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Arch. Hist. Exact Sci. (2009) 63:81125DOI 10.1007/s00407-008-0030-1

Ehrenfests adiabatic theory and the old quantumtheory, 19161918

Enric Prez

Received: 1 July 2008 / Published online: 4 September 2008 Springer-Verlag 2008

Abstract I discuss in detail the contents of the adiabatic hypothesis, formulated byEhrenfest in 1916. I focus especially on the paper he published in 1916 and 1917in three different journals. I briefly review its precedents and thoroughly analyze itsreception until 1918, including Burgerss developments and Bohrs assimilation ofthem into his own theory. I show that until 1918 the adiabatic hypothesis did not playan important role in the development of quantum theory.

Abbreviations

AHQP Archive for history of quantum physics. For the catalogue, see KUHN et al.(1967)

EA Ehrenfest archive. In the Rijksarchief voor de Geschiedenis van de Natuurwe-tenschappen en van Geneeskunde, Leiden. For the catalogue, see WHEATON(1977). I quote from the microfilm version included in the AHQP

HPE Huisbibliotheek van Paul Ehrenfest. In the Institut Lorentz, Leiden

Communicated by R. Stuewer.

E. Prez (B)Departament de Fsica Fonamental, Universitat de Barcelona,c. Mart i Franqus 1, 08028 Barcelona, Spaine-mail: [email protected]

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 822 Adiabatic invariants and the theory of quanta . . . . . . . . . . . . . . . . . 83

2.1 The adiabatic invariance of phase integrals . . . . . . . . . . . . . . . . 932.1.1 Burgerss trilogy . . . . . . . . . . . . . . . . . . . . . . . . . . 972.1.2 The developments by Krutkow and Kramers . . . . . . . . . . . . 100

2.2 Reception, before Bohr . . . . . . . . . . . . . . . . . . . . . . . . . . 1022.2.1 Smekal on the validity of Boltzmanns principle . . . . . . . . . . 1032.2.2 Planck on the asymmetric spinning top . . . . . . . . . . . . . . 1052.2.3 Sommerfeld on light dispersion . . . . . . . . . . . . . . . . . . 106

3 Bohrs principle of mechanical transformability . . . . . . . . . . . . . . . . 1093.1 The unpublished theory of 1916 . . . . . . . . . . . . . . . . . . . . . 1103.2 On the quantum theory of line spectra . . . . . . . . . . . . . . . . . . 113

4 The role of the adiabatic hypothesis in the old quantum theory . . . . . . . . 120Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

1 Introduction

Ehrenfests adiabatic hypothesis is usually mentioned in histories of the old quantumtheory,1 but no works are specifically devoted to it. Only Martin J. Klein has studiedit in any detail in his monograph on Paul Ehrenfests life and work.2 Klein covers theperiod up to 1916, but he does not treat Ehrenfests adiabatic hypothesis extensively;in particular, he does not analyze in detail the paper in which Ehrenfest formulatedit and developed its applications in 1916 and 1917 or its reception. He also does notdiscuss how Niels Bohr adapted and used it in 1916 and 1918 in his own work on thequantum theory.3

In earlier papers, Luis Navarro and I retraced Ehrenfests path to his adiabatichypothesis based upon our study of his notebooks, correspondence, and publications.4We argued that Ehrenfest, by analyzing Plancks black-body theory, discovered theessential role of adiabatic invariants in quantization. He centered his study on Wiensdisplacement law, since it could be deduced classically but remained valid in thequantum realm. We pinpointed the origin of Ehrenfests adiabatic hypothesis in aremarkable paper that he published in 1911, in which he deduced the necessity ofdiscontinuity and other important results. He then generalized his results for theblack-body system to diatomic molecules and other mechanical systems, and in 1913formulated a primitive version of his adiabatic hypothesis, although Einstein wasthe first physicist to call it such in 1914. That same year Ehrenfest established thenecessary conditions for a statistical treatment that would satisfy Boltzmanns principle

1 See, for instance, DARRIGOL (1992), JAMMER (1966), and HUND (1984).2 KLEIN (1985).3 Olivier Darrigol analyzes this adaptation in DARRIGOL (1992, pp. 85149).4 NAVARRO and PREZ (2004, 2006).

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Ehrenfests adiabatic hypothesis and the old quantum theory, 19141916 83

and demonstrated that Plancks, Debyes, and his own work on diatomic moleculesfulfilled those conditions.

I now turn to a new and definitive contribution of Ehrenfest in 19161917 in whichhe presented a major part of his research on the quantum theory during more than theprevious 10 years.5 In addition to collecting all his prior results, Ehrenfest extendedthem, showing their compatibility with Sommerfelds and Plancks quantizations ofsystems with several degrees of freedom.

Prior to the publication of his new contribution, Ehrenfests papers on quantumtheory had made little impact on the field. But then Sommerfeld and Planck appliedthe adiabatic hypothesis to particular cases.6 Bohr, who had sensed its usefulnessin 1916 prior to the publication of Ehrenfests papers, altered its form slightly andincluded it as a fundamental principle in his own works, thus ensuring its prominencein the years that followed.7 He called it the principle of mechanical transformability.Only after Bohr included Ehrenfests adiabatic hypothesis in his famous 1918 paper,On the Quantum Theory of Line Spectra,8 did it become widely known.

Ehrenfests research on the adiabatic hypothesis declined to almost nothing afterBohr exploded on the scene, as can be seen from his personal notebooks.9 He publishedonly one more paper on the subject, in 1923, in a special issue of Die Naturwissenschaf-ten commemorating the tenth anniversary of Bohrs quantum theory of the atom,10 inwhich he discussed his route to it and Bohrs use of it. Thus, after 1918 Ehrenfestsadiabatic hypothesis was intimately bound up with Bohrs work, and many physicistsused it in that context in various ways.11

In this paper, the third part of our trilogy on Ehrenfests adiabatic hypothesis, I willcomplete our analysis of its role in the development of the old quantum theory beforeBohr appeared on the scene.

2 Adiabatic invariants and the theory of quanta

Ehrenfest published his new contribution on adiabatic transformations in three jour-nals, the Proceedings of the Amsterdam Academy, first in Dutch and soon thereafterin English, in June 1916;12 in the Annalen der Physik,13 where it was received on 22July 1916, acquired a postscript on 6 September 1916, and was published in October;and in the Philosophical Magazine in June 1917,14 in which he incorporated somecorrections that had appeared as footnotes in the earlier versions and eliminated somenonessential observations and demonstrations.

5 EHRENFEST (1916a,b, 1917).6 See Sects. 2.2.2 and 2.2.3.7 See Sect. 3.8 BOHR (1918a).9 See, for instance, ENB:123 and the following notebooks; in EA, microf. AHQP/EHR-4.10 EHRENFEST (1923).11 See Sect. 4.12 EHRENFEST (1916a).13 EHRENFEST (1916b).14 EHRENFEST (1917).

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Klein included the English version in the Proceedings of the Amsterdam Aca-demy, presumably the earliest version, in his edition of Ehrenfests Collected Scienti-fic Works,15 but whenever possible I will quote from the version in the PhilosophicalMagazine, which was printed by Bartel L. van der Waerden, because of its superiorliterary quality.16

Ehrenfest had found most of the results in this paper earlier, but it also containedsome remarkable innovations. In general, as we will see, after introducing and for-mulating his adiabatic hypothesis, which he now calls as such, he relates it to variousquantum hypotheses and then connects it to the statistical interpretation of the secondlaw of thermodynamics. In other words, he presents a systematic and careful treatmentof his adiabatic hypothesis, showing for the first time its harmonious relationship toSommerfelds quantization conditions and Boltzmanns principle.

A. Formulation of the hypothesisEhrenfest formulates his adiabatic hypothesis cautiously, although he hopes to base

quantum theory on it. In view of the increasing number of physical phenomena thatcan be explained by classical mechanics and electrodynamics, on the one hand, andby quantum hypotheses, on the other, Ehrenfest seeks a more general point of viewthat will delimit these theories as clearly as possible. The case of Wiens displacementlaw shows that, without introducing quanta, it is possible to calculate the variation ofblack-body energy distribution, as well as the work performed on a radiating cavityduring a reversible adiabatic compression.17 Thus:

It might be possible that also in more general cases, when we do not restrictourselves to harmonic motions, the reversible adiabatic transformations shouldbe treated in a classical way, whereas in the calculation of other processes (e.g.an isothermal addition of heat) the quanta come into play.18

Let q1, . . . , qn be the coordinates of a given mechanical system whose potentialenergy depends not only on the qi , but also on certain parameters a1, a2, . . . that varyinfinitely slowly; the kinetic energy may be a homogeneous quadratic function ofthe velocities qi , the coefficients of which are functions of the q and may be of thea1, a2, . . .19 In this kind of system, the motions (a) can be transformed into differentmotions (a) by an infinitely slow change from the initial values of the parametersa1, a2, . . . to final values a1, a2, . . . This is a reversible adiabatic transformation,and the motions (a) and (a) are adiabatically related.

Next, following a form of presentation he uses throughout his paper, Ehrenfestmakes two observations. First, concerning the meaning of the term reversible, in per-iodic motions the parameters are returned to their initial values and the original motionis recovered; this also can occur in multiply periodic motions (see below), but for

15 KLEIN (1959, pp. 378399).16 VAN DER WAERDEN (1968, pp. 7993).17 NAVARRO and PREZ (2004, pp. 110126).18 EHRENFEST (1917); in VAN DER WAERDEN (1968, p. 79).19 EHRENFEST (1917); in VAN DER WAERDEN (1968, p. 81).

