Planck, Origin and Development of Quantum Theory

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THE ORIGIN AND DEVELOPMENTOF THEQUANTUM THEORY

BY

MAX PLANCKTRANSLATED BY

H. T. CLARKE AND L, SILBERSTEIN

BEING THE

NOBEL PRIZE ADDRESSDELIVERED BEFORE

THE ROYAL SWEDISH ACADEMY OF SCIENCESAT STOCKHOLM, 2 JUNE, 1920

OXFORDAT THE CLARENDON PRESS

1922


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THE ORIGIN AND DEVELOPMENTOF THE

, * J > JQUANTUM THEORYBY

MAX PLANCKTRANSLATED BY

H. T. CLARKE AND L. SILBERSTEIN

BEING THE

NOBEL PRIZE ADDRESSDELIVERED BEFORE

THE ROYAL SWEDISH ACADEMY OF SCIENCESAT STOCKHOLM, 2 JUNE, 1920

OXFORDAT THE CLARENDON PRESS

1922


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OXFORD UNIVERSITY PRESSLondon Edinburgh Glasgow CopenhagenNew York Toronto Melbourne Cape Town

Bombay Calcutta Madras ShanghaiHUMPHREY MILFORDPublisher to the University


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THE ORIGIN AND DEVELOPMENT OFTHE QUANTUM THEORYMY task this day is to present an address dealing with

the subjects of my publications. I feel I can best dis-charge this duty, the significance of which is deeplyimpressed upon me by my debt of gratitude to thegenerous founder of this Institute, by attempting to sketchin outline the history of the origin of the Quantum Theoryand to give a brief account of the development of this theoryand its influence on the Physics of the present day.When I recall the days of twenty years ago, when theconception of the physical quantum of ' action ' was firstbeginning to disentangle itself from the surrounding massof available experimental facts, and when I look back uponthe long and tortuous road which finally led to its disclosure,this development strikes me at times as a new illustrationof Goethe's saying, that 'man errs, so long as he is striving '.And all the mental effort of an assiduous investigator must$ indeed appear vain and hopeless, if he does not occasionallyrun across striking facts which form incontrovertible proofof the truth he seeks, and show him that after all he hasmoved at . least one step nearer to his objective. Thepursuit o*f a goal, the brightness of which is undimmed byinitial failure, is an indispensable condition, though by nomeans a guarantee, of final success.

In my own case such a goal has been for many yearsthe solution of the question of the distribution of energy inthe normal spectrum of radiant heat. The discovery byGustav Kirchhoff that the quality of the heat radia-tion produced in an enclosure surrounded by anyA 2

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emitting or absorbing bodies whatsoever, all at the sametemperature, is entirely independent of the nature of suchbodies (I) 1 , established the existence of a universal function,which depends only upon the temperature and the wave-length, and is entirely independent of the particular pro-perties of the substance. And the discovery of this re-markablefunction promised a deeper insight into the relationbetween energy and temperature, which is the principalproblem of thermodynamics and therefore also of theentire field of molecular physics. The only road to thisfunction was to search among all the different bodiesoccurring in nature, to select one of which the emissive andabsorptive powers were known, and to calculate the energydistribution in the heat radiation in equilibrium with thatbody. This distribution should then, according to KirchhofFslaw, be independent of the nature of the body.A most suitable body for this purpose seemed H. Hertz'srectilinear oscillator (dipole) whose laws of emission for agiven frequency he had just then fully developed (2). Ifa number of such oscillators be distributed in an enclosuresurrounded by reflecting walls, there would take place, in *analogy with sources and resonators in the cas* * e .sound, wan exchange of energy by means of the S^Mf*1 ana U>reception of electro-magnetic wavee^ -and finJ^jEjfeat isknown as black body radiation corresponding td*&ffchhoffslaw should establish itself in the vacuum-enclosure. I ex-pected, in a way which certainly seems at the present daysomewhat naive, that the laws of classical electrodynamicswould suffice, if one adhered sufficiently to generalities andavoided too special hypotheses, to account in the main for

