion: a circular “pencil” of white light could be decom- posed after passage through a prism into an elliptical spot displaying the colors of the spectrum; if a particular color (say, blue) was selected by passing the light through a small hole at the appropriate spot, the light remained blue in subsequent refractions through other prisms; and the light at the blue end of the spectrum was more bent (more refrangible, in Newton’s terminology) than the light at the red end. In terms of the index of refraction n, which had already been introduced in quantitative accounts of refrac- tion by Snell and Descartes, the existence of a variable refrangibility indicated that the index of refraction, which was a quantitative expression of the extent to which a light ray underwent bending in passage from air into a transparent medium, was a number greater than one that increased monotonically as one went from the red to the blue end of the spectrum in typical refractive media (such as prism glass). With the advent of the wave theory 150 years later, and the ascription of color to the varying wavelengths λ (inversely related to frequency ν by λ = c/ν, where c is the speed of light), the task of physical optics became clear. Any complete theory of the interaction of light with a material medium through which it passes must be capable of explaining quantitatively the functional dependence of the index of refraction on frequency, n(ν). By the close of the nineteenth century, Maxwell’s electromagnetic theory of light was the dominant paradigm, and the interaction of light with matter was presumed to be determined by the effect of the oscillating electric field generated by a light wave as it passes over the constituent charged “ions” of opposite and, in bulk, cancelling signs in an electrically neutral medium. The point of view generally accepted for the underlying process can be seen in the following excerpt from the second edition of Drude’s textbook on optics, in the first section of Chapter 5, “The Dispersion of Bodies:” One arrives at a theory which adequately represents the observed phenomena by intro- ducing the assumption that the smallest parts . . . of a body possess the possibility of characteristic vibrations. These [vibrations] are excited to a greater or lesser sense to the extent that their [natural] period lies nearer or further from that of the light vibrations impinging from outside. Such vibrations, stimulated by a light wave (i.e., an oscillating electrical force) are immediately understandable if we generalize the conception (made necessary by electrolysis) that each molecule of a body is composed out of positive or negative atoms, the so-called ions. These ions are not in general identical with those obtained in electrolysis and they are to be referred to by a different word, for example electrons (Drude 1906, pp. 362–363). 8.4.1 The Lorentz–Drude theory of dispersion The idea that physical optics had become a study of the interaction of light with harmon- ically bound “mobile ions” within the atoms or molecules of a material substance had already been used by Lorentz a decade before the second edition of Drude’s textbook 14 See the dissertation of Marta Jordi Taltavull (2017) for a history of dispersion in the period of interest to us (1870–1925).
- 42. The dispersion of light: a gateway to a new mechanics 21 quoted here, to explain the normal Zeeman effect (cf. Section 7.2.1), or splitting of spectral lines when an external magnetic field was applied to the radiation emitted by excited atoms (e.g., in a heated or electrically excited sample). What concerns us here is a somewhat different instance of the interaction of light with matter: the absorption and coherent re-emission of light waves of a given frequency and phase by atoms in their normal (i.e., ground state). By the time of Drude, it was accepted that this process was fully described by the theory of Maxwellian light emission from bound charged parti- cles (with some natural oscillatory frequency ν0) undergoing forced simple harmonic motion as a consequence of the oscillating electric field (with frequency ν) they experi- ence as a classical electromagnetic field passes over them. In elementary physics courses, one learns that the induced displacement of a simple harmonic oscillator of natural fre- quency ν0 when subject to an external harmonic force of frequency ν is proportional to 1/(ν2 −ν2 0 ), if the harmonic oscillator is undamped. The fact that the oscillator responds with an amplitude that goes to infinity when the external frequency ν is tuned to the natural frequency ν0 is an artifact of the neglect of damping: including damping intro- duces an additional term in the denominator that protects the induced displacement from actually becoming infinite. From the induced displacement of a charged “ion”, one proceeds directly to the induced electric polarization and thence, by a well-known formula in classical electromagnetic theory, to a formula for the index of refraction.15 For now, we ignore the issue of damping, and refer to the effect of the term 1/(ν2 −ν2 0 ) as the Sellmeier resonance pole associated with the presence of oscillators of frequency ν0, giving due homage to Wolfgang Sellmeier’s work in 1872 in relating the index of refraction of light in media to forced simple