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nonperiodic motions (for instance, hyperbolic trajectories) the original motionscannot be restored by restoring the initial values of the parameters ai . Second, Ehrenfestnotes that the definition of reversible adiabatic transformation has to be generalizedto consider, for example, a system immersed in a magnetic field or an electromagneticsystem.

For periodic and multiply periodic motions, let B{ao} be the allowed motions of asystem for a fixed set of parameters. Then:

For general values a1, a2, . . . of the parameters, those and only those motionsare allowed which are adiabatically related to the motions which were allowedfor the special values ao1, a

o2, . . . (i.e. which can be transformed into them, or

may be derived from them in an adiabatic reversible way).20

The adiabatic hypothesis then takes on a pioneering role in applying the quantumhypothesis, since the only requirement is to find an adiabatic transformation that linksone system to another system in which the allowed quantum motions are known, asin the case of Wiens displacement law.

To use the adiabatic hypothesis correctly, it is necessary to identify the adiabaticinvariants of the system to be quantized, that is, the quantities that are unchanged undera transformation of a motion (a) to an adiabatically related motion (a).

If an adiabatic invariant for the allowed motions B{ao}, belonging to thespecial values ao1, ao2, . . ., possesses the distinct numerical values , , . . .,it possesses exactly the same values for the allowed motions belonging toarbitrary values of the parameters a1, a2, . . . 21

Ehrenfest now introduces an invariant for periodic systems. Let a system be such thatfor certain arbitrarily selected values a1, a2, . . . and for any initial conditions qo1 , qo2 . . .it has only periodic motions whose period P depends on the ai , qoi and qoi . In this case,as Ehrenfest proves in the Proceedings version of his paper, the time integral over onecomplete period of twice the kinetic energy T is an adiabatic invariant:

P

0

dt 2T = 0, (1)

where indicates the difference between infinitely close values of the integral cor-responding to two adiabatically related motions. In a footnote, Ehrenfest offers otherexamples of adiabatic invariants, all being cyclical momenta (conjugate momenta ofcoordinates that do not appear explicitly in the Hamiltonian). He already presented theinvariant (1) in 1913; he now adds a geometric interpretation that allows him to relatehis adiabatic hypothesis to the quantum hypotheses of Planck, Sommerfeld, Debye,Bohr, and others. He introduces Sommerfelds action integral:

20 EHRENFEST (1917); in VAN DER WAERDEN (1968, p. 82), his emphasis.21 EHRENFEST (1917); in VAN DER WAERDEN (1968, p. 83), his emphasis.

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P

0

dt 2T =P

0

dtn

h=1phqh =

nh=1

dqh ph =

nh=1

dqhd ph,

so that

2T

=n

h=1

dqhd ph . (2)

Ehrenfest interprets this double integral as follows: If the system has a periodic motion,its trajectory in phase space is a closed curve in 2n-dimensional (q, p)-space, whosen projections on the two-dimensional planes (q1, p1), (q2, p2), . . . , (qn, pn) are nclosed curves. That is,

dqnd pn (3)

is the area enclosed by the (nth) projected curve. The numerical value of (2) does notdepend on the coordinate system. Furthermore, using suitable coordinates, not only isthe total sum (2) an adiabatic invariant, so too is each of the n addends that compriseit.

In one-dimensional systems there are no adiabatic invariants that are independentof (1). This result is equivalent to that presented by Paul Hertz in 1910:22 for acertain series of values ao1, a

o2, . . . of a n-dimensional system corresponding to a

motion represented by a curve in (q, p)-space on a hypersurface of constant energy(q, p, ao) = o, this hypersurface defines a 2n-dimensional hypervolume

dq1 d pn = Vo. (4)

If a reversible adiabatic transformation that links the series ao1, ao2, . . . to the seriesa1, a2, . . . is now performed, it will modify not only the energy of the system (byvirtue of the work exerted on it) but also the position (form) of the hypersurfacesin phase space. However, in accordance with Hertzs theorem, the hypervolume Vdefined by these hypersurfaces will not vary (see Fig. 1):

V = Vo. (5)

Hertzs proof was applicable only to ergodic systems, such as the one-dimensionalharmonic oscillator.23 In general, systems with two or more degrees of freedom willnot be ergodic.

22 HERTZ (1910).23 This explanation does not appear in the Philosophical Magazine version. See KLEIN (1959, p. 385,footnote 1), and EHRENFEST (1916b, p. 335, footnote 1).

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,

,

p

q

Fig. 1 Adiabatic transformation of Plancks ellipse (, ) into a different one (, ), with E (q, p, , ) =12

(2q2 + 2 p2

). The form changes, but the area does not. The area is an adiabatic invariant whose value

is given by A = 2 E

. This figure does not appear in Ehrenfests paper, but he describes it in a note addedto the versions in the Annalen and the Philosophical Magazine24

B. One-dimensional systemsIn quantum physics, one-dimensional systems embody Plancks hypothesis, which

states that a harmonic resonator of frequency o satisfies the equation:

o=

dqd p = 0, h, 2h, . . . (6)

In this case, Ehrenfest recalls that

2To

= o

. (7)

Now consider the equation of motion of an anharmonic oscillator,

q = (2oq + a1q2 + a2q3 +

), (8)

which for a1 = a2 = = 0, reduces to that of harmonic oscillator. Thus, sincean adiabatic transformation can be devised that connects the two, the anharmonicoscillator can execute only those motions that satisfy:

2T

= 0, h, 2h, . . . (9)

Ehrenfest concludes:

24 EHRENFEST (1917); in VAN DER WAERDEN (1968, p. 89, footnote *); EHRENFEST (1916b, p. 340,footnote 2)

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So by means of the adiabatic hypothesis we have derived Debyes hypothesis onthe values of

d pdq for nonharmonical vibrations from Plancks hypothesis of

energy elements.25

Ehrenfest thus established, for the first time, the connection between his adiabatichypothesis and Debyes quantum hypothesis. Debye in 1912 had obtained the sameresult as Ehrenfest had for a system of diatomic molecules, and Ehrenfest now showsthe equivalence of both procedures.26

He also presents the adiabatic transformation as applied to a dipole that he publishedtwice in 1913, and points out the aperiodicity of the motion that delimits oscillationsand rotations. He now proposes that a conical pendulum or a system subjected to amagnetic field can avoid it.27

C. Multidimensional systems

For multidimensional systems, Ehrenfest shows, in one of his most important inno-vations, that Sommerfelds quantum hypothesis, in particular its application to themotion of a point particle of mass m around a Newtonian center of attraction, satisfieshis adiabatic hypothesis. Let (r, a1, a2, . . .) be the potential of an attractive centralforce, so that the equations of motion in polar coordinates (r = q1, = q2) are:

mr mr 2 + ddr

= 0 and ddt

(mr2

)= 0, (10)

where the second equation shows that:

p2 = mr2 = const. (11)Eliminating from the first equation, the radial equation of motion,

mr = p22

mr3 d

dr, (12)

has the same form as the equation of motion of a point mass oscillating in one-dimension between two fixed values of r in a potential:

= p22

2mr2+ (r, a1, a2, . . .) . (13)

The adiabatic hypothesis shows that this motion has an adiabatic invariant

2T11

=

dq1d p1. (14)

25 EHRENFEST (1917); in VAN DER WAERDEN (1968, p. 86).26 Box 1 below displays all of the relations that Ehrenfest presented between his adiabatic hypothesis andvarious quantum hypotheses.27 See Bohrs discussion in the second paragraph of the text corresponding to our footnote 109. See alsoKLEIN (1985, p. 272, footnote 10).

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Equation 11 can also be interpreted in terms of the adiabatic hypothesis, since

2T22

= p2q2(q22

) = 2p2. (15)

Sommerfeld, by imposing his quantization conditions, had found that

dq1d p1 = 0, h, . . . , nh

dq2d p2 = 0, h, . . . , nh

, (16)

which are perfectly compatible with Ehrenfests adiabatic hypothesis as representedby (14) and (15). In other words, Sommerfeld had quantized two adiabatic invariants.

Ehrenfest also indicates that these same adiabatic invariants exist for periodicmotions in a Coulomb potential,

= ar,

and in an elastic restoring potential,

= ar2

2,

as well as for multiply periodic motions in an even more general potential, (r, a).Only in the first two cases, however, 1 = 2 = , so that the two adiabatic invariantscan be combined to yield

2 (T1 + T2)

= 2T

, (17)

for which Sommerfelds quantization conditions can be shown to be equivalent toPlancks hypothesis, which therefore also agreed with Ehrenfests adiabatic hypothe-sis.

Ehrenfest believed that the adiabatic invariants of even more general multiply perio-dic motions, especially those in anisotropic force fields, would elucidate the coordinatesystems in which Sommerfelds conditions could be applied. In Sommerfelds finalrefinement of his theory, where he considers the increase of electron mass with velo-city, a certain ambiguity arises in relation to the limits of the integrals (16) (the electronorbits are no longer closed).28 Ehrenfest proposes to investigate the quantities that areadiabatic invariants in the hope of removing this ambiguity.