1 The numbers in brackets refer to the notes at the end of thearticle.


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(5)the expected phenomena and thus lead to the desired goal.I thus first developed in as general terms as possible thelaws of the emission and absorption of a linear resonator,as a matter of fact by a rather circuitous route which mighthave been avoided had I used the electron theory whichhad just been put forward by H. A. Lorentz. But as I hadnot yet complete confidence in that theory I preferred toconsider the energy radiating from and into a sphericalsurface of a suitably large radius drawn around theresonator. In this connexion we need to consider onlyprocesses in an absolute vacuum, the knowledge of which,however, is all that is required to draw the necessary con-clusions concerning the energy changes of the resonator.The outcome of this long series of investigations, of

which some could be tested and were verified by com-parison with existing observations, e. g. the measurementsof V. Bjerknes(3) on damping, was the establishment ofa general relation between the energy of a resonator ofa definite free frequency and the energy radiationof the corresponding spectral region in the surroundingfield in equilibrium with it (4). The remarkable resultwas obtained that this relation is independent of thenature of the resonator, and in particular of its coefficientof damping a result which was particularly welcome,since it introduced the simplification that the energy of theradiation could be replaced by the energy of the resonator,so that a simple system of one degree of freedom could besubstituted for a complicated system having many degreesof freedom.

But this result constituted only a preparatory advancetowards the attack on the main problem, which nowtowered up in all its imposing height. The first attempt to


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master it failed : for my original hope that the radiationemitted by the resonator would differ in some characteristicway from the absorbed radiation, and thus afford thepossibility of applying a differential equation, by the integra-tion of which a particular condition for the composition ofthe stationary radiation could be reached, was not realized.The resonator reacted only to those rays which were emittedby itself, and exhibited no trace of resonance to neighbour-ing spectral regions.

Moreover, my suggestion that the resonator might beable to exert a one-sided, i. e. irreversible, action on theenergy of the surrounding radiation field called forth theemphatic protest of Ludwig Boltzmann (5), who with hismore mature experience in these questions succeeded inshowing that according to the laws of the classicaldynamics every one of the processes I was consideringcould take place in exactly the opposite sense. Thusa spherical wave emitted from a resonator when reversedshrinks in concentric spherical surfaces of continually de-creasing size on to the resonator, is absorbed by it, and sopermits the resonator to send out again into space theenergy formerly absorbed in the direction from which itcame. And although I was able to exclude such singularprocesses as inwardly directed spherical waves by theintroduction of a special restriction, to wit the hypothesisof ' natural radiation ', yet in the course of these investiga-tions it became more and more evident that in the chainof argument an essential link was missing which shouldlead to the comprehension of the nature of the entirequestion.The only way out of the difficulty was to attack the

problem from the opposite side, from the standpoint of


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(7)thermodynamics, a domain in which I felt more at home.And as a matter of fact my previous studies on the secondlaw of thermodynamics served me here in good stead, inthat my first impulse was to bring not the temperature butthe entropy of the resonator into relation with its energy,more accurately not the entropy itself but its secondderivative with respect to the energy, for it is thisdifferential coefficient that has a direct physical significancefor the irreversibility of the exchange of energy betweenthe resonator and the radiation. But as I was at that timetoo much devoted to pure phenomenology to inquire moreclosely into the relation between entropy and probability,I felt compelled to limit myself to the available ex-perimental results. Now, at that time, in 1899, interestwas centred on the law of the distribution of energy,which had not long before been proposed by W. Wien (6),the experimental verification of which had been under-taken by F. Paschen in Hanover and by 0. Lummer andE. Pringsheim of the Reichsanstalt, Charlottenburg. Thislaw expresses the intensity of radiation in terms of thetemperature by means of an exponential function. Oncalculating the relation following from this law betweenthe entropy and energy of a resonator the remarkableresult is obtained that the reciprocal value of the abovedifferential coefficient, which I shall here denote by J?, isproportional to the energy (7). This extremely simplerelation can be regarded as an adequate expression ofWien's law of the distribution of energy ; for with the de-pendence on the energy that of the wave-length is alwaysdirectly given by the well-established displacement law ofWien (8).

Since this whole problem deals with a universal law of


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(8)nature, and since I was then, as to-day, pervaded witha view that the more general and natural a law is thesimpler it is (although the question as to which formulationis to be regarded as the simpler cannot always be definitelyand unambiguously decided), I believed for the time thatthe basis of the law of the distribution of energy couldbe expressed by the theorem that the value of E is pro-portional to the energy (9). But in view of the resultsof new measurements this conception soon proved un-tenable. For while Wien's law was completely satisfactoryfor small values of energy and for short waves, on the onehand it was shown by 0. Lummer and E. Pringsheimthat considerable deviations were obtained with longerwaves (10), and on the other hand the measurements carriedout by H. Eubens and F. Kurlbaum with the infra-redresidual rays (Eeststrahlen) of fluorspar and rock salt (11)disclosed a totally different, but, under certain circum-stances, a very simple relation characterized by the pro-portionality of the value of E not to the energy but to thesquare of the energy. The longer the waves and the greaterthe energy (12) the more accurately did this relation hold.Thus two simple limits were established by directobservation for the function E : for small energies propor-

