The Origin and Development of the Quantum Theory
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Title: The Origin and Development of the Quantum Theory
Author: Max Planck
Translator: Hans Thacher Clarke
Ludwik Silberstein
Release date: September 7, 2010 [eBook #33663]
Most recently updated: April 10, 2026
Language: English
Other information and formats: www.gutenberg.org/ebooks/33663
*** START OF THE PROJECT GUTENBERG EBOOK THE ORIGIN AND DEVELOPMENT OF THE QUANTUM THEORY ***
THE ORIGIN AND DEVELOPMENT
OF THE
QUANTUM THEORY
BY
MAX PLANCK
TRANSLATED BY
H. T. CLARKE AND L. SILBERSTEIN
[Illustration]
BEING THE
NOBEL PRIZE ADDRESS
DELIVERED BEFORE
THE ROYAL SWEDISH ACADEMY OF SCIENCES
AT STOCKHOLM, 2 JUNE, 1920
[Illustration]
OXFORD
AT THE CLARENDON PRESS
1922
OXFORD UNIVERSITY PRESS
London Edinburgh Glasgow Copenhagen
New York Toronto Melbourne Cape Town
Bombay Calcutta Madras Shanghai
HUMPHREY MILFORD
Publisher to the University
THE ORIGIN AND DEVELOPMENT OF THE QUANTUM THEORY
My task this day is to present an address dealing with the subjects
of my publications. I feel I can best discharge this duty, the
significance of which is deeply impressed upon me by my debt of
gratitude to the generous founder of this Institute, by attempting
to sketch in outline the history of the origin of the Quantum Theory
and to give a brief account of the development of this theory and its
influence on the Physics of the present day.
When I recall the days of twenty years ago, when the conception of the
physical quantum of ‘action’ was first beginning to disentangle itself
from the surrounding mass of available experimental facts, and when
I look back upon the long and tortuous road which finally led to its
disclosure, this development strikes me at times as a new illustration
of 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 occasionally run across striking facts
which form incontrovertible proof of the truth he seeks, and show him
that after all he has moved at least one step nearer to his objective.
The pursuit of a goal, the brightness of which is undimmed by initial
failure, is an indispensable condition, though by no means a guarantee,
of final success.
In my own case such a goal has been for many years the solution of
the question of the distribution of energy in the normal spectrum of
radiant heat. The discovery by Gustav Kirchhoff that the quality of the
heat radiation produced in an enclosure surrounded by any emitting or
absorbing bodies whatsoever, all at the same temperature, is entirely
independent of the nature of such bodies[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
properties of the substance. And the discovery of this remarkable
function promised a deeper insight into the relation between energy
and temperature, which is the principal problem of thermodynamics and
therefore also of the entire field of molecular physics. The only road
to this function was to search among all the different bodies occurring
in nature, to select one of which the emissive and absorptive powers
were known, and to calculate the energy distribution in the heat
radiation in equilibrium with that body. This distribution should then,
according to Kirchhoff’s law, be independent of the nature of the body.
A most suitable body for this purpose seemed H. Hertz’s rectilinear
oscillator (dipole) whose laws of emission for a given frequency he
had just then fully developed[2]. If a number of such oscillators be
distributed in an enclosure surrounded by reflecting walls, there
would take place, in analogy with sources and resonators in the case
of sound, an exchange of energy by means of the emission and reception
of electro-magnetic waves, and finally what is known as black body
radiation corresponding to Kirchhoff’s law should establish itself in
the vacuum-enclosure. I expected, in a way which certainly seems at the
present day somewhat naïve, that the laws of classical electrodynamics
would suffice, if one adhered sufficiently to generalities and avoided
too special hypotheses, to account in the main for the expected
phenomena and thus lead to the desired goal. I thus first developed in
as general terms as possible the laws of the emission and absorption
of a linear resonator, as a matter of fact by a rather circuitous
route which might have been avoided had I used the electron theory
which had just been put forward by H. A. Lorentz. But as I had not yet
complete confidence in that theory I preferred to consider the energy
radiating from and into a spherical surface of a suitably large radius
drawn around the resonator. In this connexion we need to consider only
processes in an absolute vacuum, the knowledge of which, however, is
all that is required to draw the necessary conclusions 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 comparison with existing
observations, e. g. the measurements of V. Bjerknes[3] on damping,
was the establishment of a general relation between the energy of a
resonator of a definite free frequency and the energy radiation of the
corresponding spectral region in the surrounding field in equilibrium
with it[4]. The remarkable result was obtained that this relation is
independent of the nature of the resonator, and in particular of its
coefficient of damping--a result which was particularly welcome, since
it introduced the simplification that the energy of the radiation could
be replaced by the energy of the resonator, so that a simple system of
one degree of freedom could be substituted for a complicated system
having many degrees of freedom.