28 See Sect. 2.1.

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D. Singular motionsEhrenfest tentatively tries to connect the difficulties involved in the quantiza-

tion of nonperiodic motions to singular motions (motions with infinite periods thatappear when performing adiabatic transformations between periodic or multiperiodicmotions). He offers a new example of a singular motion to consider alongside the limi-ting case of motion between vibrational and rotational of a diatomic molecule. Thisnew example arises in the reversible adiabatic transformation of a mass in an aniso-tropic field of force into an isotropic field of force whose potential energy is given by:

= 12

(21

21 + 2222

), (18)

where 1 and 2 are the two position coordinates. Ehrenfest uses Sommerfelds quan-tum conditions to solve the isotropic case (1 = 2) for a central field, and Plancksquantum hypothesis for the anisotropic case for a noncentral field. In the isotropiccase, the only motions that are permitted are those that satisfy the following twoequations:

mr2 = nh2

(angular momentum) (19)

= (n + n) h, (total energy) (20)

where the integers n and n correspond to the quantization of the radial and angularvariables, respectively. In the anisotropic case, the only motions that are permitted arethose that satisfy the following equations:

11

= n1h and 22

= n2h, (21)

where n1 and n2 are integers. If 1 and 2 now approach the common value infinitelyslowly, where the relations (21) are adiabatic invariants, then the total energy of thesystem satisfies the equation:

= (n1 + n2) h. (22)

In a note added in the proofs, Ehrenfest thanked Epstein for pointing out that equa-tions (20) and (22) are incompatible, since for a circular motion the former requiresn = 0 (for arbitrary n), while the latter requires n1 = n2, which means that in theformer case / may be any multiple of h since n can take on any value, while inthe latter case, only even multiples of / are allowed (since n1 = n2). This, howe-ver, has seemingly nothing to do with singular motions. Instead, Ehrenfest wants toemphasize that in the transformation of an anisotropic into an isotropic oscillator, adouble limiting process takes place: The angular momentum is constant in the finalmotion but not in the initial motion, and when the frequencies are equal to each other,the angular momentum will oscillate more and more slowly between the values

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2n1n2 h.

The final value of the angular momentum to which the isotropic oscillator settlesdown is between these two limits.

Box 1

Connections between the adiabatic hypothesis and various quantum hypotheses, asshown by Ehrenfest in 1916. The symbol [] means an adiabatic transformationOne dimension Planck harmonic oscillators Debye anharmonic oscillators

Diatomic molecules as harmonic oscillators Diatomicmolecules as (Ehrenfest) rotors

Two dimensions Elastic or Newtonian or Coulomb central fields (simple per-iodic motions): the adiabatic invariants coincide with Som-merfelds conditions

Any central field of the form (r) (extension to multiply per-iodic motions): each integral in (14) and (15) will be an adia-batic invariant in some coordinate system

Plancks anisotropic two-dimensional oscillator Sommer-felds isotropic two-dimensional oscillator (in the isotro-pic case, Plancks quantization condition does not coincideexactly with Sommerfelds conditions)

To Ehrenfest, problems like this reveal the necessity of complementing the adiaba-tic hypothesis to deduce, for example, the quantization conditions for arbitrary centralforces from Plancks quantum hypothesis. He also thinks that they are connected to thedifficulties that appear when applying concepts such as reversible adiabatic changeor adiabatic invariant to nonperiodic motions. In them the required slowness of thetransformation cannot be defined properly.

E. The adiabatic hypothesis and Boltzmanns principleEhrenfest also considers the relationship of his adiabatic hypothesis to Boltzmanns

principle. Boltzmanns statisticalmechanical deduction of the second law of thermo-dynamics was based upon his stipulation of what regions of (q, p)-space were equallyprobable.29 Ehrenfest had already explained in 1914 that Boltzmann had assigned auniform weight to the entire -space (the phase space of a molecule), but in quantumtreatments the weight function G loses that uniformity:

G (q, p, a) = const. (23)

Ehrenfest now asks how the weight function G(q, p, a), that is, the allowed regionsin phase space, must be restricted so that Boltzmanns principle remains valid.

For molecules with one degree of freedom (resonators vibrating harmonically oranharmonically), the problem can be solved completely:

29 NAVARRO and PREZ (2006, pp. 237252).

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An ensemble of such-like molecules (resonators) will fulfil Boltzmanns rela-tion between entropy and probability if, and only if, the allowed motions aredetermined by means of the adiabatically invariant condition:

2T

=

dqd p = fixed numerical values 1,2, . . .30

As we have seen, Plancks and Debyes quantum hypotheses satisfy this condition.However:

As yet I have not been able to tell if also for molecules of more than one degree offreedom the same necessary and sufficient relations hold between the adiabaticinvariants on the one hand and the fulfilment of Boltzmanns theorem on theother.31

Ehrenfest notes that already in 1913 he had shown that starting from the most pro-bable distribution of states and undertaking a reversible adiabatic process, anothermost probable state is obtained in some cases, regardless of whether there has beenan interaction between the molecules, for example, in a cavity filled with black-bodyradiation or in an ideal gas in the absence of an external field, in both of which casesthe pressure depends upon the total energy of the system and not on its distributionamong different degrees of freedom. Ehrenfest now declares without proof that thetransformation from one most probable state to another one holds for systems withone degree of freedom if and only if a relationship of the following type holds betweenthe energy and adiabatic invariant:

= A (a) 2T

+ B (a), (24)

an expression that does not appear in the Annalen and Philosophical Magazine versionsof his paper; however, he adds an explanatory note:

Imagine an ideal gas with rigid ellipsoidal molecules; the walls of the vesselare replaced by a field of force which reflects only the centre of gravity of themolecules (the reflexion is perfectly elastic); if the gas is compressed adiabati-cally, without collisions between the molecules taking place, the kinetic energyof the translatory motion is increased, but not the energy of the rotatory motion.If collisions between the molecules do take place, this is otherwise.32

In the first case, in a system whose components can mutually interact, an adiabaticcompression cannot transform a most probable distribution into another one. Headds another example:

Point molecules move up and down along a straight line between two fixed pointsA and B, uninfluenced by any force. An elastic field of force is excited infinitely

30 EHRENFEST (1917); in VAN DER WAERDEN (1968, p. 89), his emphasis.31 EHRENFEST (1917); in VAN DER WAERDEN (1968, p. 90), his emphasis.32 EHRENFEST (1917); in VAN DER WAERDEN (1968, p. 91), his emphasis, footnote *.

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slowly, so that in the end the molecules perform harmonic vibrations about thecentre of the line.

In each case, the work done in introducing the elastic force will depend upon theposition of the molecules so that the final distribution of their velocities will differfrom the most probable one.

In the Annalen version, Ehrenfest challenges Einsteins use of the adiabatichypothesis in the 1914 article in which he gave it its name:33

I believe that this last mentioned circumstance [that, in general, a most probabledistribution does not remain as such after a reversible adiabatic transformation]invalidates the deduction of Boltzmanns theorem of the entropy and the pro-bability for the case of general quantized systems that Einstein recently tried tooffer on the basis of the adiabatic hypothesis.34

Paradoxically, Einstein apparently helped Ehrenfest recognize the difference betweeninteracting and noninteracting systems, since Ehrenfest had not pointed out this dis-tinction prior to 1914.35

2.1 The adiabatic invariance of phase integrals

Ehrenfest added a postscript to his paper on 6 September 1916, between the date onwhich the Annalen received it on 22 July and its publication in October, during whichperiod Epsteins, Debyes, and Schwarzschilds contributions appeared, as follows:

The beautiful researches of Epstein, Schwarzschild, and others which haveappeared in the meantime, show the great importance [that] the cases whichare integrable by means of Stckels method of separation of the variableshave for the development of the theory of quanta. Hence the question arises:How far are the different parts into which these authors separate the integral ofaction according to Stckels method adiabatic invariants? In the problem treatedby Sommerfeld this is the case, as is shown in Sect. 7.36

(Ehrenfest also mentioned Debye in earlier versions of his paper in the Proceedingsand Annalen.)

Ehrenfest probably decided to write his 1916 paper following the publication ofSommerfelds conditions, since Sommerfeld seems not to have been familiar with hisresearch. In fact, 2 years earlier, Ehrenfest learned in his correspondence with Planckthat not even Planck had known of his 1913 paper on the BoltzmannClausiusSzilymechanical theorem.37 Sommerfeld seems to have learned about Ehrenfests adiabatic

33 EINSTEIN (1914).34 EHRENFEST (1916b, p. 343).35 NAVARRO and PREZ (2006, pp. 253258).36 EHRENFEST (1917); in VAN DER WAERDEN (1968, p. 93).37 Planck to Ehrenfest, 28 November 1914; in EA, microfilm AHQP/EHR-24, Sect. 7. EHRENFEST(1913).

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hypothesis from Bohr, who mentioned it in a letter of 19 March 1916,38 in which hecalled Sommerfelds attention to the relevance of adiabatic transformations for hisresearch and referred him to Ehrenfests 1913 paper.

Planck and Sommerfeld were then both deeply involved in the project of findinga more general quantum hypothesis that was not limited to one-dimensional systems.In 1916, when Ehrenfest wrote his paper, most of Plancks and Sommerfelds newcontributions had already been published.

Planck had proposed his second quantum theory in 1911, in which resonators couldhave any energy, but the emitted energy was quantized. The discontinuity in the emis-sion probability appeared in certain regions of phase space: It was nonzero at theborders of the elementary cells. In 1915 he developed a quantum theory devoted tosystems with several degrees of freedom.39

That same year Sommerfeld published his quantization conditions independentlyof Wilson and Ishiwara,40 who, however, obtained no new results with them. TheSommerfeld conditions are:

pi dqi = ni h (i = 1, 2, . . . , f ), (25)

where f is the number of degrees of freedom of the system, and the integral extendsover one complete period of pi (qi ). In polar coordinates (, r) they become

pd = nh and

pr dr = nr h, (26)

where is the angular coordinate, n = 1, 2, 3, . . . , and r is the radial coordinate,and nr = 0, 1, 2, . . . From these conditions, and by assuming that the electron in thehydrogen atom moves at a relativistic velocity and in elliptical orbits, Sommerfeldwas led to an interpretation of the fine structure of the hydrogen spectrum.