tionality to the energy, for large energies proportionality tothe square of the energy. Nothing therefore seemedsimpler than to put in the general case E equal to the sumof a term proportional to the first power and anotherproportional to the square of the energy, so that the firstterm is relevant for small energies and the second for largeenergies ; and thus was found a new radiation formula (13)which up to the present has withstood experimentalexamination fairly satisfactorily. Nevertheless it cannot


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(9)be regarded as having been experimentally confirmed withfinal accuracy, and a renewed test would be mostdesirable (14).But even if this radiation formula should prove to be

absolutely accurate it would after all be only an interpola-tion formula found by happy guesswork, and would thusleave one rather unsatisfied. I was, therefore, from theday of its origination, occupied with the task of giving ita real physical meaning, and this question led me, alongBoltzmann's line of thought, to the consideration of therelation between entropy and probability ; until after someweeks of the most intense work of my life clearness beganto dawn upon me, and an unexpected view revealed itselfin the distance.

Let me here make a small digression. Entropy,according to Boltzmann, is a measure of a physical prob-ability, and the meaning of the second law of thermo-dynamics is that the more probable a state is, the morefrequently will it occur in nature. Now what one measuresare only the differences of entropy, and never entropyitself, and consequently one cannot speak, in a definiteway, of the absolute entropy of a state. But neverthelessthe introduction of an appropriately defined absolutemagnitude of entropy is to be recommended, for the reasonthat by its help certain general laws can be formulatedwith great simplicity. As far as I can see the case is herethe same as with energy. Energy, too, cannot itself bemeasured ; only its differences can. In fact, the conceptused by our predecessors was not energy but work, andeven Ernst Mach, who devoted much attention to the lawof conservation of energy but at the same time strictlyavoided all speculations exceeding the limits of observation,A 8


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(10)always abstained from speaking of energy itself. Similarlyin the early days of thermochemistry one was content todeal with heats of reaction, that is to say again withdifferences of energy, until Wilhelm Ostwald emphasizedthat many complicated calculations could be materiallyshortened if energies instead of calorimetric numbers wereused. The additive constant which thus remained un-determined for energy was later finally fixed by the

^ I relativistic law of the proportionality between energy andinertia (15).As in the case of energy, it is now possible to definean absolute value of entropy, and thus of physical prob-ability, by fixing the additive constant so that togetherwith the energy (or better still, the temperature) the entropyalso should vanish. Such considerations led to a compara-tively simple method of calculating the physical probabilityof a given distribution of energy in a system of resonators,which yielded precisely the same expression for entropy asthat corresponding to the radiation law (16); and it gave meparticular satisfaction, in compensation for the manydisappointments I had encountered, to learn from LudwigBoltzmann of his interest and entire acquiescence in my

i new line of reasoning.To work out these probability considerations the know-

ledge of two universal constants is required, each of whichhas an independent meaning, so that the evaluation ofthese constants from the radiation law could serve as ana posteriori test whether the whole process is merelya mathematical artifice or has a true physical meaning.The first constant is of a somewhat formal nature ; it isconnected with the definition of temperature. If tempera-ture were defined as the mean kinetic energy of a molecule


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(11)in a perfect gas, which is a minute energy indeed, thisconstant would have the value (17). But in the con-ventional scale of temperature the constant assumes(instead of f) an extremely small value, which naturally isintimately connected with the energy of a single molecule,so that its accurate determination would lead to thecalculation of the mass of a molecule and of associatedmagnitudes. This constant is frequently termed Boltz-mann's constant, although to the best of my knowledgeBoltzmann himself never introduced it (an odd circum-stance, which no doubt can be explained by the fact thathe, as appears from certain of his statements (18), neverbelieved it would be possible to determine this constantaccurately). Nothing can better illustrate the rapidprogress of experimental physics within the last twentyyears than the fact that during this period not only one,but a host of methods have been discovered by means ofwhich the mass of a single molecule can be measured withalmost the same accuracy as that of a planet.While at the time when I carried out this calculation on (

the basis of the radiation law an exact test of the value thusobtained was quite impossible, and one could scarcely hopeto do more than test the admissibility of its order ofmagnitude, it was not long before E. Eutherford andH. Geiger (19) succeeded, by means of a direct count of thea-particles, in determining the value of the electrical ele-mentary charge as 4 65 . 10~10 , the agreement of which withmy value 4 69 . 10~