But this result constituted only a preparatory advance towards the
attack on the main problem, which now towered up in all its imposing
height. The first attempt to master it failed: for my original hope
that the radiation emitted by the resonator would differ in some
characteristic way from the absorbed radiation, and thus afford the
possibility of applying a differential equation, by the integration
of which a particular condition for the composition of the stationary
radiation could be reached, was not realized. The resonator reacted
only to those rays which were emitted by itself, and exhibited no trace
of resonance to neighbouring spectral regions.
Moreover, my suggestion that the resonator might be able to exert
a one-sided, i. e. irreversible, action on the energy of the
surrounding radiation field called forth the emphatic protest of Ludwig
Boltzmann[5], who with his more mature experience in these questions
succeeded in showing that according to the laws of the classical
dynamics every one of the processes I was considering could take place
in exactly the opposite sense. Thus a spherical wave emitted from a
resonator when reversed shrinks in concentric spherical surfaces of
continually decreasing size on to the resonator, is absorbed by it,
and so permits the resonator to send out again into space the energy
formerly absorbed in the direction from which it came. And although
I was able to exclude such singular processes as inwardly directed
spherical waves by the introduction of a special restriction, to wit
the hypothesis of ‘natural radiation’, yet in the course of these
investigations it became more and more evident that in the chain
of argument an essential link was missing which should lead to the
comprehension of the nature of the entire question.
The only way out of the difficulty was to attack the problem from the
opposite side, from the standpoint of thermodynamics, a domain in
which I felt more at home. And as a matter of fact my previous studies
on the second law of thermodynamics served me here in good stead, in
that my first impulse was to bring not the temperature but the entropy
of the resonator into relation with its energy, more accurately not
the entropy itself but its second derivative with respect to the
energy, for it is this differential coefficient that has a direct
physical significance for the irreversibility of the exchange of
energy between the resonator and the radiation. But as I was at that
time too much devoted to pure phenomenology to inquire more closely
into the relation between entropy and probability, I felt compelled
to limit myself to the available experimental results. Now, at that
time, in 1899, interest was 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 undertaken by F. Paschen
in Hanover and by O. Lummer and E. Pringsheim of the Reichsanstalt,
Charlottenburg. This law expresses the intensity of radiation in terms
of the temperature by means of an exponential function. On calculating
the relation following from this law between the entropy and energy
of a resonator the remarkable result is obtained that the reciprocal
value of the above differential coefficient, which I shall here denote
by _R_, is proportional to the energy[7]. This extremely simple
relation can be regarded as an adequate expression of Wien’s law of
the distribution of energy; for with the dependence on the energy that
of the wave-length is always directly given by the well-established
displacement law of Wien[8].
Since this whole problem deals with a universal law of nature, and
since I was then, as to-day, pervaded with a view that the more general
and natural a law is the simpler it is (although the question as to
which formulation is to be regarded as the simpler cannot always be
definitely and unambiguously decided), I believed for the time that the
basis of the law of the distribution of energy could be expressed by
the theorem that the value of _R_ is proportional to the energy[9]. But
in view of the results of new measurements this conception soon proved
untenable. For while Wien’s law was completely satisfactory for small
values of energy and for short waves, on the one hand it was shown by
O. Lummer and E. Pringsheim that considerable deviations were obtained
with longer waves[10], and on the other hand the measurements carried
out by H. Rubens and F. Kurlbaum with the infra-red residual rays
(_Reststrahlen_) of fluorspar and rock salt[11] disclosed a totally
different, but, under certain circumstances, a very simple relation
characterized by the proportionality of the value of _R_ not to the
energy but to the square of the energy. The longer the waves and the
greater the energy[12] the more accurately did this relation hold.