harmonic motion of massive point particles. The obvious generalization, in which charged particles of various distinct natural oscil- lation frequencies νi (with i = 0, 1, 2, . . .) are present, leads to a formula for the index of refraction as unity plus a sum of Sellmeier terms with resonance poles Ci/(ν2 − ν2 i ). Formulas of this type were indeed found to describe very accurately the measured index of refraction for many transparent media, including the case of “anomalous dispersion”, which occurred when one moved from one side to the other of a resonance pole, thereby interrupting the normal continuous monotonic increase of the index of refraction with increasing frequency. Thus, in the closing years of classical physics, a perfectly reason- able “explanation” of the frequency dependence of the index of refraction n(ν), the core theoretical problem of optical dispersion, seemed at hand. In one respect, however, the Lorentz–Drude derivation of the Sellmeier formula led to empirical difficulties. The location νi of the resonance poles did indeed agree with the observed location of discrete lines in the emission and absorption spectrum 15 A full physical explanation of the dispersion phenomenon goes something like this: incident monochro- matic light (of a fixed frequency ν) passing over a charged oscillator will induce a forced oscillation of the particle, with the same frequency and in a fixed phase relation to the incident wave. The oscillation of a charged particle necessarily results in emitted electromagnetic radiation of the same frequency, so the out- going light from the particle is a superposition of the incident wave and the additional electromagnetic field produced by the particle itself. The coherent superposition of an incident wave with the emitted stimulated radiation of a uniform array of such charged oscillators results in a wave of the same frequency but, typically, smaller wavelength—that is, a wave of slower phase velocity c/n, with the index of refraction n(ν) > 1.
- 43. 22 Introduction to Volume 2 of the dispersing medium, as one would expect if these frequencies corresponded to natural frequencies of oscillations of the mobile charges in the medium. On the other hand, the numerator factors Ci in the Sellmeier poles were, quite understand- ably, directly related to the number density Ni of electrons capable of executing harmonic vibrations at the corresponding natural frequency νi, and could be imme- diately extracted from measurements by fitting the observed index of refraction to the Lorentz–Drude version of the Sellmeier formula. It turned out that the electron num- ber densities extracted in this manner were usually orders of magnitude smaller than expected on the basis of the known densities of atoms in the samples under inves- tigation. This problem remained a severe one through the years of the old quantum theory, and really was resolved properly only with the emergence of modern quantum theory. 8.4.2 Dispersion theory and the Bohr model The advent of the Bohr model for atomic hydrogen in 1913, and its spectacular success in explaining the Balmer formula, meant at the same time a “Kuhn loss”16 for the previ- ous success of the classical electron theory approach to optical dispersion. Electrons were no longer thought to be bound by harmonic forces in atoms, at fixed equilib- rium positions in the normal (i.e., ground) state of the atom, and then perturbed by the electromagnetic field of a passing light wave into sympathetic oscillations around these equilibrium points. Instead, the normal state of an atom consisted of a station- ary nucleus surrounded by electrons executing a discrete set of selected quasi-Keplerian orbits. In fact, after Bohr’s talk at the September 1913 meeting in Birmingham of the British Association for the Advancement of Science, where his new model for hydro- gen received its first widespread exposure (see Volume 1, pp. 200–201), Lorentz had asked how Bohr’s concepts of stationary states connected by quantum transitions could possibly be compatible with the classical dispersion theory (see Dresden 1987, p. 147). The Lorentz–Drude theory of dispersion had proven to be extremely successful at the empirical level, but clearly would have to be radically altered to take this new picture of atomic structure into account. In particular, the treatment of dispersion would have to be compatible with the new, utterly non-classical principle introduced by Bohr to explain the hydrogen spectrum: namely, that electrons in orbits corresponding to sta- tionary states were subject to a radiation veto, preventing them from emitting Maxwellian electromagnetic radiation, despite the obvious presence of acceleration in these bounded orbits. By early 1915, a new approach to dispersion had been introduced by Arnold Sommer- feld and Peter Debye (a former student of Sommerfeld). The rather ad-hoc procedure advocated by these authors was to insist on the application of classical electromagnetic radiation laws for the small perturbations around Bohr orbits induced by an incident light 16 This is the phenomenon, discussed by Thomas S. Kuhn (1962, Ch. 9), that a new theory (or paradigm) cannot explain something that was explained by the theory (or paradigm) it replaced. See Section 15.2, and Midwinter and Janssen (2013).
