THE

FORMATION AND LOGIC
OF

QUANTUM MECHANICS
Volume I
The Formation of Atomic Models

MITUO TAKETANI

Translated from the Japanese and with
Explanatory Notes by Masayuki Nagasaki

World Scientific


THE

FORMATION AND LOGIC
OF

QUANTUM MECHANICS
Vol.1
The Formation of Atomic Models

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THE

FORMATION AND LOGIC
OF

QUANTUM MECHANICS
Vol.1
The Formation of Atomic Models

MITUO TAKETANI

Translated from the Japanese and with
Explanatory Notes by Masayuki Nagasaki

V f e World Scientific
wb

New Jersey • London • Sine
Singapore • Hong Kong
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Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Farrer Road, Singapore 912805
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THE FORMATION AND LOGIC OF QUANTUM MECHANICS:
Vol. I. The Formation of Atomic Models
Copyright © 2001 by World Scientific Publishing Co. Pte. Ltd.
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Preface to the English Edition

This book is the English edition of The Formation and Logic of Quantum

Mechanics written in Japanese, which consists of the following three volumes:
Vol. I

—

Vol. II

—

Vol. Ill

—

The Formation of Atomic Models — by Taketani, with
Explanatory Notes by Nagasaki added to its republished
edition,
The Way to Quantum Mechanics — by Taketani and
Nagasaki,
The Establishment and Logic of Quantum Mechanics
— by Taketani and Nagasaki.

This book was planned by Taketani soon after the World War II, and Vol. I
was published in 1948 by a Japanese publishing company, which became unfortunately insolvent afterward. The republication of Vol. I and the publication
of the volumes to be connected were not accomplished for a long time in spite
of the hope of many readers, because of various circumstances, of which some
descriptions are given in the mentioned explanatory notes. Volume I was republished in 1972 by means of photocomposition, to which Explanatory Notes
by Nagasaki were added at Taketani's request, mainly for the sake of students
and readers born after the World War II. Volumes II and III were accomplished
by the collaboration of Taketani and Nagasaki and published respectively in
1991 and 1993.

The aim of this book is to analyze through what intricate logical process
the quantum theory was developed, and to elucidate by what logic quantum
mechanics thus established is governed. The method of our analysis is based
V

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Preface to the English

VI

Edition

on the three stage theory of scientific cognition, which was presented by Taketani to solve the ineffectiveness of the Machist view of science in treating the
difficulties met with in nuclear physics at that time. According to the Machist
view, scientific cognition is to know some mathematical laws by means of the
systematization of given sensuous experience. The three stage theory gives, on
the contrary, the view that scientific cognition proceeds through the series of
coiling turns of the three stages, namely, the phenomenological, substantialistic
and essentialistic stages.
The latter view was presented in 1934 by Taketani a little before the proposal by H. Yukawa of the meson theory, and was developed in a close connection with the works by H. Yukawa, S. Sakata and Taketani of constructing
the meson theory. H. Yukawa describes this situation in his preface with the
title "The way I proceeded" to the book written by S. Takauchi in Japanese
"Order and Chaos — On Hideki Yukawa!'. (Jul. 1, 1974, Kosakusya Pub. Co.)
as follows: "In the innermost recesses of my heart there was existing for a long
time a great doubt. It started growing in my heart a little after my graduation
in 1929. It was, as is described in the chapter with the title "Tenki" (Turningpoint) of my autobiography "Tabibito" (Traveler), a fundamental doubt of, or
a feeling of dissatisfaction with, the quantum theory of field. Briefly expressed,
it was the consciousness of an essentialistic problem. Afterward, shelving this

problem for a while, I proceeded to the meson theory which was full of substantialistic nature".
About the three stage theory, the reader will find more explanations in
Concluding Remarks of Vol. I and in Explanatory Notes added to Vol. I. Some
descriptions of the three stage theory are also given here and there in the text
of the present book, in relation to our concrete analyses of the process of the
formation of quantum mechanics, and in connection with our elucidation of
the logic of quantum mechanics. In the preface to each volume mentions are
made of the outline of our way of analyzing the problems in the epoch treated
in the volume concerned.
The three stage theory is in sharp contrast to the Machist theory in attaching importance to the stereo-structural nature of the logic comprehended
in scientific cognition, instead of the plane-projected view, so to say, of scientific cognition taken in the latter theory. From the analyses given in the
present book, the reader will see how it is important, in discussing quantum
mechanics as well as its formation, to distinguish the stereo-structural logic
from the plane-projected view based on the formal logic.