Thus two simple limits were established by direct observation for the
function _R_: for small energies proportionality to the energy, for
large energies proportionality to the square of the energy. Nothing
therefore seemed simpler than to put in the general case _R_ equal
to the sum of a term proportional to the first power and another
proportional to the square of the energy, so that the first term is
relevant for small energies and the second for large energies; and
thus was found a new radiation formula[13] which up to the present has
withstood experimental examination fairly satisfactorily. Nevertheless
it cannot be regarded as having been experimentally confirmed with
final accuracy, and a renewed test would be most desirable[14].
But even if this radiation formula should prove to be absolutely
accurate it would after all be only an interpolation formula found by
happy guesswork, and would thus leave one rather unsatisfied. I was,
therefore, from the day of its origination, occupied with the task of
giving it a real physical meaning, and this question led me, along
Boltzmann’s line of thought, to the consideration of the relation
between entropy and probability; until after some weeks of the most
intense work of my life clearness began to dawn upon me, and an
unexpected view revealed itself in the distance.
Let me here make a small digression. Entropy, according to Boltzmann,
is a measure of a physical probability, and the meaning of the second
law of thermodynamics is that the more probable a state is, the more
frequently will it occur in nature. Now what one measures are only the
differences of entropy, and never entropy itself, and consequently one
cannot speak, in a definite way, of the absolute entropy of a state.
But nevertheless the introduction of an appropriately defined absolute
magnitude of entropy is to be recommended, for the reason that by its
help certain general laws can be formulated with great simplicity. As
far as I can see the case is here the same as with energy. Energy,
too, cannot itself be measured; only its differences can. In fact, the
concept used by our predecessors was not energy but work, and even
Ernst Mach, who devoted much attention to the law of conservation
of energy but at the same time strictly avoided all speculations
exceeding the limits of observation, always abstained from speaking
of energy itself. Similarly in the early days of thermochemistry one
was content to deal with heats of reaction, that is to say again with
differences of energy, until Wilhelm Ostwald emphasized that many
complicated calculations could be materially shortened if energies
instead of calorimetric numbers were used. The additive constant which
thus remained undetermined for energy was later finally fixed by the
relativistic law of the proportionality between energy and inertia[15].
As in the case of energy, it is now possible to define an absolute
value of entropy, and thus of physical probability, by fixing the
additive constant so that together with the energy (or better still,
the temperature) the entropy also should vanish. Such considerations
led to a comparatively simple method of calculating the physical
probability of a given distribution of energy in a system of
resonators, which yielded precisely the same expression for entropy as
that corresponding to the radiation law[16]; and it gave me particular
satisfaction, in compensation for the many disappointments I had
encountered, to learn from Ludwig Boltzmann of his interest and entire
acquiescence in my new line of reasoning.
To work out these probability considerations the knowledge of two
universal constants is required, each of which has an independent
meaning, so that the evaluation of these constants from the radiation
law could serve as an a posteriori test whether the whole process is
merely a mathematical artifice or has a true physical meaning. The
first constant is of a somewhat formal nature; it is connected with
the definition of temperature. If temperature were defined as the mean
kinetic energy of a molecule in a perfect gas, which is a minute
energy indeed, this constant would have the value ⅔[17]. But in the
conventional scale of temperature the constant assumes (instead of ⅔)
an extremely small value, which naturally is intimately connected with
the energy of a single molecule, so that its accurate determination
would lead to the calculation of the mass of a molecule and of
associated magnitudes. This constant is frequently termed Boltzmann’s
constant, although to the best of my knowledge Boltzmann himself never
introduced it (an odd circumstance, which no doubt can be explained by
the fact that he, as appears from certain of his statements[18], never
believed it would be possible to determine this constant accurately).