Ehrenfest wrote to Sommerfeld soon after these results were published, pointingout their close connection to the application of his adiabatic hypothesis to the spe-cific heat of hydrogen molecules.41 Ehrenfest noted that Sommerfelds quantizationof the electrons elliptical orbits was just an extension of his own hypothesis fromone-dimensional to two-dimensional systems. He suggested that Sommerfeld shouldclosely examine the quantization of momentum to determine the invariant followingan adiabatic transformation of the electron mass, an investigation that Ehrenfest saidhe had already begun himself.42 Sommerfeld was not enthusiastic about pursuingEhrenfests adiabatic hypothesis,43 although Bohrs praise of it awakened his inter-est; he did, however, encourage his student Epstein to give a talk on Ehrenfests

38 Bohr to Sommerfeld, 19 March 1916; in HOYER (1981, pp. 603604).39 PLANCK (1915, 1916).40 SOMMERFELD (1916a), WILSON (1915), and ISHIWARA (1915). I quote the version Sommerfeldsent to the Annalen in 1916, instead of the original, from 1915.41 Ehrenfest to Sommerfeld, April/May 1916; in SOMMERFELD (2000, pp. 555557).42 See Sect. 2C.43 Sommerfeld to Ehrenfest, 30 May 1916; in SOMMERFELD (2000, p. 561).

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and Burgerss adiabatic hypotheses44 at the Munich colloquium on 2 March 1917.In any case, Ehrenfest himself soon suggested that his adiabatic hypothesis mighthelp to determine which system of coordinates should be used for the application ofSommerfelds formulae.45

Throughout 1916, after Ehrenfest had published his paper, new developmentsimproved Sommerfelds theory, and the ambiguities that once threatened its functio-nality were almost overcome. In the postscript quoted above, Ehrenfest was referringto these developments.46

Epstein and Schwarzschild achieved very similar results by following differentpaths,47 using HamiltonJacobi theory to determine two essential conditions for thecorrect use of Sommerfelds formulae, namely, that the number of degrees of freedommust be finite, and that the HamiltonJacobi equation must be separable.48

If we perform a canonical transformation of a mechanical system with coordi-nates q1, . . ., qn and momenta p1, . . ., pn into a coordinate system in which all ofthe variables are constants of motion, its generating function S, called Hamiltonsprincipal function, is given by the HamiltonJacobi equation:

H(

q1, . . . , qn; Sq1

, . . . ,Sqn

)+ S

t= 0. (27)

If the Hamiltonian H does not depend explicitly on time, the principal function S canbe split up as follows:

S (qi , i , t) = W (qi , i ) 1t, (28)where 1 is the energy, the remaining i are independent constants of motion, andHamiltons characteristic function W does not depend upon time t . The time-independent HamiltonJacobi equation, according to (27), then is:

H(

qi ,Wqi

)= 1. (29)

In those systems in which the variables can be separated, the problem can be greatlysimplified. In general, a coordinate qi and its conjugate momentum pi can be separatedas a function f (qi , pi ) that contains no other variables. A system of n dimensions canbe completely separable only if the principal function can be expressed as

S =

iSi (qi ;1, . . . , n; t),

44 Register volume for Mnchener physikalisches Mittwochs-Colloquium; in AHQP, microf. AHQP-20.45 EHRENFEST (1917); in VAN DER WAERDEN (1968, p. 88).46 See footnote 36.47 EPSTEIN (1916a,b) and SCHWARZSCHILD (1916).48 What follows is based mostly on BERGIA and NAVARRO (2000, pp. 325332). For a more detaileddiscussion see BORN (1925). For a contemporary discussion of HamiltonJacobi theory see GOLDSTEINet al. (2002).

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Fig. 2 Projection of a multiplyperiodic motion in the(qi , pi )-plane, where these twocoordinates describe a periodictrajectory

pi

qii i

so that there are n HamiltonJacobi equations:

Hi(

qi ; Siqi

;1, . . . , n)

+ Sit

= 0.

Any system with an arbitrary number of degrees of freedom in which the HamiltonJacobi equation is separable can have multiply periodic motions, so that while theoverall motion may not be periodic, each coordinate doublet (qi (t), pi (t)) describesa closed trajectory on the corresponding (qi , pi 4)-plane; in other words, each pi is afunction only of its corresponding qi (see Fig. 2):

pi = Fi (qi , 1, . . . , n) . (30)

Epstein and Schwarzschild pointed out that Sommerfelds conditions (25) onlyapplied when the variables in the HamiltonJacobi equation are separable. Still, insome cases the HamiltonJacobi equation can be separated in more than one systemof coordinates, namely, in so-called degenerate systems in which there is a relationof the following type,

li=1

mi i = 0, (31)

between the characteristic frequencies i of the system (mi are integers). If the fre-quencies are incommensurable, that is, if no such relation exists, the separation ofvariables is unambiguous (it can only be performed in one system of coordinates) andthe system is nondegenerate. But if there are k-independent relations of the type (31)between the frequencies of the different couples (qi , pi ), the system has a degene-racy of order k, and a suitable system of coordinates in which to apply Sommerfeldsconditions cannot be determined unambiguously.

Schwarzschild introduced the action-angle variables (Ji , i ), with

Ji =

pi dqi (i = 1, 2, . . .) (32)

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and

i = W Ji

(i = 1, 2, . . .), (33)

where, if the variables are separable,

W =i=1

Wi (qi , J1, J2, . . . , Jn) (34)

is the characteristic function that transforms the variables (p, q) into the variables(J, ). The Hamiltonian H then depends only upon the action variables, or

H = H (J1, J2, . . . , Jn) = E, (35)

where E is the energy. Integrating the equations of motion then yields

i = i t + i , (36)

where t is the time and the i are constants determined by the initial conditions. Theaction variables are then quantized according to Ji = ni h. For degenerate systemsa canonical transformation can be made to other action-angle variables with fewerfrequencies; in the Kepler problem one action variable and one angular variable aresufficient (see Fig. 3).

Epstein and Schwarzschild thus generalized the Sommerfeld conditions, and inthat way explained the influence of an external electric field on atomic spectra (theStark effect). Sommerfeld incorporated their contributions into his work, publishingan extensive paper on atomic spectra in the Annalen in 1916.49 That same year he alsoapplied them to the normal Zeeman effect.50

2.1.1 Burgerss trilogy

Soon after Ehrenfest had written his paper, he realized it was necessary to study thepossible invariance of the phase integrals in adiabatic processes. He assigned this taskto his student Burgers in the summer of 1916, and Lorentz presented his initial resultsin a paper entitled Adiabatic invariants of mechanical systems. I51 to AmsterdamAcademy that November.

This was the first part of a trilogy whose second and third parts were presented on 21December 1916 and on 27 January 1917.52 Burgerss results provided great insight intothe adiabatic hypothesis and quantization rules. As he recalled in his autobiographicalnotes:

49 SOMMERFELD (1916a).50 SOMMERFELD (1916b).51 BURGERS (1917a).52 BURGERS (1917b,c).

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Fig. 3 Keplerian orbits for a relativistic electron. A central force causes motion in a plane, which inspherical coordinates produces a degenerate motion, since the periods of both the angular and azimuthalvariables are identical. If the relativistic correction (which adds a term proportional to r3 to the force)is not taken into account, the motion will be completely degenerate (simply periodic), in which case theorbit will be closed, since the periods of the radial coordinate r and the angular coordinate are identical.In the relativistic case, a canonical transformation can be made to new variables in which the motion isnondegenerate (as described by two pairs of action-angle variables). If a magnetic field is introduced, thedegeneracy will disappear completely (the plane of the orbit precesses) and three pairs of action-anglevariables are required to describe the motion. Schwarzschild showed that quantization must be applied, inevery case, to the action variables whose frequencies are incommensurable

After having given a proof for [the] general case without degeneration, I couldshow that in the degenerate case the remaining independent phase-integralsstill were invariants. Later I constructed a new proof with the aid of the transfor-mation to action and angular variables, as used by Schwarzschild, and treated inE.T. Whittakers Analytical Dynamics.53

Burgerss first task was to demonstrate the invariance of phase integrals in thenondegenerate case. He considered systems with n degrees of freedom in which thecoordinates q and p did not increase indefinitely but remained between certain extremevalues (Assumption A). The Hamiltonian should depend on these coordinates and onsome parameters a that experience infinitely slow variations. He limited his conside-rations to multiply periodic motions (Assumption B), that is, to systems for which

pk =

Fk(qk, 1, . . . , n, a

), (37)

where the m are integration constants, an expression that is equivalent to Eq. 30 if(Fk)

2 = Fk . He assumed that the function Fk has at least two roots, k and k , andthat at a certain instant qk lies between k and k , which agrees with Assumption A,and prescribes that qk describes a periodic trajectory (see Fig. 2). The phase integralsthus are:

53 J. M. Burgers, Autobiographical Notes; in AHQP, microf. AHQP/OHI-1.

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Ik(1, . . . , n, a

)=

dqk pk = 2

k

k

dqk

Fk (qk). (38)

Burgerss goal is to calculate the variation of the integral Ik when the system undergoesan adiabatic process. He concluded, following a difficult calculation, that if no relationof the type

j

m j j1 = 0 (39)

is satisfied, where j = 1, . . . , n and where the are what he calls the mean motions,which in a footnote he relates to Schwarzschilds angle variables, then:

Ik = 0. (40)Burgers published an abridged version of his paper in the Annalen and a transla-

tion of it in the Philosophical Magazine,54 where it immediately followed Ehrenfestspaper on adiabatic invariants. In these widely read abridged versions, Burgers men-tioned some calculations and proofs that he already had developed in his paper in theProceedings of the Amsterdam Academy, but he also presented some conclusions thathe would publish in his subsequent papers in the Proceedings that would deal withthe invariants of a motion and a more general treatment of the problem using action-angle variables. In particular, in his Philosophical Magazine version, Burgers addeda comment in which he questioned the validity of his results,55 which he included inthe third part of his trilogy.