- 44. The dispersion of light: a gateway to a new mechanics 23 wave of small amplitude passing over the atom, while the overall Bohr veto of transitions out of a stationary state orbit (on the time scale of the orbital period, or the period of the incident light) was maintained. Dispersion measurements were available for hydrogen gas, which consists of hydro- gen molecules, not separated atoms. Debye adopted Bohr’s 1913 model of a hydrogen molecule, in which two electrons circulated around the line between two hydrogen nuclei, at opposite sides of a circular orbit, and subjected the electrons to the small electric field of an electromagnetic wave of frequency ν (Debye 1915). It is hardly surprising that the resulting formula for the index of refraction obtained by Debye did not have the classic Sellmeier form, in two respects. In addition to resonant pole terms of Sellmeier form, Ci/(ν2 i −ν2 ), there were terms with resonant poles at com- plex frequencies, corresponding to unstable perturbations of the Bohr orbit. Moreover, the frequencies of the real Sellmeier poles νi were completely unrelated to the frequen- cies associated with lines in the spectrum of molecular hydrogen, which by the Bohr frequency condition were associated with the energy difference between Bohr station- ary states (divided by Planck’s constant). The Debye–Sommerfeld calculations instead related these frequencies to the Fourier components of the initial orbit being followed by the electron(s), and not to any other stationary state orbits which the electron(s) might occupy after a quantum jump. By 1922, Paul Epstein could assert confidently, after an extremely careful rederivation of the Sommerfeld–Debye results, using the full machinery of the canonical perturbation theory of multiply periodic systems, that, given the failure of the theory to yield locations of maximum dispersion and absorption as required by the “Kirchhoff principle”—namely, at the same frequencies corresponding to emission lines of the material—the “conclusion seems unavoidable” that the foun- dations of a hybrid dispersion theory of this sort were incorrect (Epstein 1922c; see Section 10.2.1). Progress in reconciling the Bohr view of an atom in terms of electrons following a dis- crete set of classical paths, distinguished by appropriate quantization conditions, with the Lorentz–Drude–Sellmeier theory, which seemed at least phenomenologically to describe very well the passage of light through transparent media, would require the combination of two ingredients that became available by the end of the second decade of the twenti- eth century. The first ingredient was Bohr’s correspondence principle, in the extended version covering both frequencies and intensities of transitions. We have already seen that the correspondence principle was (at best) a deceptive red herring in the search for the rationale underlying the periodic character of atomic structure as revealed in Mendeleev’s arrangement of the elements. This periodicity in the end could only be explained once the Pauli exclusion principle had been asserted. However, the correspon- dence principle would turn out to be critically important in the developments leading up to the Kramers–Heisenberg theory of optical dispersion—one of the few quantita- tive predictions of the old quantum theory to survive completely intact the transition to modern quantum mechanics. The second ingredient essential to progress in dispersion theory was Einstein’s quan- tum radiation theory of 1916/1917 (see Section 3.6). Einstein’s remarkable insight was to explain the emergence of the classical laws describing the emission and absorption of
- 45. 24 Introduction to Volume 2 electromagnetic radiation from charged particles (at the macroscopic level) in terms of statistical phenomena (at the microscopic, atomic level). For example, a bound electron executing an orbit would classically, because of the inevitable accelerations in its motion, continuously emit radiation, thereby losing energy and dropping to a lower state. Instead of this, Einstein proposed that electrons in an excited state k with energy Ek could make a quantum jump to a state i of lower energy Ei by emitting a light quantum with a frequency given by the Bohr frequency condition, i.e., νki = (Ek −Ei)/h. The probability per unit time of such an emission arose from two distinct sources: a spontaneous emission probability Ai k, which was always present (even in the absence of external radiation), and a stimulated emission probability Bi kρ, proportional to the presence of electromagnetic radiation incident on the atom with energy density ρ, at the frequency of the light quanta emitted in the transition k → i. Similarly, in the presence of electromagnetic radiation, the inverse process of absorption, in which an electron would use light energy absorbed from the environment to execute the reverse transition i → k, would not occur continuously, as in classical theory, but randomly and stochastically, with a probability per unit time of an absorption event given by Bk i ρ. The A and B coefficients appearing in Einstein’s theory were not calculable with the tools at his disposal, but remarkably, using very general arguments of thermodynamic equilibrium, Einstein was able to recover both the Bohr frequency condition and the Planck black-body radiation law for the dependence of ρ on frequency on the basis of these very simple statistical assumptions. In a paper entitled “The