Nothing can better illustrate the rapid progress of experimental
physics within the last twenty years than the fact that during this
period not only one, but a host of methods have been discovered by
means of which the mass of a single molecule can be measured with
almost 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 thus obtained was quite
impossible, and one could scarcely hope to do more than test the
admissibility of its order of magnitude, it was not long before E.
Rutherford and H. Geiger[19] succeeded, by means of a direct count of
the α-particles, in determining the value of the electrical elementary
charge as 4.65·10⁻¹⁰, the agreement of which with my value 4.69·10⁻¹⁰
could be regarded as a decisive confirmation of my theory. Since then
further methods have been developed by E. Regener, 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 interpretation of the
second universal constant of the radiation law, which, as the product
of energy and time (amounting on a first calculation to 6.55·10⁻²⁷
erg·sec.) I called the elementary quantum of action. While this
constant was absolutely indispensable to the attainment of a correct
expression for entropy--for only with its aid could be determined
the magnitude of the ‘elementary region’ or ‘range’ of probability,
necessary for the statistical treatment of the problem[21]--it
obstinately withstood all attempts at fitting it, in any suitable form,
into the frame of the classical theory. So long as it could be regarded
as infinitely small, that is to say for large values of energy or long
periods of time, all went well; but in the general case a difficulty
arose at some point or other, which became the more pronounced the
weaker and the more rapid the oscillations. The failure of all attempts
to bridge this gap soon placed one before the dilemma: either the
quantum of action was only a fictitious magnitude, and, therefore,
the entire deduction from the radiation law was illusory and a mere
juggling with formulae, or there is at the bottom of this method of
deriving the radiation law some true physical concept. If the latter
were the case, the quantum would have to play a fundamental rôle in
physics, heralding the advent of a new state of things, destined,
perhaps, to transform completely our physical concepts which since the
introduction of the infinitesimal calculus by Leibniz and Newton have
been founded upon the assumption of the continuity of all causal chains
of events.
Experience has decided for the second alternative. But that the
decision should come so soon and so unhesitatingly was due not to the
examination of the law of distribution of the energy of heat radiation,
still less to my special deduction of this law, but to the steady
progress of the work of those investigators who have applied the
concept of 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 action seemed capable of explaining
readily a series of remarkable properties of light action discovered
experimentally, such as Stokes’s rule, the emission of electrons,
and the ionization of gases[22], and on the other hand, by the
identification of the expression for the energy of a system of
resonators with the energy of a solid body, derived a formula for
the specific heat of solid bodies which on the whole represented it
correctly as a function of temperature, more especially exhibiting
its decrease with falling temperature[23]. A number of questions were
thus thrown out in different directions, of which the accurate and
many-sided investigations yielded in the course of time much valuable
material. It is not my task to-day to give an even approximately
complete report of the successful work achieved in this field; suffice
it to give the most important and characteristic phase of the progress
of the new doctrine.
First, as to thermal and chemical processes. With regard to specific
heat of solid bodies, Einstein’s view, which rests on the assumption
of a single free period of the atoms, was extended by M. Born and Th.
von Karman to the case which corresponds better to reality, viz. that
of several free periods[24]; while P. Debye, by a bold simplification
of the assumptions as to the nature of the free periods, succeeded in
developing a comparatively simple formula for the specific heat of
solid bodies[25] which excellently represents its values, especially
those for low temperatures obtained by W. Nernst and his pupils, and
which, moreover, is compatible with the elastic and optical properties
of such bodies. But the influence of the quanta asserts itself also
in the case of the specific heat of gases. At the very outset it was
pointed out by W. Nernst[26] that to the energy quantum of vibration
must correspond an energy quantum of rotation, and it was therefore
to be expected that the rotational energy of gas molecules would
also vanish at low temperatures. This conclusion was confirmed by
measurements, due to A. Eucken, of the specific heat of hydrogen[27];
and if the calculations of A. Einstein and O. Stern, P. Ehrenfest,
and others have not as yet yielded completely satisfactory agreement,
this no doubt is due to our imperfect knowledge of the structure of
the hydrogen atom. That ‘quantized’ rotations of gas molecules (i.
e. satisfying the quantum condition) do actually occur in nature
can no longer be doubted, thanks to the work on absorption bands in
the infra-red of N. Bjerrum, E. v. Bahr, H. Rubens and G. Hettner,
and others, although a completely exhaustive explanation of their
remarkable rotation spectra is still outstanding.