In the second part, Burgers loosened an earlier restriction that he had called Assump-tion C, which was responsible for vetoing the presence of commensurability relations[his (39) and our (31), which are, of course, equivalent]. He now demonstrated thatin adiabatic processes in which there are such relations but which remain unchanged,that is, in processes in which no new relations of the type in (39) appear or disappear,certain linear combinations of Ik are also adiabatic invariants. In strictly periodic andtherefore completely degenerate motions, the only invariant is the sum of all of thephase integrals. Ehrenfest had already reached this conclusion in certain cases in 1916,and Bohr, as we will see, later used it in dealing with degenerate systems.

Burgers devoted the third part of his trilogy on adiabatic invariants to a reformulationof Schwarzschilds results on action-angle variables. Schwarzschild had establishedthat:56

2

0

Jk dk = 2 Jk = nkh + const. (41)

54 BURGERS (1917d,e).55 BURGERS (1917d, p. 520).56 Burgers uses a different notation. The paper is BURGERS (1917c).

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Action-angle variables can always be used if the variables are separable, but thereverse is not true. Burgers now showed that separability is unnecessary to selectthe variables as adiabatic invariants, so he broadened Schwarzschilds and Epsteinsearlier results. Burgers concluded:

In the present paper it has been supposed that the mean motions i are allincommensurable. The i are, however, functions of the parameters. Hence ifthe a are varied, the i change too, and their ratios pass through rational values.It has still to be investigated, whether this may give rise to difficulties. (Thisapplies also to the demonstrations given in the preceding papers).57

Burgers thus questions the validity of his results: he does not demonstrate if the factthat the i vary under the effect of the transformations and thus they pass throughstates in which they satisfy relations of commensurability invalidates his results.

2.1.2 The developments by Krutkow and Kramers

Burgerss papers were not the only attempts to develop Ehrenfests hypothesis. Howe-ver, they were the deepest and most successful. When Bohr popularized the adiabatichypothesis, Burgerss work on it also became widely known, since it showed thatEhrenfests adiabatic hypothesis was compatible with the quantization rules.

Two other former students of Ehrenfest, Krutkow and Kramers, tried to consolidatethe role of adiabatic transformations in the quantum theory. Krutkow only knew a littleabout Burgerss work, and did not go beyond it. Kramers only wrote an unpublishedmanuscript on the adiabatic hypothesis that he probably sent to Ehrenfest. I now brieflyturn to these two attempts to extend Ehrenfests adiabatic hypothesis in 1916.

A month before Burgers completed his doctoral thesis, on 2 October 1918, hissupervisor Ehrenfest wrote to him that:

Today I have received a long letter from Krutkow in St. Petersburg dated 14September 1918 (!!!). They are often hungry and cold, beyond desperation,but all my friends are still alive and they are working and even publishing (!).Krutkow, for instance, concerning your work about the adiab.[atic] invariants.My colloquium is still working, always lively, though people do not know ifthey will survive the following winter.58

A few weeks later, Ehrenfest received a manuscript from Krutkow entitled Contribu-tion to the theory of adiabatic invariants,59 which Lorentz subsequently presented tothe Amsterdam Academy. Krutkows goal was to find a general procedure for deter-mining the adiabatic invariants of a system. He argued that a quantity had to satisfythree conditions to be quantized: (1) it had to be a function of the integrals of motion;(2) it had to be an adiabatic invariant, and (3) its meaning had to be independent ofthe chosen system of coordinates. Regarding condition (2), he declared that:

57 BURGERS (1917c, p. 169).58 Ehrenfest to Burgers, 2 October 1918; in AHQP, microf. AHQP-75. For prior collaborations betweenEhrenfest and Krutkow, see NAVARRO and PREZ (2004, pp. 130132).59 KRUTKOW (1919).

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Assuming that the adiabatic influence may be calculated by the methods ofmechanics this condition follows directly from the fact, that the quantum-quantityvaries abruptly, whereas the external influence may be infinitely small; the quan-tizable quantity therefore cannot vary at all, it must be an adiabatic invariant. . .

This condition imposes on us the task to find the adiabatic invariants ofa given mechanical system and to look for a general method of solving theadiabatic problem: A method of that kind was unknown so far; the adiabaticinvariants had to be guessed at and their adiabatic invariability had to be testeda posteriori.60

Ehrenfest had not developed such a general method, but Krutkows results did not gobeyond Burgerss. Indeed, as Ehrenfest wrote to Burgers on 13 December 1918, ifKrutkow had known about the third part of Burgerss trilogy, he probably would nothave sent his manuscript to him.61 Still, Ehrenfest decided to publish it anyway becauseof its great human value, the desperate conditions under which Krutkow wrote it.

Kramers left Leiden for Copenhagen in the summer of 1916 to become Bohrs firstassistant. Although Kramers applied the adiabatic hypothesis in a much later paper onthe helium atom, in 1923,62 he published nothing on it earlier. He did, however, writea 21-page manuscript entitled On the adiabatic changes of mechanical systems,63probably during the summer of 1917, soon after Burgers papers had appeared, sincea copy of it has been recently found in the Ehrenfest Professional Library in Leiden,64indicating that he sent it to his former teacher Ehrenfest.

Kramerss goal was to extend the range of Burgerss proof of the adiabatic invarianceof phase integrals. He first characterizes multiply periodic motions in cases in whichthere is degeneracy by means of action-angle variables, and later proves that anadiabatic change doesnt alter the mean values of the Js,65 claiming that he wasfollowing Bohrs work of 1916, presumably meaning aspects of it which later appearedin Bohrs 1918 paper, On the Quantum Theory of Line Spectra. In any case, Kramersconsiders a multiply periodic motion that undergoes an adiabatic transformation andcalculates the change in the action variable, that is, in a phase integral, in the sameway that Burgers had.

Kramers then makes an original contribution. He does not suppose that a uniquecoordinate system exists in which the Hamiltonian is separable during the transforma-tion. He proves that the values of the phase-integrals in a cond.[itionally] per.[iodic]system are equal to the mean values of the phase integrals which belong to a neigh-bouring cond. per. system.66 He designates

Ji

60 KRUTKOW (1919, pp. 11121113).61 Ehrenfest to Burgers, 13 December 1918; in AHQP, microf. AHQP-75.62 KRAMERS (1923).63 Notes on adiabatic invariance; in AHQP, microf. AHQP-27, Sect. 10.64 Manuscript, On the adiabatic invariants of mechanical systems; in HPE, document EB06.65 His emphasis.66 His emphasis.

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as the average variation of Ji during a change in coordinates and concludes that it iszero except in the usual degenerate cases.

At the end of his manuscript, Kramers assessesbut does not provethe validityof his results assuming a rigorous relativistic treatment of them. In a corollary to hisfirst proof, he also shows that a small adiabatic change that goes to zero leaves thephase integrals unchanged, even if the motion ceases to be multiply periodic duringthe change, since the variables still could be separable. He emphasizes the importanceof this property, arguing that it justifies the procedures followed by others who hadassumed that the motions were multiply periodic and had dropped some terms in theircalculations. Kramers thus also attempted to strengthen the validity of Ehrenfestsadiabatic hypothesis.

I summarize the contributions of Burgers, Krutkow, and Kramers in Box 2.

Box 2

Contributions to the development and improvement of the adiabatic hypothesis bet-ween 1916 and 1918

BURGERS (19161917) Demonstrates the adiabatic invariance of thephase integrals of nondegenerate multiply periodicmotions (Ik) and of the action variables (Jk)

Finds adiabatic invariants for degenerate motionsin which the number of commensurability relationsdoes not change during the transformation

Discusses the validity of the entire treatment, sincehe cannot prove that the inevitable passage throughmotions with commensurability relations during thetransformations is unproblematic

KRUTKOW (1918) Proposes a method to determine the adiabatic inva-riants of a system and particularizes it to multiplyperiodic, cyclic, and ergodic cases; he does not studydegeneracy

KRAMERS(1917, unpublished)

Extends the adiabatic invariance of the actionvariables (Jk) to processes in which there is a changein the coordinate system in which the variables areseparable

Extends the invariance of the phase integrals(without proof) to relativistic systems

2.2 Reception, before Bohr

I now turn to the reception of Ehrenfests adiabatic hypothesis and of Burgerss contri-butions to it prior to Bohrs publication of his paper, On the Quantum Theory of lineSpectra, in 1918.

I begin with a brief description of two papers of 1918 by a young Viennese physicist,Smekal, exceptional case, since he was the only physicist who noted the relevance of

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the adiabatic hypothesis for the foundations of quantum statistics. His two papers hadno or little perceptible influence at the time, but in 1925 he published a long article inthe Encyklopdie der Mathematischen Wissenschaften entitled General Foundationsof Quantum Statistics and Quantum Theory,67 in which he again used the adiabatichypothesis. That article, however, is beyond the scope of my paper.