10 could be regarded as a decisive con-firmation of my theory. Since then further methods havebeen developed by E. Eegener, R A. Millikan, and others (20),which have led to a but slightly higher value.Much less simple than that of the first was the interpreta-


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(12)tion of the second universal constant of the radiation law,which, as the product of energy and time (amounting on afirst calculation to 6 55 . 10~27 erg. sec.) I called the elemen-tary quantum of action. While this constant was abso-lutely indispensable to the attainment of a correct expressionfor entropy for only with its aid could be determined themagnitude of the ' elementary region ' or ' range ' of prob-ability, necessary for the statistical treatment of theproblem (21) it obstinately withstood all attempts at fit-ting it, in any suitable form, into the frame of the classicaltheory. So long as it could be regarded as infinitely small,that is to say for large values of energy or long periods oftime, all went well; but in the general case a difficultyarose at some point or other, which became the more pro-nounced the weaker and the more rapid the oscillations.The failure of all attempts to bridge this gap soon placedone before the dilemma : either the quantum of action wasonly a fictitious magnitude, and, therefore, the entire de-duction from the radiation law Was illusory and a merejuggling with formulae, or there is at the bottom of thismethod of deriving the radiation law some true physicalconcept. If the latter were the case, the quantum wouldhave to play a fundamental role in physics, heralding theadvent of a new state of things, destined, perhaps, to trans-form completely our physical concepts which since theintroduction of the infinitesimal calculus by Leibniz andNewton have been founded upon the assumption of thecontinuity of all causal chains of events.Experience has decided for the second alternative. Butthat the decision should come so soon and so unhesitatinglywas due not to the examination of the law of distributionof the energy of heat radiation, still less to my special


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(13)deduction of this law, but to the steady progress of thework of those investigators who have applied the conceptof the quantum of action to their researches.The first advance in this field was made by A. Einstein,who on the one hand pointed out that the introduction of

the quanta of energy associated with the quantum of actionseemed capable of explaining readily a series of remarkableproperties of light action discovered experimentally, suchas Stokes's rule, the emission of electrons, and the ioniza-tion of gases (22), and on the other hand, by the identificationof the expression for the energy of a system of resonatorswith the energy of a solid body, derived a formula for thespecific heat of solid bodies which on the whole representedit correctly as a function of temperature, more especiallyexhibiting its decrease with falling temperature (23). Anumber of questions were thus thrown out in differentdirections, of which the accurate and many-sided investiga-tions yielded in the course of time much valuable material.It is not my task to-day to give an even approximatelycomplete report of the successful work achieved in thisfield ; suffice it to give the most important and character-istic phase of the progress of the new doctrine.

First, as to thermal and chemical processes. With regardto specific heat of solid bodies, Einstein's view, which restson the assumption of a single free period of the atoms, wasextended by M. Born and Th. von Karman to the casewhich corresponds better to reality, viz. that of several freeperiods (24) ; while P. Debye, by a bold simplification ofthe assumptions as to the nature of the free periods, suc-ceeded in developing a comparatively simple formula forthe specific heat of solid bodies (25) which excellently repre-sents its values, especially those for low temperatures


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(14)obtained by W. Nernst and his pupils, and which, moreover,is compatible with the elastic and optical properties of suchbodies. But the influence of the quanta asserts itself alsoin the case of the specific heat of gases. At the veryoutset it was pointed out by W. Nernst(26) that to theenergy quantum of vibration must correspond an energyquantum of rotation, and it was therefore to be expectedthat the rotational energy of gas molecules would alsovanish at low temperatures. This conclusion was confirmedby measurements, due to A. Eucken, of the specific heat ofhydrogen (27) ; and if the calculations of A. Einstein andO. Stern, P. Ehrenfest, and others have not as yet yieldedcompletely satisfactory agreement, this no doubt is due toour imperfect knowledge of the structure of the hydrogenatom. That l quantized' rotations of gas molecules (i.e.satisfying the quantum condition) do actually occur innature can no longer be doubted, thanks to the work onabsorption bands in the infra-red of N. Bjerrum, E. v. Bahr,H. Rubens and G. Hettner, and others, although a com-pletely exhaustive explanation of their remarkable rotationspectra is still outstanding.