Since all affinity properties of a substance are ultimately determined
by its entropy, the quantic calculation of entropy also gives access to
all problems of chemical affinity. The absolute value of the entropy
of a gas is characterized by Nernst’s chemical constant, which was
calculated by O. Sackur by a straightforward combinatorial process
similar to that applied to the case of the oscillators[28], while H.
Tetrode, holding more closely to experimental data, determined, by a
consideration of the process of vaporization, the difference of entropy
between a substance and its vapour[29].
While the cases thus far considered have dealt with states of
thermodynamical equilibrium, for which the measurements could yield
only statistical averages for large numbers of particles and for
comparatively long periods of time, the observation of the collisions
of electrons leads directly to the dynamic details of the processes in
question. Therefore the determination, carried out by J. Franck and G.
Hertz, of the so-called resonance potential or the critical velocity
which an electron impinging upon a neutral atom must have in order to
cause it to emit a quantum of light, provides a most direct method
for the measurement of the quantum of action[30]. Similar methods
leading to perfectly consistent results can also be developed for the
excitation of the characteristic X-ray radiation discovered by C. G.
Barkla, as can be judged from the experiments of 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 on exposure to light-rays, or
X-rays, and here, too, the energy quanta following from the action
quantum and the vibration period play a characteristic rôle, as was
early recognized from the striking fact that the velocity of the
emitted electrons depends not upon the intensity[31] but only on
the colour of the impinging light[32]. But quantitatively also the
relations to the light quantum, pointed out by Einstein (p. 13),
have proved successful in every direction, as was shown especially
by R. A. Millikan, by measurements of the velocities of emission of
electrons[33], while the importance of the light quantum in inducing
photochemical 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 overwhelming
proof of the existence of the quantum of action, the quantum hypothesis
received its strongest support from the theory of the structure of
atoms (Quantum Theory of Spectra) proposed and developed by Niels Bohr.
For it was the lot of this theory to find the long-sought key to the
gates of the wonderland of spectroscopy which since the discovery of
spectrum analysis up to our days had stubbornly refused to yield. And
the way once clear, a stream of new knowledge poured in a sudden flood,
not only over this entire field but into the adjacent territories of
physics and chemistry. Its first brilliant success was the derivation
of Balmer’s formula for the spectrum series of hydrogen and helium,
together with the reduction of the universal constant of Rydberg to
known magnitudes[35]; and even the small differences of the Rydberg
constant for these two gases appeared as a necessary consequence of the
slight wobbling of the massive atomic nucleus (accompanying the motion
of electrons around it). As a sequel came the investigation of other
series in the visual and especially the X-ray spectrum aided by Ritz’s
resourceful combination principle, which only now was recognized in its
fundamental significance.
But whoever may have still felt inclined, even in the face of this
almost overwhelming agreement--all the more convincing, in view of
the extreme accuracy of spectroscopic measurements--to believe it to
be a coincidence, must have been compelled to give up his last doubt
when A. Sommerfeld deduced, by a logical extension of the laws of the
distribution of quanta in systems with several degrees of freedom, and
by a consideration of the variability of inert mass required by the
principle of relativity, that magic formula before which the spectra
of both hydrogen and helium revealed the mystery of their ‘fine
structure’[36], as far as this could be disclosed by the most delicate
measurements possible up to the present, those of F. Paschen[37]--a
success equal to the famous discovery of the planet Neptune, the
presence and orbit of which were calculated by Leverrier [and Adams]
before man ever set eyes upon it. Progressing along the same road, P.