quantum-theoretic meaning of the number of disper- sion electrons,” Rudolf Ladenburg (1921), an experimental physicist in Breslau (now Wrocław), made a significant step forward in the interpretation of the Sellmeier for- mula by combining the Einstein radiation theory with an argument inspired by the correspondence principle equating two distinct approaches, one quantum and the other classical, to calculating the radiation emitted by a charged oscillator. By this point it was known that the numerator factors in the Sellmeier poles simply could not be asso- ciated directly with the “number of dispersion electrons” N, if this was to correspond to the number of valence electrons in the dispersing sample, which were presumably the electrons participating in the dispersion of incident light. Ladenburg showed that by equating the classical formula for the radiation rate of a charged Planckian oscilla- tor immersed in ambient black-body radiation to the Einstein formula for spontaneous emission, one could show that the number of dispersion electrons N was not equal to the actual number of electrons N, but rather given by N multiplied by a factor involving the Einstein A coefficient for the transition associated with the Sellmeier pole in question. This factor was in general a real number that was certainly not constrained to be equal to unity. However, while it could be extracted from measurements, this factor could not at this stage be predicted by theory, as it involved the Einstein A coefficient, the relation of which to the underlying quantum dynamics of atomic electrons was as yet unknown. The year 1923 saw the tenth anniversary of Bohr’s remarkable trilogy establishing the planetary Rutherford model, supplemented by quantum conditions on the electron orbits, as the basis for atomic theory. The journal Die Naturwissenschaften devoted an
- 46. The dispersion of light: a gateway to a new mechanics 25 entire issue to articles commemorating Bohr’s great contribution. Ladenburg, in col- laboration with his Breslau colleague Fritz Reiche, a theoretical physicist, contributed an article on “Absorption, scattering, and dispersion in the Bohr atomic model” to this issue (Ladenburg and Reiche 1923). Bohr himself had long been puzzled by the infa- mous paradox whereby the poles in the well-established Sellmeier dispersion formula were located at the quantum transition frequencies, rather than the frequencies corre- sponding to actual Fourier components of the electron motion in a Bohr stationary state orbit. In a long paper entitled “On the application of the quantum theory to atomic structure,” submitted in November 1922, Bohr had emphasized that “dispersion must be so conceived that the reaction of the atom on being subjected to radiation is closely connected with the unknown mechanism which is responsible for the emission of radi- ation on the transition between stationary states” (Bohr 1923d, p. 162). Although Bohr had no very definite ideas to suggest that would help to flesh out the quantitative nature of this connection, the problem of dispersion comes up on several occasions in the cor- respondence between Bohr and Ladenburg in the Spring of 1923, and may have been the impetus for Ladenburg’s return to the subject, this time together with Reiche, in an attempt to deepen the connections he had already found in 1921 between the Einstein radiation theory and dispersion phenomena (Duncan and Janssen 2007, p. 585). Ladenburg and Reiche (1923) fully adopted the guiding methodology of the sharp- ened correspondence principle, by which quantum transition frequencies and inten- sities) were asymptotically related to the Fourier frequencies and squares of Fourier amplitudes, respectively, of the classical motion of the radiating electron, in the event that the motion was multiply periodic (i.e., a superposition of a finite set of basic frequen- cies together with their harmonic overtones). This principle could be used to guess the analytic form of the Einstein spontaneous radiation A coefficient, which Ladenburg had shown two years earlier to be related to the numerator factor in the Sellmeier dispersion formula—and hence, to the number N of dispersion electrons. The extended/sharpened correspondence principle was most easily applied to the simplest dynamical system representing a bound charged particle, the Planckian sim- ple harmonic oscillator of natural frequency ν, as in this case there were no harmonic overtones, just the fundamental frequency ν, which meant that only quantum transi- tions with unit change in the quantum number n of the stationary state were allowed, ∆n = ±1. When the correspondence argument was applied to a system of N such oscil- lators (moving in an isotropic three-dimensional potential) in the quantum state n, the authors found that the number of dispersion particles needed to account for the numer- ator factor in the Sellmeier formula was N = ((n + 3)/3)N. Evidently, this meant that the number of dispersion particles N agreed with the actual number N of particles only when the system was in its ground state (all particles with n = 0). For a system with the oscillators all in an excited state n > 0, the “number of dispersion particles” was evidently a number (not necessarily an integer!) greater than the actual number of par- ticles. This discrepancy would be resolved a year later with the derivation of a “new, improved” dispersion formula by Hans Kramers. Ladenburg and Reiche, in their attempts to find a new language to describe the admit- tedly bizarre situation where the number of apparent “dispersion electrons” could not