Since all affinity properties of a substance are ultimatelydetermined by its entropy, the quantic calculation of en-tropy also gives access to all problems of chemical affinity.The absolute value of the entropy of a gas is characterizedby Nernst's chemical constant, which was calculated byO. Sackur by a straightforward combinatorial process simi-lar to that applied to the case of the oscillators (28), whileH. Tetrode, holding more closely to experimental data,determined, by a consideration of the process of vaporiza-tion, the difference of entropy between a substance and itsvapour (29).


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(15)While the cases thus far considered have dealt with

states of thermodyiiamical equilibrium, for which the mea-surements could yield only statistical averages for largenumbers of particles and for comparatively long periods oftime, the observation of the collisions of electrons leadsdirectly to the dynamic details of the processes in question.Therefore the determination, carried out by J. Franck andG. Hertz, of the so-called resonance potential or the criticalvelocity which an electron impinging upon a neutral atommust have in order to cause it to emit a quantum of light,provides a most direct method for the measurement of thequantum of action (30). Similar methods leading to per-fectly consistent results can also be developed for theexcitation of the characteristic X-ray radiation discoveredby C. G. Barkla, as can be judged from the experimentsof D. L. Webster, E. Wagner, and others.The inverse of the process of producing light quanta by

the impact of electrons is the emission of electrons onexposure to light-rays, or X-rays, and here, too, the energyquanta following from the action quantum and the vibra-tion period play a characteristic role, as was early recognizedfrom the striking fact that the velocity of the emittedelectrons depends not upon the intensity (31) but only onthe colour of the impinging light (32). But quantitativelyalso the relations to the light quantum, pointed out byEinstein (p. 13), have proved successful in every direction,as was shown especially by K. A. Millikan, by measure-ments of the velocities of emission of electrons (33), whilethe importance of the light quantum in inducing photo-chemical reactions was disclosed by E. Warburg (34).Although the results I have hitherto quoted from the most

diverse chapters of physics, taken in their totality, form an


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(16)overwhelming proof of the existence of the quantum ofaction, the quantum hypothesis received its strongest sup-port from the theory of the structure of atoms (QuantumTheory of Spectra) proposed and developed by Niels Bohr.For it was the lot of this theory to find the long-sought keyto the gates of the wonderland of spectroscopy which sincethe discovery of spectrum analysis up to our days had stub-bornly refused to yield. And the way once clear, a streamof new knowledge poured in a sudden flood, not only overthis entire field but into the adjacent territories of physicsand chemistry. Its first brilliant success was the derivationof Balmer's formula for the spectrum series of hydrogen andhelium, together with the reduction of the universal con-stant of Eydberg to known magnitudes (35) ; and even thesmall differences of the Eydberg constant for these twogases appeared as a necessary consequence of the slightwobbling of the massive atomic nucleus (accompanying themotion of electrons around it). As a sequel came theinvestigation of other series in the visual and especiallythe X-ray spectrum aided by Kitz's resourceful combinationprinciple, which only now was recognized in its funda-mental significance.

But whoever may have still felt inclined, even in theface of this almost overwhelming agreement all the moreconvincing, in view of the extreme accuracy of spectro-scopic measurements to believe it to be a coincidence,must have been compelled to give up his last doubt whenA. Sommerfeld deduced, by a logical extension of the lawsof the distribution of quanta in systems with several degreesof freedom, and by a consideration of the variability ofinert mass required by the principle of relativity, thatmagic formula before which the spectra of both hydrogen


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(17)and helium revealed the mystery of their ' fine structure ' (36),as far as this could be disclosed by the most delicatemeasurements possible up to the present, those ofF. Paschen (37) a success equal to the famous discoveryof the planet Neptune, the presence and orbit of whichwere calculated by Leverrier [and Adams] before manever set eyes upon it. Progressing along the same road,P. Epstein achieved a complete explanation oftheStark effectof the electrical splitting of spectral lines (38), P. Debye ob-tained a simple interpretation of the K-series(39) of the X-rayspectrum investigated byManne Siegbahn, and then followeda long series of further researches which illuminated withgreater or less success the dark secret of atomic structure.