Epstein achieved a complete explanation of the Stark effect of the
electrical splitting of spectral lines[38], P. Debye obtained a simple
interpretation of the K-series[39] of the X-ray spectrum investigated
by Manne Siegbahn, and then followed a long series of further
researches which illuminated with greater or less success the dark
secret of atomic structure.
After all these results, for the complete exposition of which many
famous names would here have to be mentioned, there must remain for
an observer, who does not choose to pass over the facts, no other
conclusion than that the quantum of action, which in every one of the
many and most diverse processes has always the same value, namely
6.52·10⁻²⁷ erg·sec.[40], deserves to be definitely incorporated into
the system of the universal physical constants. It must certainly
appear a strange coincidence that at just the same time as the idea of
general relativity arose and scored its first great successes, nature
revealed, precisely in a place where it was the least to be expected,
an absolute and strictly unalterable unit, by means of which the amount
of action contained in a space-time element can be expressed by a
perfectly definite number, and thus is deprived 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 been established. Nay, the
path which research has yet to cover to reach that goal is perhaps
not less long than that from the discovery of the velocity of light
by Olaf Römer to the foundation of Maxwell’s theory of light. The
difficulties which the introduction of the quantum of action into the
well-established classical theory has encountered from the outset have
already been indicated. They have gradually increased rather than
diminished; and although research in its forward march has in the
meantime passed over some of them, the remaining gaps in the theory are
the more distressing to the conscientious theoretical physicist. In
fact, what in Bohr’s theory served as the basis of the laws of action
consists of certain hypotheses which a generation ago would doubtless
have been flatly rejected by every physicist. That with the atom
certain quantized orbits [i. e. picked out on the quantum principle]
should play a special rôle could well be granted; somewhat less easy
to accept is the further assumption that the electrons moving on these
curvilinear orbits, and therefore accelerated, radiate no energy. But
that the sharply defined frequency of an emitted light quantum should
be different from the frequency of the emitting electron would be
regarded by a theoretician who had grown up in the classical school as
monstrous and almost inconceivable.
But numbers decide, and in consequence the tables have been turned.
While originally it was a question of fitting in with as little strain
as possible a new and strange element into an existing system which
was generally regarded as settled, the intruder, after having won an
assured position, now has assumed the offensive; and it now appears
certain that it is about to blow up the old system at some point.
The only question now is, at what point and to what extent this will
happen. If I may express at the present time a conjecture as to the
probable outcome of this desperate struggle, everything appears to
indicate that out of the classical theory the great principles of
thermodynamics will not only maintain intact their central position in
the quantum theory, but will perhaps even extend their influence. The
significant part played in the origin of the classical thermodynamics
by mental experiments is now taken over in the quantum theory by P.
Ehrenfest’s hypothesis of the adiabatic invariance[41]; and just as
the principle introduced by R. Clausius, that any two states of a
material system are mutually interconvertible on suitable treatment by
reversible processes, formed the basis for the measurement of entropy,
just so do the new ideas of Bohr show a way into the midst of the
wonderland he has discovered.
There is one particular question the answer to which will, in my
opinion, lead to an extensive elucidation of the entire problem. What
happens to the energy of a light-quantum after its emission? Does it
pass outwards in all directions, according to Huygens’s wave theory,
continually increasing in volume and tending towards infinite dilution?
Or does it, as in Newton’s emanation theory, fly like a projectile in
one direction only? In the former case the quantum would never again
be in a position to concentrate its energy at a spot strongly enough
to detach an electron from its atom; while in the latter case it would
be necessary to sacrifice the chief triumph of Maxwell’s theory--the
continuity between the static and the dynamic fields--and with it the
classical theory of the interference phenomena which accounted for
all their details, both alternatives leading to consequences very
disagreeable to the modern theoretical physicist.