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Preface to the English

Edition

vn

The three stage theory was somewhat introduced internationally, by means
of the publication of the English translation of some related original papers
in Supplement No. 50 (1971), Progress of Theoretical Physics (Kyoto). A
part of the mentioned publication in English was published in German as
" Wissenschaftliche Tascenbiichen Bd. 221" (Akademie-Verlag, Berlin, 1980).
L. M. Brown and H. Rechenberg give also a certain introduction to the three

stage theory in their recent book " The Origin of the Concept of Nuclear Forces"
(Institute of Physics Publishing, Bristol and Philadelphia, 1996).
The present English edition of our book has been prepared by Nagasaki
as the translator. The quotations from cited papers written in German and
French are translated also into English by Nagasaki. His thanks are due to
Miss T. Ohsuga of Laboratory of Theoretical Physics, Rikkyo University for
her assistance in typewriting.
Mituo Taketani
Masayuki Nagasaki
Dec. 24, 1997

Postcript added in proof: I, M. Nagasaki, would like to thank Prof. M.
Konuma of Musashi Institute of Techniligy for his interest in the present book,
and for his kindness to introduce it to Prof. K. K. Phua of World Scientific
Publishing Co. Pte. Ltd. I am grateful to Prof. Phua for his goodwill to
publish it from the mentioned publishig company. My thanks are also due to
Dr. L. Y. Wong as Scientific Editor of the same company for his good offices
in its publication.
I add by myself the present postscript, because, to my great sorrow, Prof.
Taketani passed away on April 22, 2000. Prof. Taketani was of the same
mind as me to express our sincere thanks of Prof. Konuma, Prof. Phua and
Dr. Wong in our Preface to the English Edition at the time of reading the first
Proof.
Masayuki Nagasaki
September 24, 2000

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Preface

Atomic physics is in every respect one of the greatest achievements of human
beings, and is recognized as the one to give human thought a great revolution
and a firm foundation. Nevertheless, there has been little work of making clear
its logical structure to put the outcome in the property of thought. Natural
scientists, who are generally not versed in logical expression and elucidation
of high level, have not been able to clarify the significance of the works they
did, so that they understand it superficially and their interpretations are quite
separated from their works. Philosophers on the other hand have not analyzed
the logical structure of physics itself, and have selected fragments of one or
another physicist's words, using those irresponsibly for the sake of their own
theories.
Therefore, such a work must be done first of all as to trace the correct
process of formation of the atomic theory, and thereby to draw its logic. This
is however surely a hard task. I hope that my present book will serve as a
footing for this aim.
The present book consists of Vol. I, The Formation of Atomic Models,
Vol. II, The Formation of Quantum Mechanics, and Vol. Ill, The Logic of
Quantum Mechanics.
Volume I treats the process up to the establishment of Rutherford's model
of the atom, in which our Nagaoka made an important work, so great a contribution to world physical society made by Japanese physics in its young days
that we can never forget it. Nevertheless, its actual fact is generally unknown.
Also in foreign countries, almost no detailed book on the history of the
formation of atomic models before the time of Rutherford has not been seen,
ix

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Preface

X

because this subject was regarded as settled already. However, the period
mentioned is very important when it is seen from the viewpoint of epistemdolgy. Every epistemological confusion arises from having no appreciation of
this period.
In the present book particular attention is paid to reveal clearly on what
assumption each paper bases, and what conclusion is deduced by means of
what reasoning as well as from what fact, for the purpose of clarifying step by
step the process of development in it.
The method of carrying out this purpose has been described in detail in
my book "Problems in Dialectic". I shall be glad if the reader refers to this
book.
I intend that descriptions shall represent the features of each original paper
as far as possible, and also that words shall be faithful to the original papers.
In this respect the present book may lack plainness and unity such as those
found in textbooks. As to this point, I would like to recommend the reader to
consult some textbook of quantum mechanics, such as that by S. Tomonaga, for
example, for supplementary information. The knowledge which I suppose to be
necessary to understand the present book, is that which is treated in textbooks
of general physics for undergraduate students, and that of the Maxwell theory
of electromagnetics.
As a work on the history of science, the present one makes only the fundamental study as an initial step. It is still necessary to make clear the relation
of science to society and the interaction of science with general philosophical
thought. I plan for a study on "the experimental foundation of quantum mechanics" as to the former, and a study on "the role played by idealism and
materialism in the formation of atomic physics" as to the latter.
The book "The History of Quantum Mechanics", a great work by the late