After all these results, for the complete exposition ofwhich many famous names would here have to be men-tioned, there must remain for an observer, who does notchoose to pass over the facts, no other conclusion than thatthe quantum of action, which in every one of the manyand most diverse processes has always the same value,namely 6 52 . 10~27 erg. sec. (40), deserves to be definitelyincorporated into the system of the universal physical con-stants. It must certainly appear a strange coincidence thatat just the same time as the idea of general relativity aroseand scored its first great successes, nature revealed, pre-cisely in a place where it was the least to be expected, anabsolute and strictly unalterable unit, by means of whichthe amount of action contained in a space-time element canbe expressed by a perfectly definite number, and thus isdeprived of its former relative character.Of course the mere introduction of the quantum of action

does not yet mean that a true Quantum Theory has beenestablished. Nay, the path which research has yet to cover


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(18)to reach that goal is perhaps not less long than that fromthe discovery of the velocity of light by Olaf Romer to thefoundation of Maxwell's theory of light. The difficultieswhich the introduction of the quantum of action into thewell-established classical theory has encountered from theoutset have already been indicated. They have graduallyincreased rather than diminished ; and although researchin its forward march has in the meantime passed oversome of them, the remaining gaps in the theory are themore distressing to the conscientious theoretical physicist.In fact, what in Bohr's theory served as the basis of thelaws of action consists of certain hypotheses which a genera-tion ago would doubtless have been flatly rejected byevery physicist. That with the atom certain quantizedorbits [i.e. picked out on the quantum principle] should playa special role could well be granted ; somewhat less easyto accept is the further assumption that the electronsmoving on these curvilinear orbits, and therefore accel-erated, radiate no energy. But that the sharply dennedfrequency of an emitted light quantum should be differentfrom the frequency of the emitting electron would be re-garded by a theoretician who had grown up in the classicalschool as monstrous and almost inconceivable.

But numbers decide, and in consequence the tables havebeen turned. While originally it was a question of fittingin with as little strain as possible a new and strange ele-ment into an existing system which was generally regardedas settled, the intruder, after having won an assured posi-tion, now has assumed the offensive ; and it now appearscertain that it is about to blow up the old system at somepoint. The only question now is, at what point and towhat extent this will happen. If I may express at the


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(19)present time a conjecture as to the probable outcome ofthis desperate struggle, everything appears to indicate thatout of the classical theory the great principles of thermo-dynamics will not only maintain intact their central positionin the quantum theory, but will perhaps even extend theirinfluence. The significant part played in the origin of theclassical thermodynamics by mental experiments is nowtaken over in the quantum theory by P. Ehrenfest's hypo-thesis of the adiabatic invariance (41) ; and just as theprinciple introduced by K. Clausius, that any two states ofa material system are mutually interconvertible on suitabletreatment by reversible processes, formed the basis for themeasurement of entropy, just so do the new ideas of Bohrshow a way into the midst of the wonderland he hasdiscovered.

There is one particular question the answer to whichwill, in my opinion, lead to an extensive elucidation of theentire problem. What happens to the energy of a light-quantum after its emission ? Does it pass outwards in alldirections, according to Huygens's wave theory, continuallyincreasing in volume and tending towards infinite dilution ?Or does it, as in Newton's emanation theory, fly like a pro-jectile in one direction only? In the former case thequantum would never again be in a position to concentrateits energy at a spot strongly enough to detach an electronfrom its atom ; while in the latter case it would be neces-sary to sacrifice the chief triumph of Maxwell's theory thecontinuity between the static and the dynamic

fields andwith it the classical theory of the interference phenomenawhich accounted for all their details, both alternativesleading to consequences very disagreeable to the moderntheoretical physicist.


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(20)Whatever the answer to this question, there can be no

doubt that science will some day master the dilemma, andwhat may now appear to us unsatisfactory will appear froma higher standpoint as endowed with a particular harmonyand simplicity. But until this goal is reached the problemof the quantum of action will not cease to stimulateresearch, and the greater the difficulties encountered inits solution the greater will be its significance for thebroadening and deepening of all our physical knowledge.


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NOTESThe references to the literature are not claimed to be in any way

complete, and are intended to serve only for a preliminary orientation.(1) G. Kirchhoff, Uber das Verhaltnis zwischen dem Emissionsver-

mogen und dem Absorptionsvermogen der Korper fur Warme undLicht. Gesammelte Abhandlungen. Leipzig, J. A. Barth, 1882, p. 597( 17).

