Paul Ehrenfest Vol. I The Making of a Theoretical
Physicist,
by M.J. Klein,
Chapter 11. THE ADIABATIC PRINCIPLE (pp. 274 – 279)
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(Bohr’s Atomic Model)
The Wolfskehl week took place in May 1913,
several months too early for any discussion of the most significant new work on
the quantum theory, the first of Niels Bohr's papers, "On the Constitution
of Atoms and Molecules."12 Bohr, whose boldness and originality
were comparable only to Einstein's, had written an extremely rich and suggestive
work. It was the first phase of an attempt to develop a theory of atomic and
molecular structure and to use this theory to understand a whole range of
physical and chemical properties of matter.13 In this first paper
Bohr had studied the binding of a single electron by a nucleus with a single
positive charge to form a hydrogen atom. His work was based on Rutherford's
model of the atom as a small massive nucleus surrounded by light electrons.
Among other things Bohr worked out the frequencies of the spectral lines that
could be emitted or absorbed by hydrogen on the basis of his theory. Although
this was not a part of his original program, it turned out to be the most
strikingly successful aspect of the theory. Bohr's results agreed remarkably
well with experiment, not only for the general structure of the spectrum, but
also for its single numerical constant. As Bohr pointed out, Planck's constant
was just what was needed to complete the determination of the scale of atomic
dimensions.
Since Bohr was working on a quantum
theory of atomic systems, he had to face the same question that his fellow
theorists were puzzling over: how was the quantum theory of Planck's oscillator
to be extended to more general systems? It is characteristic of Bohr that he
did not try to give a simple answer to this difficult question.'' He saw that
in Planck's theory of the oscillator the motions were restricted to certain
discrete values of the energy. When the oscillator emitted or absorbed energy
it was by means of a definite transition from one of these allowed states to
another, and no radiation was emitted or absorbed while the oscillator remained
in one of its allowed states. Planck's requirement that the energy be capable
of only certain discrete values was, in effect, a way of achieving stability, a
stability utterly foreign to classical electrodynamics in which an oscillating
charge (or any accelerated charge) would necessarily radiate energy. It was
apparently this feature of the quantum theory, the stability imposed by the
existence of discrete stationary states, that impressed Bohr most strongly, for
it was the stability of atomic systems which seemed both so characteristic and
so hard to understand.
At the most general level Bohr postulated that
atomic systems possessed a series of allowed stationary states, in which no
radiation occurred. While the system remained in such a state one could apply
ordinary mechanics to discuss its dynamical equilibrium, but the process of transition
from one such state to another could no be treated that way. For the transition
process Bohr assumed that the radiation emitted was "homogeneous," of
a single frequency ν. This frequency ν was to
be related to the energy E emitted,
the energy difference between the initial and final states, by Planck's
relationship, E = hν.
The first of those assumptions amounted to a severe restriction on the validity
of mechanics, to which the concept of stationary state is foreign. Bohr
described the second as being “in obvious contrast to the ordinary ideas of
electrodynamics,” but he was notoriously mild in his language: it was absolutely
heretical. He renounced any attempt to identify the frequency of the emitted
light with the frequency of some motion of the sources of the electromagnetic
field, the electrically charged particles in the atom.
Bohr also made a more specific assumption
about the way in which the quantum of action must be used to determine the
stationary states. He actually did this in several different ways in this first
paper, beginning with the assumption that the frequencyν of the light
emitted when a free electron at rest is bound into a stationary state, is just
half the frequency ν' of the electron's revolution in its circular
orbit in that state. Bohr then used "Planck's relationship" to set
the energy E emitted equal to nhν or (nhν’/2) with n an integer. (Bohr shifted his
interpretation of the equation E = nhν’/2 in the course of his paper, originally
treating n as the number of quanta
emitted, there being a different number for each stationary state, but then
later treating E as one quantum of
the appropriately higher frequency.) It was also in curse of his probing into
the significance of this assumption that Bohr introduced a mode of argument he
would later use more and more. This was the requirement that any result obtained
from the quantum theory would have to agree with the result of a classical
calculation "in the region of slow vibrations,” where the classical theory
was in agreement with experiment.