Mr. Kiyoshi Amano who was my respected senior, has recently been published
to my delight. I hope that the reader will read through it once, since into it
plenty of descriptions of social background and episodes are woven into, and it
is plain to be readable by general readers. I would like to thank Mr. Amano
in this occasion for his having given me much knowledge.
As the present book is the first step in my study I am afraid that it may
have insufficient points. I would like to get suggestions from those who know
well about circumstances of those times.
The present book is written in a specialized way with the use of mathematical equations. But they are put in for the sake of contributing to more

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Preface

XI

understanding. Those who are not familiar with mathematical equations may
well skip the mathematical parts to get the outline of reasoning.
I would like to acknowledge Mr. S. Tomonaga, Mr. S. Sakata and other my
seniors and co-workers for their good wishes and advice. I am indebted very
much to Mr. S. Nakamura for his help in getting the literature. My thanks
are due to Mr. G. Tominga, Miss Yuminami and Miss Ghi for their help about
the literature.
I would like to thank Mr. Y. Kuyoshi of the Ginza Pub. Co. for his
goodwill to the present book of unpopular subject and for his good offices in
its publication. I am grateful to Mr. M. Minakami and Mr. M. Watanabe of
the same company for their acceptance of my personal circumstances to offer
convenience. I intended to write out the present book in August of last year,
but I spent almost one year more in finishing my work. I would like to express

my sincere thanks to the workers of the printing office for the trouble they
have taken for me.
The author
May 8, 1948

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Contents

Preface to the English Edition

v

Preface

ix

Volume I.

T h e Formation of Atomic Models

Chapter 1. Quantum of Radiation
1.
Thermodynamical Investigation of Black Body Radiation
§1. The prehistory of thermal radiation
§2. Kirchhoff's law

§3. Growth of atomistic viewpoint
§4. Stefan-Boltzmann's law
§5. Wien's displacement law
2.
Atomistic Investigations of Black Body Radiation
§1. Application of atomism
§2. Wien's distribution law
§3. Planck's effort
§4. Foundation by Planck of Wien's distribution formula
§5. Rayleigh's critique based on the theory of vibration
§6. Breakdown of the classical theory
§7. Planck's new distribution formula
§8. Planck's effort to found the new formula
§9. Discovery of energy quantum
3.
Einstein's Light Quantum
xiii

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1
3
3
3
4
6
6
8
12
12

14
16
17
21
23
24
26
27
33


xiv

4.

Contents

§1.
§2.
§3.
§4.
§5.
§6.
§7.
§8.
§9.
§10.
§11.
The


Photoelectric effect
33
Einstein's critique of the classical theory
34
Derivation of the radiation formula from the classical theory . 35
Einstein's consideration of Planck's theory
37
Two limits of Planck's formula
38
Meaning of Wien's formula
38
Treatment of radiation by means of the kinetic theory of gases 41
Stokes' law and the light quantum
43
Photoelectric effect and the light quantum
44
Ionization of molecule
47
Einstein's analysis of Planck's theory
48
Light Quantum and the Theory of Relativity
50

Chapter 2. The Formation of Atomic Models
1.
On Line Spectra
§1. Experiments on line spectra
§2. Mathematical law of line spectra
§3. Attempts of deriving spectral laws from dynamical
models

2.
Ether Model of the Atom
§1. Atomism
§2. Vortex atoms
3.
Clarification by the Theory of Electrons
§1. Lorentz's theory of electrons
§2. The Zeeman effect
§3. Explanation by Lorentz's theory of electrons
§4. Experimental verification by Zeeman of Lorentz's theory . . .
§5. Measurement of e/m
§6. Larmor's theory
4.
Model of the Atom without Nucleus
§1. Study of cathode rays by J. J. Thomson
§2. Thomson's atomic model in terms of corpuscles
§3. On the electric charge of electron
§4. Kelvin's model of the atom
§5. Permanence and magnetism of atoms
§6. The Thomson model of the atom
5.
Model of the Atom with Nucleus

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55
55
55
60
66

73
73
75
77
77
85
88
91
93
94
100
100
106
110
115
127
138
162


Contents

xv

§1. Lenard's Dynamiden atom
§2. Nagaoka's Saturn-like atom
§3. Arguments against Schott's critique
6.
Examinations of Both Models and the Determination of Atomic
Model

§1. About the number of electrons in an atom-I
§2. Origin of atomic mass
§3. Investigations of oscillations in atoms and of intensities of
spectral lines
§4. The determination by Rutherford
§5. About the number of electrons in an atom-II
Concluding Remarks
Explanatory Notes

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162
168
179
182
182
195
197
207
221
231
234


Volume I

The Formation of Atomic Models

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Chapter 1

Quantum of Radiation

1.
§1.

