The origin and development of the quantum theory

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THE ORIGIN AND DEVELOPMENT

OF THE

QUANTUM THEORY

BY

MAX PLANCK

TRANSLATED BY

H. T. CLARKE AND L, SILBERSTEIN

BEING THE

NOBEL PRIZE ADDRESS

DELIVERED BEFORE

THE ROYAL SWEDISH ACADEMY OF SCIENCES
AT STOCKHOLM, 2 JUNE, 1920

OXFORD
AT THE CLARENDON PRESS

1922

THE ORIGIN AND DEVELOPMENT

OF THE

, * J > J

QUANTUM THEORY

BY

MAX PLANCK

TRANSLATED BY

H. T. CLARKE AND L. SILBERSTEIN

BEING THE

NOBEL PRIZE ADDRESS

DELIVERED BEFORE

THE ROYAL SWEDISH ACADEMY OF SCIENCES
AT STOCKHOLM, 2 JUNE, 1920

OXFORD

AT THE CLARENDON PRESS
1922

OXFORD UNIVERSITY PRESS

London Edinburgh Glasgow Copenhagen

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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 dis-
charge 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 o*f 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 radia-
tion produced in an enclosure surrounded by any

A 2

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emitting or absorbing bodies whatsoever, all at the same
temperature, is entirely independent of the nature of such
bodies (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-
markable 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 KirchhofFs
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 cas* * e. sound, • w
an exchange of energy by means of the S^Mf*1 ana U>
reception of electro-magnetic wavee^ -and finJ^jEjfeat is
known as black body radiation corresponding td*&ffchhoff s
law should establish itself in the vacuum-enclosure. I ex-
pected, in a way which certainly seems at the present day
somewhat naive, 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

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

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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 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 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 integra-
tion 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 neighbour-
ing 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 de-
creasing 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 investiga-
tions 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

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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 ex-
perimental 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 under-
taken by F. Paschen in Hanover and by 0. 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 J?, 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 de-
pendence 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

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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 E is pro-
portional to the energy (9). But in view of the results
of new measurements this conception soon proved un-
tenable. 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 0. Lummer and E. Pringsheim
that considerable deviations were obtained with longer
waves (10), and on the other hand the measurements carried
out by H. Eubens and F. Kurlbaum with the infra-red
residual 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 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 E : for small energies propor-
tionality to the energy, for large energies proportionality to
the square of the energy. Nothing therefore seemed
simpler than to put in the general case E 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

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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 interpola-
tion 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 prob-
ability, and the meaning of the second law of thermo-
dynamics 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,

A 8

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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 un-
determined for energy was later finally fixed by the
^ I 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 prob-
ability, 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 compara-
tively 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
i new line of reasoning.

To work out these probability considerations the know-
ledge 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 tempera-
ture were defined as the mean kinetic energy of a molecule

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in a perfect gas, which is a minute energy indeed, this
constant 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 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 Boltz-
mann's constant, although to the best of my knowledge
Boltzmann himself never introduced it (an odd circum-
stance, 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. Eutherford and
H. Geiger (19) succeeded, by means of a direct count of the
a-particles, in determining the value of the electrical ele-
mentary charge as 4 • 65 . 10~10, the agreement of which with
my value 4 • 69 . 10~10 could be regarded as a decisive con-
firmation of my theory. Since then further methods have
been 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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tion 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 elemen-
tary quantum of action. While this constant was abso-
lutely 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 prob-
ability, necessary for the statistical treatment of the
problem (21) — it obstinately withstood all attempts at fit-
ting 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 pro-
nounced 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 de-
duction 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 role in physics, heralding the
advent of a new state of things, destined, perhaps, to trans-
form 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

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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 ioniza-
tion 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 investiga-
tions 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 character-
istic 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, suc-
ceeded in developing a comparatively simple formula for
the specific heat of solid bodies (25) which excellently repre-
sents its values, especially those for low temperatures

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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 l 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 com-
pletely 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 en-
tropy 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 simi-
lar 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 vaporiza-
tion, the difference of entropy between a substance and its
vapour (29).

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While the cases thus far considered have dealt with
states of thermodyiiamical equilibrium, for which the mea-
surements 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 per-
fectly 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 vibra-
tion period play a characteristic role, 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 K. A. Millikan, by measure-
ments of the velocities of emission of electrons (33), while
the 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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overwhelming proof of the existence of the quantum of
action, the quantum hypothesis received its strongest sup-
port 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 stub-
bornly 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 con-
stant of Eydberg to known magnitudes (35) ; and even the
small differences of the Eydberg 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 Kitz's resourceful combination
principle, which only now was recognized in its funda-
mental 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 spectro-
scopic 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

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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 ob-
tained 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 men-
tioned, 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 con-
stants. 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, pre-
cisely 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

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to reach that goal is perhaps not less long than that from
the discovery of the velocity of light by Olaf Romer 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 genera-
tion 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 role could well be granted ; somewhat less easy
to accept is the further assumption that the electrons
moving on these curvilinear orbits, and therefore accel-
erated, radiate no energy. But that the sharply denned
frequency of an emitted light quantum should be different
from the frequency of the emitting electron would be re-
garded 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 ele-
ment into an existing system which was generally regarded
as settled, the intruder, after having won an assured posi-
tion, 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

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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 thermo-
dynamics 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 hypo-
thesis of the adiabatic invariance (41) ; and just as the
principle introduced by K. 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 pro-
jectile 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 neces-
sary 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.

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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, Uber das Verhaltnis zwischen dem Emissionsver-
mogen und dem Absorptionsvermogen der Korper fur Warme 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 :

b

U=a.e~r.
Since

1_ dS

T~ dlf

where 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 the
resonator with the natural vibration period i/f is 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.

(22)

(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) :

_ d*S U2

(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 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 :

If we put, therefore, T = V, then k = §. 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,
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 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.

(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 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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