While Bohr did not arrive at his
results this way, he found "a very simple interpretation" of his
stationary states “by help of symbols taken from the ordinary mechanics.” He
warned, however, that “there obviously can be no question of a mechanical
foundation of the calculations given in this paper." The
"interpretation" Bohr referred to was the remark that the angular
momentum of the electron in its nth
circular orbit around the nucleus would be precisely nh/2π. This interpretation, in which Bohr was
surely influenced by the prior work of J. W. Nicholson,15 quickly
took a central position in Bohr’s work on atomic problems. It is, in fact,
remarkable to see how the idea of quantized angular momentum, (to use an anachronistic
term), which appears so guardedly on page 15 of Bohr's paper, after much
discussion of other ways of interpreting his generalization of Planck's
equation, becomes the key idea in his thinking about atomic and molecular
problems throughout the rest of his series of three papers. His “main
hypothesis” in studying "the permanent states" of atomic and molecular
systems is that "the angular momentum of every electron round the centre
of its orbits” will be h/2πin this permanent state.
(Reactions to Bohr’s Model from his Contemporaries)
5. Bohr's
contemporaries reacted to his work in a great variety of ways. Sommerfeld,16
for example, wrote Bohr to thank him for his "extremely interesting
work." He remarked that he had been thinking for some time about the
problem of expressing the Rydberg constant in terms of Planck's h and had even "talked about it to
Debye a few years ago." He was "for the present still rather
skeptical about atom models in general," but felt "nevertheless"
that "the calculation of this constant is indisputably a great
achievement." Sommerfeld was interested enough to want to apply Bohr's
methods to the Zeeman effect, and asked whether Bohr had any plans along this
line.
This was one characteristic reaction – characteristic, that is, of
Sommerfeld. Elsewhere there were doubts, as at Goettingen, where Richard
Courant was "laughed at for taking such fantastic ideas seriously."17
Ernest Rutherford, Bohr’s great mentor, found his ideas "very ingenious,"
but thought that "the mixture of Planck`s ideas with the old
mechanics" made it "very difficult to form a physical idea of what is
the basis of it."18 James Jeans presented Bohr's ideas in a
favorable light to the. British Association in September 1913. Having been
convinced of the necessity of quanta by Poincare's work, Jeans was ready to
take the every large next step needed to accept Bohr's new theory.19
We have Einstein's reaction to Bohr's work in the words of George Hevesy.
When Hevesy told him that new experiments confirmed Bohr's explanation of the
origin of the Pickering lines (as produced by ionized helium), "the big
eyes of Einstein looked still bigger and he told me ‘Then it is one of the
greatest discoveries.’”20 In another letter Hevesy described
Einstein as being particularly impressed by one point: "’Then the
frequency of the light does not depend at all on the frequency of the electron’
- (I understood him so??) And this is ann enormous
achievement. The theory of Bohr must be then wright.”21 (The
picturesque spelling is Hevesy`s.) Just this point would be a major objection
to Bohr's work for most physicists. Here is what Erwin Schroedinger had to say
about it in 1926: "The frequency discrepancy in the Bohr model, on the
other hand, seems to me, (and had indeed seemed to me since 1914), to be
something so monstrous, that I should like to characterize the excitation of
light in this way as really almost inconceivable.”22
Schroedinger's words of 1926 are not
terribly different from those that Paul Ehrenfest used in expressing his first
reaction to Bohr's paper. In a letter to Lorentz written in August 1913,
Ehrenfest commented: "Bohr's work on the quantum theory of the Balmer
formula (in the Phil. Mag.), has
driven me to despair. If this is the way to reach the goal. I must give up
doing physics."23 This was not just a casual remark.
Ehrenfest's notebooks and correspondence. clearly indicate that he did not try
to work himself into a full understanding and acceptance of Bohr's approach for
several years. Once he met Bohr and absorbed the essence of his ideas and
attitudes in direct conversation, Ehrenfest became an enthusiastic booster and
close friend, but that had to wait for the end of the World War. In 1913 and
1914 Ehrenfest continued to work on his own approach to the problems of the
quantum theory. uninfluenced by Bohr's contribution.
(Ehrenfest’s Reaction)
6. One of the
questions that had troubled Ehrenfest for years was what the quantum theory
might do to the statistical foundations of thermodynamics. Boltzmann's derivation
of the second law of thermodynamics as a theorem in statistical mechanics had
used as one of its essential links the assumption that equal volumes in phase
space were to be assigned equal statistical weights. It was of the essence of
the quantum theory, however, that this classical uniform weighting of phase
space was to be abandoned, and only certain particular regions of phase space
were to be allowed, that is, to be given nonzero weigh. In the case of Planck's
oscillators, for example, the allowed domains were those particular ellipses of
constant energy whose enclosed areas were integral multiples of h.