Thermodynamical Investigation of Black Body Radiation 1 )
The prehistory

of thermal

radiation

Let us first take investigations of the emission or absorption of thermal energy
by a body at a certain temperature by means of radiation into or from space,
namely, investigations of the fact, for example, that the hotter a solid body
is heated, the more intensive the heat rays emitted by it becomes, and at a
temperature slightly above 500° C (that is at 525° C) it begins to emit red rays
of light which are perceptible by eyes, the emitted rays becoming whiter and
whiter at higher temperatures. About the investigation of radiations of such
nature, there was early Newton's well-known law of cooling (1701), and there
were found afterward Lambert's law (1760) of the relation of the flow of light
to the plane passed through by it, and Prevost's law (1792) of heat exchange,
which says that every one of bodies at one and the same temperature exchanging heat among them by means of radiation, emits an amount of radiation
equal to that of radiation it receives from the other bodies.

In 19th century the theory of heat developed fast, and the knowledge of
thermal radiation advanced rapidly along with the establishment of the wave
theory of light, so that the identity of the ray of heat with the ray of light was
' ' O n this subject there has already been published an excellent book by Kiyoshi Amano
"The Origin of the Theory of Thermal Radiation and Quantum Theory" (Dai-Nippon Pub.
Co., 1943; in Japanese), which the reader is recommended to refer to. In the present section, we shall thus be concerned mainly in the question of logical development. Cf. also
Takuzo Sakai "Thermal Radiation" (Iwanami Lecture Series in Physics, Vol. VI B, 1939; in
Japanese), in which the matter under consideration is described in a historical way.
3

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4

Quantum

of

Radiation

made clear. There appeared the discovery of infrared rays by Herschel (1800),
Dulong-Petit's law of cooling (1817), Ritchie's experiment on emissive power
and absorptive power (1833), Ampere's consideration of a displacement law of
thermal radiation. In thermodynamics, Carnot presented the Carnot cycle of
heat engine in 1814, and Mayer the conservation law of energy in 1942. In
the next year 1943 Joule made clear the mechanical equivalence of heat, and
then in 1947 Helmholtz declared the conservation law of energy. The second
law of thermodynamics was proposed by Clausius in 1850 and by W. Thomson
in 1951, and the kinetic theory of gases was shown by Kronig in 1856 and by

Clausius in 1857.
In this way the basis for the study of thermal radiation was nearly established, and from about the middle of the 19th century the knowledge of it was
rapidly deepened, to bring about steady progress in the theoretical research
on it. From nearly the same time the center of research moved into Germany.
This was directly connected with the rapid development of German industry
at that time, as is described in detail in Amano's book cited above. That
is, after the Prussia-France War (1870-71) as the turning point, the weight
of importance of German economy moved from agriculture to industry, and
metallurgical industry, gas industry, illumination industry, etc. demanded and
promoted the study of problems of high temperature and thermal radiation.
§2.

Kirchhoff's

law

It is Kirchhoff's law (1859) that makes the starting point in the theory of
thermal radiation. A similar law was found by Stewart in 1856 and published
in 1861, but it did not play an important role in the development of the study
made mainly in the Continent. Kirchhoff established in 1859 with Bunsen
the method of spectral analysis. In this study it was made clear that line
spectra are characteristics of the respective elements, but he proceeded in the
same year, starting from the inversion of Fraunhofer lines, to the direction of
the problem of the law of emission and absorption of radiation, that is, the
problem of the theory of thermal radiation, of which the basis for theoretical
consideration was already provided by thermodynamics, instead of going to
the direction of the problem of the cause of line spectra. 2 )
2
' T h e problem of line spectra is one of the main themes in our later chapters. The subject
making the central part of both the problems was thus given its starting point by Kirchhoff

in that year.