- 47. 26 Introduction to Volume 2 in general be brought into agreement with the actual number of radiating charged par- ticles in the optical medium of interest, introduced a concept that would play a very important role in the 1924/1925 period, and which would lead to a final, and concep- tually satisfactory, solution to the quantum dispersion problem for atomic models of the Bohr type. Namely, it was assumed that optical dispersion phenomena could be understood by treating the radiating charged particles in atoms as “substitute oscilla- tors” (Ersatzoszillatoren). In other words, as far as phenomena of absorption, emission, or coherent scattering (dispersion) were concerned, atomic electrons circulating in mul- tiperiodic, quasi-Keplerian stationary state orbits, for some reason, acted as though they were actually charged simple harmonic oscillators, with each Bohr transition r → s with emitted frequency νrs associated with a corresponding substitute oscillator with a natu- ral frequency of oscillation equal to νrs. The peculiar values of the Sellmeier numerator factors (now connected to the Einstein spontaneous emission coefficient As r) could now be adjusted on the assumption that the substitute oscillators were to be treated as having non-standard values for the electron charge e and/or mass m. The idea of such oscilla- tors as the responsible entities for the emission of radiation from atoms was not in fact entirely new: in a response to a question from Langevin in the question session follow- ing his talk at the 1921 Solvay conference (“Notes on the theory of electrons”), H. A. Lorentz had speculated one could for example imagine that besides the atoms of Bohr a [radiating] gas contains true “vibrators” [oscillators] which could provisionally store the energy lost by the atom in passing from one stationary state to another; it would be necessary in this case for the quantity of [stored] energy be exactly the quantum corresponding to the proper frequency of the vibrator [i.e., hν for an oscillator of natural frequency ν] . . . Perhaps also the atom temporarily changes itself into a vibrator (Solvay 1923, p. 24). 8.4.3 Final steps to a correct quantum dispersion formula The derivation of correct quantum theoretical expressions for the interaction of bound atomic electrons with the electromagnetic field would be accomplished, more or less independently, by two physicists working in different continents and in very different scientific environments. The first, Kramers, Bohr’s trusted assistant in Copenhagen, we have already met on several occasions in connection with the Stark effect and the Bohr– Kemble model of helium (see Sections 6.3 and 7.4). The other physicist, the young American theorist John Van Vleck, we encountered in our account of the final (failing) attempts of the old quantum theory to deal with the binding energy of two electrons in the helium atom (see Section 7.4). Indeed, the extra technical equipment essential in their attack on the helium atom, a complete facility with canonical perturbation the- ory in classical mechanics, was the essential ingredient that allowed both Kramers and Van Vleck to go beyond the work of Ladenburg and Reiche, and arrive at formulas for emission, absorption, and (in the case of Kramers, and later, Heisenberg) both coher- ent (dispersion) and incoherent (Raman) scattering, formulas that survived completely intact the transition to modern quantum mechanics. Of course, the formulas in question
- 48. The dispersion of light: a gateway to a new mechanics 27 contain elements (such as the Einstein coefficients) that could not be calculated from first principles until the full apparatus of the new theory was in place, but the form and structure of the results obtained by Van Vleck, Kramers, and Heisenberg are completely correct, and represent an inspired application of the correspondence-principle method- ology, insofar as the authors managed to “guess” the correct results—remarkably, in the absence of a consistent underlying dynamical framework. While Kramers was finishing his calculations on the binding energy of helium in the Bohr–Kemble model—he submitted his paper on the topic the last day of 1922 (Kramers 1923a)—he was also working on rotational band spectra in diatomic molecules, in collaboration with Pauli. They submitted their joint paper, “On the theory of band spectra,” the first week of January 1923 (Kramers and Pauli 1923). Sometime in the next few months, probably by early May 1923 at the latest, when Reiche wrote to Kramers asking for help on the dispersion problem,17 Kramers must have begun to think more intensely about the difficulties of reconciling the classical dispersion theory with Bohrian atomic models. One presumes after all that Kramers, working in Copen- hagen at the Bohr Institute, and in almost daily contact with his mentor, would have shared Bohr’s unease about the state of the quantum dispersion problem that Bohr had clearly described in the article on applications of quantum theory to atomic structure, which had been submitted in November of 1922. The conflict between central scientific contributions of Kramers’ two greatest heroes, Lorentz and Bohr, would certainly have engaged him directly. Reiche’s letter to Kramers on May 9, 1923, informing him of the upcoming publication of his article