2.2.1 Smekal on the validity of Boltzmanns principle

Smekals first paper, On the most probable distribution applied to the proof ofBoltzmanns principle,68 appeared in the Physikalische Zeitschrift in January 1918.Smekal thanks Ehrenfest for his critical comments on the manuscript, but we do notknow what they were. We do know, however, that Ehrenfest wrote to Bohr that summer,saying:

I could not understand what Smekal was trying to do, but that does not provemuch, since I am often a little hard-headed when it comes to things that affectme closely (otherwise, he is a young man who has moved from Graz to Berlin:towards the source of wisdom. I only wish to say: let Kramers report to you onSmekals work before mentioning it with approval).69

It seems that Ehrenfest never again referred to Smekals work. Smekal, however, heldEhrenfest and his work in the highest esteem.70 In 1922, Smekal, now a professorin Vienna, published a paper on the quantization of nonperiodic motions in whichEhrenfests adiabatic hypothesis (and Bohrs correspondence principle) played crucialroles.71

In his paper of 1918, Smekal succinctly summarizes Ehrenfests ruminations of1914 on the validity of Boltzmanns principle, highlighting the problematic normali-zation condition of the weight function G, namely,

G (q, p, a) d = 1, (42)

where d = dq1 dqnd p1 d pn . This integral, extended over all phase space,diverges both in Boltzmanns analysis (uniformG) and in subsequent quantum ana-lyses. Smekal pointed out a possible source of this problem: the distribution of afinite number of components of the system over an infinite number of finite regionsin phase space. He did not consider this condition (which in principle is implicit inconsidering Gd as an a priori probability) to be indispensable, so he proposed analternative, more physically significant one. (Ehrenfest himself was not particularlyworried about this problem, although in the last paragraph of his 1914 paper he offe-

67 SMEKAL (1925).68 SMEKAL (1918a).69 Ehrenfest to Bohr, 14 August 1918; in AHQP, microf. AHQP/BSC-2, Sect. 1.70 SMEKAL (1925).71 SMEKAL (1922).

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red a formulation that avoided it.)72 Smekal also related Ehrenfests G-condition toGibbss statistical mechanics,73 showing that fulfilling this condition constitutes thesufficient and necessary requirement for Gibbss definition of entropy in the generali-zed canonical ensemble.

One month later, in February 1918, Smekal submitted a second paper, The adia-batic hypothesis and Boltzmanns principle, to the Physikalische Zeitschrift.74 Henow attempted to strengthen the link between Ehrenfests adiabatic hypothesis andhis G-condition of 1916. Ehrenfest had proved that for one-dimensional systemsthe dependence of the quantization rules on adiabatic invariants is a necessary andsufficient condition for the statistical formulation of the second law of thermodyna-mics, and Smekal now proposed to extend Ehrenfests proof of the sufficiency of thiscondition to systems with an arbitrary number of degrees of freedomhe mistakenlyadded necessity, which he subsequently corrected75 by imposing the additionalcondition that the weight function G is zero at the walls of the gas container.

Smekal proved that the adiabatic invariance of the quantities on which the weightfunction G depends is sufficient to ensure the applicability of Boltzmanns principle toa system with a finite volume and a finite number of degrees of freedom. He presentedEhrenfests proof of 1914 (which Ehrenfest did not publish but provided to Smekalprivately) of which is the most general weight function that does not threaten thevalidity of Boltzmanns principle if the variables q and p appear in its argument in theform (q, p). Smekal concluded:

If, as the adiabatic hypothesis demands, the quantum rules are formulated on thebasis of adiabatically invariant quantities, then the quantum weights p [are a]function of the elementary regions, that is, only of the quantum numbers that puteach one of those elementary regions in order, [and] will also be adiabaticallyinvariant.76

That is, if the weight function depends upon adiabatic invariants and the quantizationrules therefore are formulated by starting from adiabatically invariant quantities, thenthe statistical foundation of the second law is not threatened. Smekal concluded:

[The adiabatic hypothesis g]uarantees the compatibility of the extension of thequantum theory to systems with several degrees of freedom with the secondlaw of thermodynamics in the form that this law adopts through Boltzmannsprinciple.77

Smekals papers seem to have had no or little influence, even though they (particu-larly the second one) clarified the pertinence of Ehrenfests adiabatic hypothesis toBoltzmanns principle. Bohr sent Smekal a copy of the first part of his 1918 paper,

72 EHRENFEST (1914); in KLEIN (1959), p. 662.73 GIBBS (1902).74 SMEKAL (1918b).75 See correction in Physikalische Zeitschrift 19 (1918), p. 200.76 SMEKAL (1918b, p. 141).77 SMEKAL (1918b, p. 142).

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On the Quantum Theory of Line Spectra, to which Smekal replied, offering Bohrsome observations,78 and insisting, for instance, on the incorrectness of consideringthe weight function Gd as an a priori probability: Its integral cannot be extendedover infinite spacea point that he had emphasized in his second paper.

2.2.2 Planck on the asymmetric spinning top

In 1914 Planck asked Ehrenfest for information on his most general calculation of theheat capacity of a diatomic gas.79 We do not know Ehrenfests answer, but whateverit was, it did not greatly affect Plancks research, since in his next paper on the theoryof rotation spectra he did not cite either Ehrenfest or his adiabatic hypothesis.80

Four years later, however, Planck did refer to Ehrenfests adiabatic hypothesis ina paper on the quantization of the motion of an asymmetric spinning top, that is,on the rotational motion of a rigid body with one fixed point and three moments ofinertia.81 He was attracted to this problem because although a complete integrationof the equations of motion was possible, finding a solution based upon separation ofvariables was not, at least not in the general case. In 1905, Kolossoff had introduceda restriction that enabled him to use separation of variables, and in 1918 Reiche hadgone a step further by applying the most suitable quantization, which set one of thethree quantum numbers to zero.82 Planck now felt that Reiches step was by no meanssatisfactory.

Planck applied a systematic method of quantization that did not require the impo-sition of restrictions at the outset, by generalizing the quantum hypothesis based uponthe structure of phase space. The distinguishable motions of a system (no motionswere prohibited in Plancks second quantum theory) would be fixed by certain sur-faces that delimit the elementary regions of equal probability. Planck saw that theresult in the case of the asymmetric spinning top is ambiguous and hence that he hadto discriminate among the different quantizations by resorting to Ehrenfests adia-batic hypothesis to privilege the adiabatically invariant quantum conditions. CitingEhrenfests and Burgerss papers of 19161917, Planck formulated the hypothesis asfollows:

According to this hypothesis, the quantum functions . . . remain invariant underany infinitely slow reversible adiabatic influence on the system.83

In the case of the spinning top he considered its three moments of inertia to undergoinfinitely slow adiabatic changes and obtained equations of motion with which he veri-fied that Reiches quantization, which selected out slow transformations, was correct.This is the only application that Planck made of Ehrenfests adiabatic hypothesis. It was

78 Smekal to Bohr, 11 May 1918; in AHQP, microf. AHQP/BSC-7, Sect. 2.79 Planck to Ehrenfest, 28 December 1914; in EA, microfilm AHQP/EHR-24, Sect. 7.80 PLANCK (1917).81 PLANCK (1918).82 REICHE (1918).83 PLANCK (1918); in PLANCK (1958, vol. 2, p. 493).

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not as fundamental as Ehrenfest sought, but it did serve as a pioneering contribution tothe conviction that among all possible quantizations, those that cannot be connectedadiabatically to known ones must be discarded.

2.2.3 Sommerfeld on light dispersion

On 16 November 1916, Sommerfeld wrote to Ehrenfest,84 saying that after readingEhrenfests 1916 paper, he found that it contained nothing that he himself had not saidalready in his own 1916 paper in the Annalen.85 He agreed that Ehrenfests adiabatichypothesis was closely related to Wiens displacement law:

The reference to Wien seems to me of much weight too, and requires comparingthe quantization required with the adiabatic hypothesis.86

Furthermore:

Which quantization is the required one can and must be shown by the spectrallines; only they have the necessary clarity and certainty.

In cases such as the incompatibility of isotropic and anisotropic oscillators, wherespectral lines have nothing to say, or in elucidating the Zeeman splitting, Ehrenfestsadiabatic hypothesis might provide insight.

Almost a year later, in October 1917, Sommerfeld again wrote to Ehrenfest, infor-ming him of the imminent publication of a new paper, Drudes theory of dispersionfrom the point of view of Bohrs model, and the constitution of H2, O2 and N2, inthe Annalen, and encouraging him to comment on its proofs, which he would send tohim shortly.87 In this paper, which built upon an earlier version of 1915, Sommerfeldused Ehrenfests adiabatic hypothesis in an attempt to find a theory of dispersion.88

The phenomenon of light dispersion was notorious for its inability to obey the quan-tum postulates.89 The classical theory of dispersion of Lorentz and Drude was builtupon Maxwells electrodynamics. The Bohr model of the hydrogen atom fundamen-tally challenged the classical theory, since the frequency of the emitted or absorbedradiation now was unrelated to the orbital frequency of the electrons. In 1915, Debyeand Sommerfeld, independently, and in 1916 Davisson made the first attempts toformulate a quantum theory of dispersion and in 1917 Sommerfeld returned to theproblem, attempting to justify his theory with the help of Ehrenfests adiabatic hypo-thesis.

In the first of three sections in his paper, Sommerfeld discusses the dispersion oflight by a Bohr gas molecule. In its second section, he treats the magnetic rotation of theplane of polarization of light in a magnetic field, a phenomenon related to the Zeeman

84 Sommerfeld to Ehrenfest, 16 November 1916; in SOMMERFELD (2000, pp. 571573).85 He was referring, I suppose, to SOMMERFELD (1916a).86 Sommerfeld to Ehrenfest, 16 November 1916; in SOMMERFELD (2000, pp. 571572).87 Sommerfeld to Ehrenfest, 10 October 1917; in SOMMERFELD (2000, pp. 576578).88 SOMMERFELD (1917). The earlier version is SOMMERFELD (1915).89 For a good historical analysis of the theory of light dispersion, see DUNCAN and JANSSEN (2007,pp. 571597).

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effect. In the third section, he focuses on the constitution of hydrogen, nitrogen, andoxygen molecules. Ehrenfests adiabatic hypothesis appears in the second section, butSommerfelds treatment of a magnetic field throughout his paper is clearly based uponit as well.