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Thermodynamical

Investigation

of Black Body

Radiation

5

Kirchhoff developed his theory on the basis of a strict thermodynamical
thought experiment to establish the concept of "black body", and made clear
the law of activity of emission and that of absorption. The concept of black
body played a central role in the later development of the theory of thermal
radiation.
Kirchhoff's law states that the ratio of the emissive power e(A,T) to the
absorptive power a(X, T) of any body which is in the thermal equilibrium at a
temperature, is irrespective of the kind of the matter making up the body, being
a function only of the wavelength A of thermal radiation and the temperature
T. A black body is then defined as an ideal body which completely absorbs
every radiation falling on it, so that the emissive power of any black body
eo(A, T) is a definite function only of A and T regardless of the nature of every
material thing making it. The ratio of the emissive power to the absorptive
power mentioned above of a body in general, is therefore identical with the
emissive coefficient of a black body eo(A,T), because the absorptive power of

any black body ao(X,T) is equal to unity. In other words, one has
a 0 (A,T) = l ,

(la)

e(A, T)/a(X, T) = a universal function = eo(A, T).

(lb)

From these properties the nature of the radiation which is in thermal equilibrium with a black body at a temperature is determined. Such radiation is
called black body radiation.
Kirchhoff's second law states that, when any body is in thermal equilibrium with thermal radiation in vacuum, the thermal radiation is identical, of
whatever kind the body is, with black body radiation.
Thus, the radiation field in a cavity enclosed with a wall which absorbs
completely but does not emit radiation, is concluded to become black body radiation if there is in the cavity an absorptive body, however small its quantity
is. This black body radiation corresponds to the state of maximum entropy.
It was left to later studies to determine the universal function F(X, T) representing e 0 (A,T), which is dependent only on A and T but is independent of
the kind of matter, that is
e 0 (A,T) = F(A,T).

(2)

In the proof of the first law use was made of Kirchhoff's reciprocal theorem.
It states that the amount of black body radiation, which emerges from a surface

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6


Quantum

of

Radiation

taken in the field of black body radiation and arrives at another such surface,
is equal to the amount of black body radiation, which emerges from the latter
surface and arrives at the former surface. The violation of this theorem means
a contradiction with the second law of thermodynamics.
Kirchhoff's law should therefore be said to be the application of the second
law of thermodynamics to thermal radiation.

§3.

Growth of atomistic

viewpoint

The foundation of the theory of thermal radiation was thus established. Then,
in 1860 the law of distribution of velocity of gas molecules was given by
Maxwell, and in 1864 the basic equations of electromagnetic field were proposed by him. At nearly the same time experimental studies of thermal radiation were advanced, and in 1864 the dependence of thermal radiation on
temperature was studied experimentally by Tyndall about platinum. In 1865
the concept of entropy was founded by Clausius, so that the theoretical basis of thermodynamics was established. In 1871 the electromagnetic theory of
light was proposed by Maxwell. In the same year Bartoli treated the problem
of light pressure, discussing its relation to the second law of thermodynamics.
In 1877 there appeared Boltzmann's famous foundation of the second law of
thermodynamics by means of statistical mechanics, and the firm foundation
was thus given to atomism, that aimed to explain phenomena from the substantialistic structure of matter in view of materialistic molecules, contrary to
the phenomenological tendency from the time of Kirchhoff, that is, energetics that rested on thermodynamical bases. Hard controversies were developed

between the two separate schools of energetics and atomism in opposition to
each other. 3 )
Through these controversies, there grew up the theory of thermal radiation
from the atomistic and statistical viewpoint, under the influence of the steady
success of atomism.

§4.

Stefan-Boltzmann's

law

In 1879 the fourth power law of the total radiation energy was presented by
Stefan, which was derived theoretically by Boltzmann in 1884.
3

' K . Amano, ibid., pp. 10, 48.

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Thermodynamical

Investigation

of Black Body

7

Radiation


This law states that the density u of the total radiation energy is proportional to the fourth power of the absolute temperature T:
u = aT4.

(3)

Some confusion arose because Stefan supposed that this law would be valid
in general without taking into account the condition of black body.4) Boltzmann derived with the method of thermodynamics the relation between the
radiation pressure and the temperature, starting from Bartoli's work mentioned above, which showed that the radiation pressure can be treated like
the gas pressure in no contradiction with the second law of thermodynamics.
Boltzmann derived also the explanation of the radiation pressure, by applying
to it Maxwell's revolutionary electromagnetic theory of light. 5 '
4

)K. Amano, ibid., p. 19.
' T h e radiation pressure p is represented by the component of Maxwell's stress normal to
a plane, a plane normal to the x-axis, say. The component is expressed as
5

Px« = ^{El

- El - El) + ^(H2X-

Hi -

Hi),

in terms of the electric and magnetic fields E(EX, Ey,Ez)
and H(HX, Hy, Hz), respectively.
Since the pressure is given by the average over the time pxx of pXx, one makes use of the

relations
j?2 _ S 2 _ p 2 _ * p2

» 2 _ fr2 _ &2 _ 1 fr2

which hold for the respective field quantities on the average, to get
Pxx = - - • — (E2 +H2)
O

= --u

=

-p.