(2) H. Hertz, Ann. d. Phys. 36, p. 1, 1889.(3) Sitz.-Ber. d. Preuss. Akad. d. Wiss. Febr. 20, 1896. Ann. d. Phys.

60, p. 577, 1897.(4) Sitz.-Ber. d. Preuss. Akad. d. Wiss. May 18, 1899, p. 455.(5) L. Boltzmann, Sitz.-Ber. d. Preuss. Akad. d. Wiss. March 3, 1898,

p. 182.(6) W. Wien, Ann. d. Phys. 58, p. 662, 1896.(7) According to Wien's law of the distribution of energy the

dependence of the energy U of the resonator upon the temperatureis given by a relation of the form :

bU=a.e~r.

Since 1_ dST~ dlfwhere S is the entropy of the resonator, we have for E as used in the

text:

(8) According to Wien's displacement law, the energy U of theresonator with the natural vibration period i/fis expressed by :

(9) Ann. d. Phys. 1, p. 719, 1900.(10) 0. Lurnmer und E. Pringsheim, Verhandl der Deutschen Physikal.

Ges., 2, p. 163, 1900.(11) H. Kubens and F. Kurlbaum, Sitz.-Ber. der Preuss. Akad d. Wiss.

Oct. 25, 1900, p. 929.


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(22)(12) It follows from the experiments of H. Rubens and F. Kurlbaum

that, for high temperatures, U=cT. Then, in accordance with themethod quoted in (7) : _ d*S U 2

(13) Put

then by integration, -- - -lo h -whence the radiation formula,

U=bc:(e- b/T-l).Cf. Verhandlungen der Deutschen Phys. Ges. Oct. 19, 1900, p. 202.

(14) Cf. W. Nernst und Th. Wulf, Verh. d. Deutsch. Phys. Ges. 21,p. 294, 1919.

(15) For the absolute value of the energy is equal to the productof the inert mass and the square of light velocity.

(16) Verhandlungen der Deutschen Phys. Ges. Dec. 14, 1900, p. 237.(17) Generally, if k be the first radiation constant, the mean kinetic

energy of a gas molecule is :

If we put, therefore, T= V, then k = . In the conventional [absoluteKelvinian] temperature scale, however, T is defined by putting thetemperature difference between boiling and freezing water equal to 100.

(18) Cf. for example L. Boltzmann, ZurErinnerung an Josef Loschmidt,Populdre Schriften, p. 245, 1905.(19) E. Rutherford and H. Geiger, Proc. Boy. Soc. A. Vol. 81, p. 162,1908.

(20) Cf. R. A. Millikan, Phys. Zeitschr. 14, p. 796, 1913.(21) The evaluation of the probability of a physical state is based

upon counting that finite number of equally probable special casesby which the corresponding state is realized ; and in order sharplyto distinguish these cases from one another, a definite concept of eachspecial case has necessarily to be introduced.

(22) A. Einstein, Ann. d. Phys. 17, p. 132, 1905.(23) A. Einstein, Ann. d. Phys. 22, p. 180, 1907.(24) M. Born und Th. v. Karman, Phys. Zeitschr. 14, p. 15, 1913.(25) P. Debye, Ann. d. Phys. 39, p. 789, 1912.(26) W. Nernst, Phys. Zeitschr. 13, p. 1064, 1912.


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(23)(27) A. Euckeri, Sitz.-Ber. d. preuss. Akad. d. Wiss. p. 141, 1912.(28) 0. Sackur, Ann. d. Phys. 36, p. 958, 1911.029) H. Tetrode, Proc. Acad. Sci. Amsterdam, Febr. 27 and March 27,1915.(30) J. Franck und G. Hertz, Verh. d. Deutsch. Phys. G-es. 16, p. 512,

1914.(31) Ph. Lenard, Ann. d. Phys. 8, p. 149, 1902.(32) E. Ladenburg, Verh. d. Deutschen Phys. G-es. 9, p. 504, 1907.(33) K. A. Millikan, Phys. Zeitschr. 17, p. 217, 1916.(34) E. Warburg, Uber den Energieumsatz bei photochemischen

Vorgangen in Gasen. Sitz.-Ber. d. preuss. Akad. d. Wiss. from 1911onwards.