Whatever the answer to this question, there can be no doubt that
science will some day master the dilemma, and what may now appear to us
unsatisfactory will appear from a higher standpoint as endowed with a
particular harmony and simplicity. But until this goal is reached the
problem of the quantum of action will not cease to stimulate research,
and the greater the difficulties encountered in its solution the
greater will be its significance for the broadening and deepening of
all our physical knowledge.
NOTES
The 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, Über das Verhältnis zwischen dem Emissionsvermögen
und dem Absorptionsvermögen der Körper für Wärme und Licht. _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 temperature is
given by a relation of the form:
[Illustration: upper U equals a dot e Superscript negative b divided
by t.]
Since
[Illustration: StartFraction 1 Over upper T EndFraction equals
StartFraction d upper S Over d upper U EndFraction,]
where _S_ is the entropy of the resonator, we have for _R_ as used in
the text:
[Illustration: upper R equals 1 colon StartFraction d squared upper S
Over d upper U squared EndFraction equals minus b upper U.]
[8] According to Wien’s displacement law, the energy _U_ of the
resonator with the natural vibration period ν is expressed by:
[Illustration: upper U equals nu dot f left parenthesis StartFraction
upper T Over nu EndFraction right parenthesis.]
[9] _Ann. d. Phys._ 1, p. 719, 1900.
[10] O. Lummer und E. Pringsheim, _Verhandl. der Deutschen Physikal.
Ges._, 2, p. 163, 1900.
[11] H. Rubens and F. Kurlbaum, _Sitz.-Ber. der Preuss. Akad. d. Wiss._
Oct. 25, 1900, p. 929
[12] It follows from the experiments of H. Rubens and F. Kurlbaum that,
for high temperatures, _U_ = _cT_. Then, in accordance with the method
quoted in [7]:
[Illustration: upper R equals 1 colon StartFraction d squared upper S
Over d upper U squared EndFraction equals minus StartFraction upper U
squared Over c EndFraction.]
[13] Put
[Illustration: upper R equals 1 colon StartFraction d squared upper
S Over d upper U squared EndFraction equals minus b upper U minus
StartFraction upper U squared Over c EndFraction,]
then by integration,
[Illustration StartFraction 1 Over upper T EndFraction equals
StartFraction d upper S Over d upper U EndFraction equals StartFraction
1 Over b EndFraction log left brace 1 plus StartFraction b c Over upper
U EndFraction right brace]
whence the radiation formula,
[Illustration: upper U equals b c colon left parenthesis e Superscript
negative b divided by upper T Baseline minus 1 right parenthesis.]
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 product of
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:
[Illustration: upper U equals three halves k upper T]
If we put, therefore, _T_ = _U_, then _k_ = ⅔. In the conventional
[absolute Kelvinian] temperature scale, however, _T_ is defined by
putting the temperature difference between boiling and freezing water
equal to 100.
[18] Cf. for example L. Boltzmann, Zur Erinnerung an Josef Loschmidt,
_Populäre Schriften_, p. 245, 1905.
[19] E. Rutherford and H. Geiger, _Proc. Roy. 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 cases by
which the corresponding state is realized; and in order sharply to
distinguish these cases from one another, a definite concept of each
special 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
[27] A. Eucken, _Sitz.-Ber. d. Preuss. Akad. d. Wiss._ p. 141, 1912.
[28] O. Sackur, _Ann. d. Phys._ 36, p. 958, 1911.
[29] H. Tetrode, _Proc. Acad. Sci. Amsterdam_, Febr. 27 and March 27,
1915.
[30] J. Franck und G. Hertz, _Verh. d. Deutsch. Phys. Ges._ 16, p. 512,
1914.
[31] Ph. Lenard, _Ann. d. Phys._ 8, p. 149, 1902.
[32] E. Ladenburg, _Verh. d. Deutschen Phys. Ges._ 9, p. 504, 1907.
[33] R. A. Millikan, _Phys. Zeitschr._ 17, p. 217, 1916.
[34] E. Warburg, Über den Energieumsatz bei photochemischen Vorgängen
in Gasen. _Sitz.-Ber. d. Preuss. Akad. d. Wiss._ from 1911 onwards.
[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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