O

o7T

The negative sign before p means that the pressure is in the opposite direction to the normal.
The last equation thus gives the relation between the radiation pressure and the density of
the total radiation energy. One then puts this relation into the second law of thermodynamics
dU + pdV
T
where U, V and S denote the total radiation energy, the total volume and the entropy
respectively. Because U = Vu, one has
dU = d(Vu) = V^-dT
dT

+ udV .


Inserting this and p = u / 3 into dS given above, and using the condition of complete differential
_ a _ 9 5 _ d dS

dVdf

~

dfdV'

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8

Quantum

of

Radiation

In the way the theoretical basis was made clear of the relation between the
total radiation energy and the temperature. Kirchhoff showed that the nature
of black body radiation is determined by a universal function F(X, T) only of
the wavelength and the temperature. He indicated the importance of finding
out this function, saying in the following way:
Though difficulties lie in the experimental determination of it, we may have
a well-founded hope of knowing it by means of experiment. This is because it
will also be almost doubtless a simple function, like all the functions hitherto
known that are independent of the nature of individual bodies have simple

forms. Only after the solution of this problem, the whole effectiveness of the
proposition proved above will become clear.6)
On this function Stefan-Boltzmann's law imposes the condition

L
Đ5.

oo

Wien's displacement

F(X, T)dX = const. ã T 4 .

(4)

law

There arises then the question of finding the dependence of this universal
function on wavelength, that is, the distribution law. Before the complete
discovery of this law, a relation between wavelength and temperature was
found by Wien7) in 1893 as a condition of this function. This relation was
derived from a thermodynamical consideration of the change in wavelength,
which is caused to black body radiation by a moving reflective wall on account
of the Doppler effect. It is called Wien's displacement law, and states that, in
the normal spectrum of black body radiation, every wavelength changes with
temperature holding the product of wavelength by temperature constant. In
other words, if A changes into Ao as T changes into To, o n e has
TX = T0X0 = const..
one gets
du

dT
from the integration of which one obtains

Au

Y'

u = aT*.

6
7

>Cf. Amano, ibid., p. 17.
>W. Wien, Situngsber. d. Berliner Akad. 1893, 55; Wied. Ann. 52 (1893), 132.

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(5)


Thermodynamical

Investigation

of Black Body

Radiation

9


To show this, Wien considers, first, in the case of a box with perfectly
reflective walls enclosing radiation, to move the walls with velocities small
compared with the light velocity so that the density of the radiation energy
may remain homogeneous over the whole space within the box. With the
use of a clever thought experiment, he proves by means of the second law
of thermodynamics that, if the volume of the box which contains initially
black body radiation at a temperature is changed, and the density of the
total radiation energy in it is made equal to that of black body radiation
at another temperature, the spectrum of the density of the radiation energy
in the box is the same as that of black body radiation at this temperature over
the whole range of the wavelength. That is, the radiation in the box preserves
the distribution law of black body radiation.
Wien treats then the case in which one of the walls is a piston moving with
a velocity v small compared with the light velocity c, to calculate the change
in wavelength, or the change in period caused by reflection in the direction
normal to the wall. Let r and r ' be the periods before and after the reflection,
respectively.
According to the Doppler effect one has the relation
'

c-2v

r.
c
Because of r = A/c and r ' = A'/c, this gives the relation
T

=

(6)


A' = ^ ^ A .
(7)
c
Now, let the dependence of the density of radiation energy in the box
on wavelength be given by the function 0(A), so that the density of radiation
energy included between A and X + d\ is expressed as 4>(\)d\. The distribution
of radiation energy after n times of reflection / n (A) is calculated to be

/„(A) = ^ A + y ) ,

(8)

where I is the shortening of wavelength in a single reflection. If the box is
assumed to be a rectangular parallelepiped, and its initial size in the direction
of the movement of the piston is denoted by a, the light is known to make n
round runs in the box while the distance x covered by the piston increases by
dx, where

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