(35) N. Bohr, Phil Mag. 30, p. 394, 1915.(36) A. Sommerfeld, Ann. d. Phys. 51, pp. 1, 125, 1916.(37) F. Paschen, Ann. d. Phys. 50, p. 901, 1916.(38) P. Epstein, Ann. d. Phys. 50, p. 489, 1916.(39) P. Debye, Phys. Zeitschr. 18, p. 276, 1917.(40) E. Wagner, Ann. d. Phys. 57, p. 467, 1918.(41) P. Ehrenfest, Ann. d. Phys. 51, p. 327, 1916.


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1thv,English by HENRY L. BROSE, with an Introaucnon uy r . ,i*wmn.. . y -~.(9 x 6), pp. xii + 90. 6s. 6d. net.A GENERAL COURSE OF PURE MATHEMATICS, from Indicesto Solid Analytical Geometry, by A. L. BOWLEY. 1913. In nine sections : Algebra ;Geometry ; Trigonometry ; Explicit Functions, Graphic Representation, Equations ;Limits ; Plane Co-ordinate Geometry ; Differential and Integral Calculus ; Imaginaryand Complex Quantities; Co-ordinate Geometry in Three Dimensions. With ninety-seven figures and answers to Examples. 8vo (9 x 6), pp. xii + 272. 75. 6d. net.A TREATISE ON STATICS WITH APPLICATIONS TOPHYSICS, by G. M. MINCHIN. Two volumes. 8vo (9 x 6).Vol. T. Equilibrium of Coplanar Forces. Seventh edition, 1915. With 245 figuresand 389 examples (300 being new to this edition). Pp.476. I2s.net.Vol. II. Non-Coplanar Forces. Fifth edition, revised by H. T. GERRANS, 1915.With sixty-one figures, and examples 90-389 of Vol. 1 + 555- PP- 378 - I2 s. net.ALGEBRA OF QUANTICS, an introduction, having as its primaryobject the explanation of the leading principles of invariant algebra, by E. B. ELLIOTT.Second edition, with additions. 1913. 8vo (9x6), pp. xvi + 416. i6s. net>PLANE ALGEBRAIC CURVES, by HAROLD HILTON. 1920. 8vo(9 x 6), pp. xvi + 388. 285. net.THEORY OF GROUPS OF FINITE ORDER, an Introduction whichaims at introducing the reader to more advanced treatises and original papers onGroups of finite order. By H. HILTON. 1908. With numerous examples and hintsfor the solution of the most difficult. 8vo (9 x 6), pp. xii + 236. 145. net.HOMOGENEOUS LINEAR SUBSTITUTIONS. A collection forthe benefit of the Mathematical Student of those properties of the HomogeneousLinear Substitution with real or complex co-efficients of which frequent use is made inthe Theory of Groups and the Theory of Bilinear Forms and Invariant Factors. ByH. HILTON. 1914. 8vo (g\ x 6), pp. viii + 184. 125. 6d. net.INTRODUCTION TO THE INFINITESIMAL CALCULUS, withapplications to Mechanics and Physics, presenting the fundamental principles andapplications of the Differential and Integral Calculus in as simple a form as possible.By G. W. CAUNT. 1914. 8vo (9^ x 6|), pp. xx + 568. ias. 6d. net.INTRODUCTION TO ALGEBRAICAL GEOMETRY, by A.CLEMENT JONES. 1912. For the beginner and candidate for a MathematicalScholarship, the syllabus for the Honour School of the First Public Examination atOxford being taken as a maximum limit. With answers and index. 8vo (9^ x 6),pp. 548. I2S. 6d. net.CREMONA'S ELEMENTS OF PROJECTIVE GEOMETRY,lated by C. LEUDESDORF. Third edition. 1913. With 251 figures. 8vo(8x6j),pp. xx + 304. 153. net.CREMONA'S GRAPHICAL STATICS, being two treatises on theGraphical Calculus and Reciprocal Figures, translated by T. H. BEARE. 1890.With 143 figures. 8vo (9x6), pp. xvi 4- 162. 8s. 6d. net.ON THE TRAVERSING OF GEOMETRICAL FIGURES, byJ. COOK WILSON. In three parts: Analytical Method, Constructive Method, andApplication of the Principle of Duality. 1905. 8vo (9x6), pp. 1 66, with Addendum.6s. 6d. net.A TREATISE ON THE CIRCLE AND THE SPHERE, by J. L.COOLIDGE. 1916. 8vo (9 x 6), pp. 604, with thirty figures. 25s.net.NON-EUCLIDEAN GEOMETRY, the Elements, by J. L. COOLIDGE.1909. 8vo (9x6), pp. 292. i6s. net.


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