The rotational motion of atomic electrons can be inferred from the dispersion andpolarization of light. Sommerfeld notes, however, that a dilemma arises when attemp-ting to quantize the rotational motion of an electron in a magnetic field: On the onehand, the quantization rules can be applied to the motion of an electron in the absenceof a magnetic field and thereby calculate its orbits, which then will be unquantized,or at least will not satisfy the same quantum rules that described its initial motion.On the other hand, the quantization rules can be applied to the electrons initial andfinal motions, in which case the ordinary laws of mechanics and electrodynamics nolonger describe the changes in its orbit when a magnetic field is applied. Related tothis dilemma is the apparent contradiction between describing spectra and dispersion.The former indicates that atoms radiate only when electrons jump between allowedorbits, while the latter indicates that atoms radiate when the electrons orbits change inresponse to the passage of light waves. If the latter is the case, Sommerfeld declares,then we can calculate these changes using Ehrenfests adiabatic hypothesis, since theinfluence of light waves on the rotational motions of the electrons can be consideredas being infinitely slow.90

Ehrenfests adiabatic hypothesis also suggests how to quantize the rotation of anelectron. In the absence of a magnetic field, its angular momentum,

p = ma2, (43)

where m is its mass, a the radius of its orbit (assumed to be circular), and itsrotational frequency, is quantized, but in the presence of a magnetic field its canonicalmomentum,

p = ma2 + e2c a2 Hz,

where e is its charge, c the velocity of light, and Hz the magnetic-field intensity per-pendicular to the plane of its orbit, is quantized. This generalization of the momentumstems from associating a vector potential with the magnetic field and calculating, inthe usual way,

p = Lq

,

where L is the Lagrangian of the system. In Ehrenfests terminology, this momentum isan adiabatic invariant and, Sommerfeld concludes, is the one that must be quantized.

In general, Sommerfeld declares:

90 SOMMERFELD (1917); in SOMMERFELD (1968, vol. 3, p. 383), his emphasis.

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Ehrenfest . . ., with his adiabatic hypothesis, has established an interesting rulewhich correctly quantized motions must fulfil, and Burgers has proved that forperiodic motions that up to now could only be treated conditionally, it gives agood result. The rule says: when varying infinitely slowly the parameters of aquantum motion allowed by the external conditions (establishing fields, etc.) anorbit is produced according to the laws of mechanics, which again is quantumallowed.91

Sommerfeld had asked Ehrenfest in his letter of 10 October 1917 if it was legitimateto apply his adiabatic hypothesis to Debyes concept of the formation of the hydrogenmolecule, namely, by bringing two hydrogen atoms together, each of whose electronshas angular momentum

p = h2

,

in an infinitely slow way to produce a hydrogen molecule whose electrons have angularmomentum 2p. Ehrenfest either answered affirmatively, or Sommerfeld convincedhimself that this indeed was a legitimate use of Ehrenfests adiabatic hypothesis,since he used it in his 1917 paper on dispersion.92 In fact, in an appendix added inits proofs, he discarded an empirical law that he had just established concerning theangular momentum of a ring of more than two electrons because it was incompatiblewith Ehrenfests adiabatic hypothesis.

Sommerfeld thus fully supported Ehrenfests adiabatic hypothesis at the end of1917 and even tried to strengthen it by extending its applicability to dispersion. Heobjected only to its name as he told Ehrenfest privately, unaware that Einstein hadsuggested it. The hypothesis of reversibility would have been more suitable, becauseit emphasizes the slowness of the transformations, or the hypothesis of reconciliation,because it provides the basis for making peace between the quantum and classicaltheories. As he wrote in the introduction to his paper:

From here we can arrive at the pleasing perspective that the contradiction between[classical] mechanics and quantum theory is not as sharp as it seemed until now:in one correct application of the quantum theory . . . the mechanical foundationsremain effective during the transit of the molecule from the initial state to thefinal state . . . Ehrenfests adiabatic hypothesis indicates the direction in which[classical] mechanics and quantum theory can meet again.93

Sommerfeld thus saw Ehrenfests adiabatic hypothesis as a bridge between the classicaland quantum theories. Bohr, on the contrary used it in 1918 as a means for guaranteeingthe stability of the quantum-mechanical stationary states in the atom.

91 SOMMERFELD (1917); in SOMMERFELD (1968, vol. 3, p. 407).92 SOMMERFELD (1917); in SOMMERFELD (1968, vol. 3, p. 428).93 SOMMERFELD (1917); in SOMMERFELD (1968, vol. 3, p. 383).

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3 Bohrs principle of mechanical transformability

In the first letter that Bohr wrote to Ehrenfest, on 5 May 1918, he enclosed a copy ofhis 1918 paper, explaining that:

As you will see, the considerations are to a large extent based on your importantprinciple of adiabatic invariance. As far as I understand, however, I considerthe problem from a point of view which differs somewhat from yours, and I havetherefore not made use of the same terminology as in your original papers. Inmy opinion the condition of the continuous transformability of the motion inthe stationary states may be considered as a direct consequence of the necessarystability of these states and to my eyes the main problem consists therefore in thejustification of the application of ordinary mechanics in calculating the effectof a continuous transformation of the system. As it appears to me it is hardlypossible to base this justification entirely on thermodynamical considerations,but it seems naturally suggested from the general agreement with experimentsobtained by calculating the motion in the stationary states themselves by meansof ordinary mechanics. I have endeavoured to show how from this point of viewthe characteristic exceptions from the principle in question seems [to] presentthemselves in a clearer light.94

Bohrs thermodynamical considerations allude to the term adiabatic, but contraryto what Bohr thought, its significance was not purely thermodynamic, as Ehrenfestexplained in his reply on 14 August,95 pointing out that its meaning is close to whatBoltzmann and Helmholtz meant in their writings on mechanics. Ehrenfest also objec-ted to Bohrs expression mechanical transformation, since that seemed to permitconnections between all kinds of motions. Moreover, if a quantum system in equili-brium, with a most probable distribution of states A, is transformed into a state B,that state in general is no longer an equilibrium state, since it is no longer the mostprobable state, because the components of the system have to interact to establish thenew equilibrium state. To Ehrenfest, adiabatic processes must describe the systemas a whole, that is, they must include such interactions. Thus, Ehrenfest conceived themeaning of adiabatic to include not only thermodynamic and mechanical processes,but statistical ones as well.

Bohrs response is unknown; only years later did he use the expression adiabaticprinciple, which was close to Ehrenfests original one.96 However, that surely wasmore of a reaction to the increasing problems that arose when trying to account forspectra in the early 1920s than reflection on Ehrenfests reasoning.

In any case, Bohrs paper of 1918 was preceded by an unpublished paper in early1916 on periodic motions in which he used some of the results of Ehrenfests quantum-theoretical research.

94 Bohr to Ehrenfest, 5 May 1918; in EA, microfilm AHQP/EHR-17, Sect. 5.95 Ehrenfest to Bohr, 14 August 1918; in AHQP, microf. AHQP/BSC-2, Sect. 1. Because he was ill,Ehrenfest did not reply immediately.96 See footnote 136.

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3.1 The unpublished theory of 1916

Bohr published his atomic model in 1913 in three successive installments in the Phi-losophical Magazine,97 and in the following years he applied it to some specificproblems, such as the Stark and Zeeman effects and relativistic corrections to hiscalculations.98 By the end of 1915 he was searching for a more general point of view:

I have thought quite a bit about the general foundation of the quantum theory;but I have been so busy with lectures during this autumn that I have not been ableto complete anything, but now I am looking forward to the Christmas holidayfor trying to better collect my thoughts.99

Around March 1916 Bohr sent a paper entitled On the application of the quantumtheory to periodic systems to the Philosophical Magazine,100 but prior to its publi-cation Sommerfelds 1916 paper arrived on his desk. Bohr tried to take Sommerfeldsnew results into account, as he later told Sommerfeld:

But when I arrived at the statistical question, discussed in the second part, I couldnot work with a satisfactory interpretation for me. In the first place, because, natu-rally, I had not the rational foundations that furnish the works from Ehrenfest.101

Bohrs paper, which was eventually published in 1921, shows that he was familiar withEhrenfests quantum-theoretical research (except for his 1914 paper on Boltzmannsprinciple, as indicated in the above quotation).102 Thus, when he acknowledged thereceipt of Sommerfelds 1916 paper on 19 March 1916, he told Sommerfeld that:

I have myself been working a good deal with quantum theory in this winterand had just finished a paper for publication in which I had attempted to showthat it was possible to give the theory a logically consistent form covering allthe different kinds of applications. In this I had made largely use of Ehrenfestsidea about adiabatic transformations which seems to me very important andfundamental, and I had discussed a great number of different phenomena, alsothe dispersion.103

Bohr had incorporated Ehrenfests idea into his unpublished paper after first explai-ning:

That an atomic system can exist permanently only in certain series of statescorresponding with a discontinuous series of values for its energy, and thatany change of the energy of the system including absorption and emission of

97 BOHR (1913).98 BOHR (1914, 1915).99 Bohr to Oseen, 20 December 1915; English translation in HOYER (1981, p. 568).100 BOHR (1916).101 Bohr to Sommerfeld, 19 December 1919; in SOMMERFELD (2004, pp. 6970).102 It appeared in BOHR (1921).103 Bohr to Sommerfeld, 19 March 1916; in HOYER (1981, p. 604).