Ehrenfest had already considered this question in detail for the quantum
theory of harmonic oscillators in his 1911 paper.24 He had shown
that Boltzmann's relationship between entropy and the number of ways of
achieving the most probable distribution remained valid precisely because
Planck had quantized the oscillator's adiabatic invariant, the ratio of its
energy to its frequency, Ehrenfest had proved more generally that if, and only
if, the statistical weight function depended only on this adiabatic invariant,
the statistical thermodynamics of the oscillator was secure.
In 1914 Ehrenfest was ready to discuss
the problem again on a much more general basis, without any special reference
to the oscillator. In a papers sent off to the Physikalische Zeitschrift in May, he dealt systematically with one
sharply defined question. For a general ideal molecular system, what
restrictions must be placed on the statistical weight function in phase space
so that Boltzmann's relationship between the thermodynamic properties of the
system and the averages of its molecular properties over the most probable
distribution remains valid?
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12. N. Bohr,
"On the Constitution of Atoms and Molecules," Phil. Mag. 26 (1913),
p. 1. The other two papers in the series appeared in the same volume, at pages
476 and 857. The three papers were reprinted in a book under the same title,
with an extremely interesting and informative introduction by L. Rosenfeld.
(Copenhagen: Munksgaard 1963). The three papers appear in German translation in
the book cited in note 10.
13. For general
discussion of Bohr's papers see Rosenfeld's introduction, op. cit., note 12.
Also see Jammer's book, op. cit., note l0e of Chapter 10, pp. 62-88, and K. M.
Meyer-Abich, Korrespondenz, Individualitaet, and Komplementaritaet, (Wiesbaden:
Franz Steiner Verlag, 1965), pp. 1-45.
14. See Bohr's
early discussion of his work in a lecture "On the Spectrum of
Hydrogen," given in Copenhagen on 20 December 1913. It is reprinted in N.
Bohr, The Theory of Spectra and Atomic Constitution, (Cambridge: University
Press, 1922), p. 1.
15. For a
discussion of Nicholson's work and its influence an Bohr see R. McCormmach,
"The Atomic Theory of John William
Nicholson," Archive for History
of Exact Sciences 3 (1966). P. 160.
16. A. Sommerfeld
to N. Bohr. 4 September 1913. Quoted in full by Rosenfeld, op. cit., note 12, p.
iii.
17. Quoted by L. Rosenfeld and E Ruedinger
in their article, "The Decisive Years 1911 – 1918," in Niels
Bohr, His Life and Work as Seen by
his Friends and Colleagues, ed. S. Rozental. ( Amsterdam: North-Holand,
1967), p. 57.
18. E. Rutherford
to N. Bohr, 20 March 1913. Quoted in full in Bohr’s Rutherford Memorial Lecture of 1958, "Reminiscences of
the Founder of Nuclear Science and of Some Developments Based on his
Work," Proc. Phys. Soc. 78
(1961), p. 1091. Reprinted in N. Bohr,
Essays 1958-1962 on Atomic Physics and Human Knowledge. (New York: John
Wiley, 1963), p. 41.
19. See J.H. Jeans,
op. cit., note 58. Chapter 10, p. 50.
20. G. Hevesy to
E. Rutherford, 14 October 1913. Quoted by Rosenfeld, op. cit., p. xiii.
21. G. Hevesy to N.
Bohr, 23 September 1913. Quoted by Rosenfeld. op. cit., p. xiii.
22. E. Schroedinger
to H.A. Lorentz, 6 June 1926. See Letters
on Wave Mechanics, E. Schroedinger,
M. Planck, A. Einstein, and H.A. Lorentz, ed. K. Przibram. transl. M.J.
Klein, (New York: Philosophical Library, 1967), p. 61.
23. - P. Ehrenfest to H. A. Lorentz, 25 August 1913.
24. See the reference in note 41 of Chapter 10, and
the discussion in section 9 of Chapter 10.
25. P. Ehrenfest, “Zum Boltzmannschen
Entropie-Wahrscheinlichkaits- Theorem,” Phys.
Z. 15 (1914), p. 657. Papers, p.
347.