with Ladenburg in the Bohr commemorative issue of Die Naturwissenschaften, might well have been the additional impulse needed for Kramers to focus on the question of the Sellmeier formula—specifically, the problem of the “number of dispersion electrons” N and the connection of N to the Einstein spontaneous emission A coefficients. What we do know with considerable certainty is that, by the time the young Amer- ican physicist John Slater arrived in Copenhagen in late December of 1923, Kramers had derived, and could show to Slater, a generalized form of the Sellmeier–Lorentz– Drude dispersion formula, expressing the electric polarization of a material exposed to an incident monochromatic electromagnetic wave as a difference of two sums, the indi- vidual terms of the sums taking the classic Sellmeier form, with poles at frequencies corresponding to allowed quantum transitions of the material (as empirically established from the location of emission and absorption lines). For materials in which all the atoms were in their lowest possible energy state (which was the case in the vast majority of dis- persion measurements), the sum of negative terms was absent, and Kramers’ formula reduces to the usual Sellmeier form as a sum of poles with positive numerators. How did Kramers arrive at this peculiar generalization of a formula that had survived 50 years of intense empirical testing? As we explain in greater detail in Chapter 10, the reconstruction of the exact chronology of Kramers’ discovery of the correct quantum dispersion formula must remain to some degree speculative, as his earliest publication 17 Reiche to Kramers, May 9, 1923 (Duncan and Janssen 2007, p. 588; see Section 10.3.3).
- 49. 28 Introduction to Volume 2 on the subject came several months after his discovery of the formula, and, somewhat misleadingly, was phrased in terms of a new theory by Bohr, Kramer, and Slater (1924a), of which it is really conceptually independent. Here we merely summarize what the best evidence shows, and follow the chronological sequence of events as best we can determine it. It seems that by late 1923 Kramers had succeeded in deriving, using the technology of action/angle variables for multiperiodic classical systems, a formula for the coherent polarization of systems of mobile charged particles induced by an applied periodic sinu- soidally varying electric field, such as that associated with an incident electromagnetic wave. As mentioned earlier, this polarization can be directly related to the (squared) index of refraction n2 of the system. In the case of a system of charged particles subject to harmonic forces, and hence, executing harmonic oscillations around their equilibrium positions, the resulting formula for n2 is then of Sellmeier form. Even for particles exe- cuting more general types of multiperiodic motion, however, Kramers found a formula giving n2 as a sum of terms of Sellmeier type, with poles located at frequencies corre- sponding to the non-vanishing Fourier components of the classical motion. However, the formula had a critical feature of great importance in the correspondence-principle con- text where it would be applied: each Sellmeier pole appeared with a continuous derivative with respect to the action variables Ji of the system. Classically, of course, the action variables of a system were continuously variable quantities, so the derivative could be calculated according to the normal rules of calculus. In the old quantum theory, however, the allowed stationary states of the system were selected by insisting that each action vari- able was an integral multiple of Planck’s constant, Ji = nih. In fact the stationary states were identified by specifying the associated set of quantum numbers {ni}. In the correspondence limit of large quantum numbers ni ≫ 1, the action variables (such as the energy) changed relatively slowly in a quantum transition when the (large) ni changed by numbers of order unity and one could asymptotically replace continuous derivatives with discrete ones. In the action/angle formalism, where a change of ni by unity involved a change of the associated action variable Ji by h, one could therefore approximate the derivative df( Ji)/dJi of a function f( Ji) by the discrete difference oper- ation (f( Ji =(ni + 1)h) − f( Ji =nih))/h. Kramers’s derivation of a quantum dispersion rule involved the replacement of the action derivatives appearing in his classical formula with discrete differences, in which each Sellmeier pole involved a difference of two terms, in which the negative term was related to the positive term by a reduction of the quantum numbers by unity (or, in the event that there were Fourier components corresponding to higher harmonics, the corresponding small whole numbers). It is clear in retrospect that Kramers’ work on dispersion in the Fall of 1923 must have involved the invocation of the following Replacement Postulate: Postulate: In classical formulas expressing the behavior of a multiperiodic system susceptible to description in terms of action/angle variables, each derivative of an action variable Ji is to be replaced by the corresponding discrete difference operation for the associated quantum number ni, divided by Planck’s constant. This apparently precise and concrete instruction represents in some sense the deep- est “sharpening” that the correspondence principle would achieve prior to the arrival of
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