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electromagnetic radiation must take place by a transition between two such states.These states are termed the stationary states of the system.104

They embodied three different aspects:

(1) As to the conditions to be fulfilled in the stationary states, (2) as to thenature of the radiation emitted or absorbed by the transitions between differentstationary states, and (3) as to the probability of the various stationary states ina distribution of statistical equilibrium.105

Adiabatic transformations connect the stationary states, which differ only in the valueof certain parameters, such as those of an applied external field. Bohr generalizesPlancks expression,

E = nh, (44)

to

T

=

T dt = 12

nh (45)

for the periodic motion of a single particle, where T is its mean kinetic energy and its frequency. To guarantee the existence of stationary states, he appeals to the relation

W = 2(

T

), (46)

where W is the total energy and denotes a small variation, which implies that statesdefined by the same integer n that satisfy Eq. 45 have the same total energy W . In otherwords, the ratio T / is constant during small changes in the motion if the total energyis constant; otherwise, radiation would be emitted and absorbed in the absence oftransitions between stationary states. Notice the similarity of Bohrs reasoning to thatof Krutkows: the adiabatic hypothesis is converted into an inescapable requirementof the quantum theory.106

Bohr emphasized that Ehrenfest stressed the crucial importance of the invariantT /, which

allows us by varying the external conditions to obtain a continuous transforma-tion through possible states from a stationary state of any periodic system to thestate corresponding with the same value of n of any other such system containingthe same number of moving particles.107

104 BOHR (1916); in HOYER (1981, p. 434).105 BOHR (1916); in HOYER (1981, p. 434).106 See footnote 60.107 BOHR (1916); in HOYER (1981, p. 436).

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Note that Bohrs goal, in contrast to Ehrenfests, was to establish a solid theoreticalfoundation, so to justify the invariant T / he does not appeal to Wiens displace-ment law, nor does he even use the term adiabatic: Equation (45) follows from thefundamental assumption of stationary states, almost as a logical consequence of it.

These considerations can be applied to harmonic-oscillator systems, such as:

two mass points of invariable distance rotating round their common centre ofgravity; if the two points carry equal and opposite electric charges, we shall callthe system an electrical doublet. . . A system of this kind may be expected toshow close analogies with diatomic molecules. . .108

Bohr comments on the singularity that Ehrenfest pointed out on the limits of the motionbetween vibrations and rotations:

As now in this critical state a single complete vibration of the doublet correspondswith nearly two entire rotations of the system, Ehrenfest concludes that (1/2)hnin [(45)] for rotating doublets must be replaced by (1/4)hn. It does not seempossible, however, in the way indicated to effect a complete transformation ofthe vibrating system to a freely rotating doublet. In the critical state the frequencyof vibration will be infinitely small and the state cannot be reached in a finitetime.109

Bohr proposed another transformation that is free of singularities:

A complete transformation, on the other hand, can be simply obtained if origi-nally the axis of the doublet instead of vibrating in a plane containing the axis ofthe field, executes small rotations in a circular cone the axis of which is parallelto that of the field. If now the field is diminished the vertical angle of this conewill gradually increase until when the field has disappeared the doublet rotatesfreely round an axis parallel to the field.

Recall that Ehrenfest proposed in his 1916 paper to use the conical pendulum toconnect vibrational and rotational motions.110

Bohr does not discuss adiabatic transformations further but turns to other topics,such as the radioactive emission of alpha and beta particles, which leads him to proposea relativistic version of (45), introducing the relativistic expression for the kineticenergy T , but not investigating its adiabatic invariance.111 In any case, in periodicsystems undergoing small changes, the kinetic energy and frequency differ very littlefrom that of the actual system, which means that

a periodic motion satisfying [(45)] for a given value of n is stable if the totalenergy of the system is smaller than that corresponding to any small variationof the motion which satisfies the same conditions.112

108 BOHR (1916); in HOYER (1981, p. 437).109 BOHR (1916); in HOYER (1981, p. 438, footnote *).110 See Sect. 2B.111 BOHR (1916); in HOYER (1981, p. 440).112 BOHR (1916); in HOYER (1981, p. 442).

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The stability of a system is therefore guaranteed by the venerable principle of attainingits state of minimum energy. In sum, the BoltzmannClausiusSzily mechanical theo-rem, which Ehrenfest rescued for quantum theory, constituted part of Bohrs ideas andreasoning as he was formulating his new quantum theory.

Sommerfelds quantization rules of 1915 were themselves refined by Epstein andSchwarzschild in 1916, and in 1917 Einstein published his pioneering paper on theemission and absorption of radiation. Bohrs 1916 work thus was immediately behindthe times, since it included nothing on multiply periodic motions and Einsteins conceptof transition probabilities.

3.2 On the quantum theory of line spectra

All of that changed when Bohr published his paper, On the Quantum Theory of LineSpectra, the first two parts of which appeared in English in the Proceedings of theRoyal Danish Academy of Sciences and Letters in 1918,113 the third in November1922,114 while the fourth was never published. Bohrs paper contains the most com-plete formulation of Ehrenfests adiabatic hypothesis, taking as its essence that infini-tely slow transformations provide continuous connections between quantum states.

A. General principles

Bohr begins with two fundamental assumptions:

1. That an atomic system can, and can only, exist permanently in a certain seriesof states corresponding to a discontinuous series of values for its energy,and that consequently any change of the energy of the system, includingemission and absorption of electromagnetic radiation, must take place by acomplete transition between two such states. These states will be denoted asthe stationary states of the system.

2. That the radiation absorbed or emitted during a transition between two sta-tionary states is unifrequentic and possesses a frequency , given by therelation

E E = h,where h is Plancks constant and where E and E are the values of the energyin the two states under consideration.115

These assumptions differ little from those of Bohrs atomic theory of 1913, but manyinnovations then follow. Bohr mentions, for example, transition probabilities betweenstationary states, a concept that allows him to avoid specifying a transition mecha-nism. Bohr notes further that at low radiation frequencies, the quantum and classicaltheories yield the same results, which enables him to establish a connection betweenthe transition probabilities and the polarization and intensity of the corresponding

113 BOHR (1918a,b).114 BOHR (1922).115 BOHR (1918a); in NIELSEN (1976, p. 71).

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spectral lineshis first use of what became known as his correspondence principle,which he first presented publicly in a lecture in Berlin on 27 April 1920.116

To Bohr an atomic system is a set of electrical particles that move under the influenceof a potential that depends only upon their relative positions. Under a slowly varyingexternal influence, however, the stability of the stationary states forces us to assumethat during such a transformation their motion differs only slightly from that in thestationary state. If, in addition, the variation proceeds slowly at a constant rate (orone that changes slowly) their motion must differ little from that of slowly movingmolecules, so

with the approximation mentioned, the motion of an atomic system in the sta-tionary states can be calculated by direct application of ordinary mechanics, notonly under constant external conditions, but in general also during a slow anduniform variation of these conditions.117

Bohr calls this the principle of mechanical transformability, and in this connectionmentions Ehrenfests work:

In these papers the principle in question is called the adiabatic hypothesis inaccordance with the line of argumentation followed by Ehrenfest in whichconsiderations of thermodynamical problems play an important part. From thepoint of view taken in the present paper, however, the above notation mightin a more direct way indicate the content of the principle and the limits of itsapplicability.118

Bohrs principle thus does not rest upon Ehrenfests thermodynamic arguments butconstitutes an extension of his above two fundamental assumptions.

Bohrs principle of mechanical transformability, despite its name, accentuates thenonmechanical character of his theory, since it deals only with infinitely slow trans-formations. It was helpful in overcoming the difficulty of how to establish energy dif-ferences between stationary states. The exact mechanism of transition, during which adefinite amount of energy is emitted or absorbed that is equal to the energy differencebetween the stationary states, was one of the blind spots of his theory. Bohrs principlealso was useful for finding the statistical weights of the stationary states. Bohr notedthat Ehrenfest had discovered a particular condition that the a priori probabilities ofthe stationary states had to satisfy during changes induced by external influences: forthe statistical interpretation of the second law of thermodynamics to remain valid, theymust remain constant during infinitely slow transformations. Thus:

If the a priori probabilities are known for the states of a given atomic system,. . .they may be deduced for any other system which can be formed from this by acontinuous transformation without passing through one of the singular systemsreferred to below.119

116 DARRIGOL (1992), who provides a complete analysis of Bohrs principle.117 BOHR (1918a); in NIELSEN (1976, p. 74).118 BOHR (1918a); in NIELSEN (1976, p. 74, footnote 1).119 BOHR (1918a); in NIELSEN (1976, p. 75).

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These singular systems were motions in which, during a transformation, the ener-gies of different stationary states coincide. In such systems the a priori probabilitiesno longer remain constantsystems (singular motions in Ehrenfests terminology)that had worried, and continued to worry Ehrenfest.

B. One-dimensional systems

Bohr now tackled the simplest case, one-dimensional systems, which allowed him toconstruct a general theory of stationary states, because if the distances between theircomponents do not increase indefinitely, their motions are always simple periodicones. He based his discussion on the BoltzmannClausiusSzily mechanical theorem(although he mentioned only Boltzmann), but he did not follow Ehrenfest exactly.

For the sake of the considerations in the following sections it will be conve-nient here to give the proof in a form which differs slightly from that given byEhrenfest, and which takes also regard to the modifications in the ordinarylaws of mechanics claimed by the theory of relativity.120

Instead of beginning with the Lagrangian of the system, as Ehrenfest had, Bohr beginswith the Hamiltonian equations for a conservative system with s degrees of freedom. Ifthe velocities of the particles and hence their masses are nonrelativistic, their canonicalmomenta are defined in the usual way; if they are relativistic, their momenta areobtained by using the relativistic expression for their kinetic energy. He defines thetotal momentum

I =

0

s1

pkqkdt, (47)

where is the period of the motion, and shows that:

I will be invariant for any fini

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