The formation and logic of quantum mechanics 3 volume set

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 fro

MITUO TAKETANI

Translated from the Japanese and with

Explanatory Notes by Masayuki Nagasaki

World Scientific

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MITUO TAKETANI

Translated from the Japanese and with

Explanatory Notes by Masayuki Nagasaki

V f e World Scientific

w b New Jersey London • Sine New Jersey • London • Singapore • Hong Kong

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World Scientific Publishing Co Pte Ltd

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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 — The Formation of Atomic Models — by Taketani, with

Explanatory Notes by Nagasaki added to its republished

edition,

Vol II — The Way to Quantum Mechanics — by Taketani and

Nagasaki,

Vol Ill — 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 unfor-tunately 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 re-

published 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 Edition

on the three stage theory of scientific cognition, which was presented by tani 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

Take-The latter view was presented in 1934 by Taketani a little before the posal by H Yukawa of the meson theory, and was developed in a close con-nection 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

pro-"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" point) of my autobiography "Tabibito" (Traveler), a fundamental doubt of, or

(Turning-a feeling of diss(Turning-atisf(Turning-action with, the qu(Turning-antum 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 sub-stantialistic 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

at-taching importance to the stereo-structural nature of the logic comprehended

in scientific cognition, instead of the plane-projected view, so to say, of entific 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

sci-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 bution 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,

contri-ix

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Preface

because this subject was regarded as settled already However, the period mentioned is very important when it is seen from the viewpoint of epistem-dolgy 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

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 matical equations But they are put in for the sake of contributing to more

mathe-www.pdfgrip.com

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 F o r m a t i o n of Atomic Models 1

Chapter 1 Quantum of Radiation 3

1 Thermodynamical Investigation of Black Body Radiation 3

§1 The prehistory of thermal radiation 3

§2 Kirchhoff's law 4

§3 Growth of atomistic viewpoint 6

§4 Stefan-Boltzmann's law 6

§5 Wien's displacement law 8

2 Atomistic Investigations of Black Body Radiation 12

§1 Application of atomism 12

§2 Wien's distribution law 14

§3 Planck's effort 16

§4 Foundation by Planck of Wien's distribution formula 17

§5 Rayleigh's critique based on the theory of vibration 21

§6 Breakdown of the classical theory 23

§7 Planck's new distribution formula 24

§8 Planck's effort to found the new formula 26

§9 Discovery of energy quantum 27

3 Einstein's Light Quantum 33

xiii

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xiv Contents

§1 Photoelectric effect 33

§2 Einstein's critique of the classical theory 34

§3 Derivation of the radiation formula from the classical theory 35

§4 Einstein's consideration of Planck's theory 37

§5 Two limits of Planck's formula 38

§6 Meaning of Wien's formula 38

§7 Treatment of radiation by means of the kinetic theory of gases 41

§8 Stokes' law and the light quantum 43

§9 Photoelectric effect and the light quantum 44

§10 Ionization of molecule 47

§11 Einstein's analysis of Planck's theory 48

4 The Light Quantum and the Theory of Relativity 50

Chapter 2 T h e Formation of Atomic Models 55

1 On Line Spectra 55

§1 Experiments on line spectra 55

§2 Mathematical law of line spectra 60

§3 Attempts of deriving spectral laws from dynamical

models 66

2 Ether Model of the Atom 73

§1 Atomism 73

§2 Vortex atoms 75

3 Clarification by the Theory of Electrons 77

§1 Lorentz's theory of electrons 77

§2 The Zeeman effect 85

§3 Explanation by Lorentz's theory of electrons 88

§4 Experimental verification by Zeeman of Lorentz's theory 91

§5 Measurement of e/m 93

§6 Larmor's theory 94

4 Model of the Atom without Nucleus 100

§1 Study of cathode rays by J J Thomson 100

§2 Thomson's atomic model in terms of corpuscles 106

§3 On the electric charge of electron 110

§4 Kelvin's model of the atom 115

§5 Permanence and magnetism of atoms 127

§6 The Thomson model of the atom 138

5 Model of the Atom with Nucleus 162

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Contents xv

§1 Lenard's Dynamiden atom 162

§2 Nagaoka's Saturn-like atom 168

§3 Arguments against Schott's critique 179

6 Examinations of Both Models and the Determination of Atomic

Model 182

§1 About the number of electrons in an atom-I 182

§2 Origin of atomic mass 195

§3 Investigations of oscillations in atoms and of intensities of

spectral lines 197

§4 The determination by Rutherford 207

§5 About the number of electrons in an atom-II 221

Concluding Remarks 231

Explanatory Notes 234

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Volume I

The Formation of Atomic Models

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Chapter 1

Q u a n t u m of Radiation

1 Thermodynamical Investigation of Black B o d y Radiation 1 )

§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 exchang-ing 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 P u b

Co., 1943; in Japanese), which the reader is recommended to refer to In the present tion, we shall thus be concerned mainly in the question of logical development Cf also

sec-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 lished, 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

estab-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 ' 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

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

equi-librium 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

equilib-rium 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

ra-diation 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)

repre-senting e0(A,T), which is dependent only on A and T but is independent of

the kind of matter, that is

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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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 pro-posed by him At nearly the same time experimental studies of thermal ra-diation 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 ba-sis 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 sub-stantialistic structure of matter in view of materialistic molecules, contrary to the phenomenological tendency from the time of Kirchhoff, that is, energet-ics 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

Thermodynamical Investigation of Black Body Radiation 7

This law states that the density u of the total radiation energy is

propor-tional to the fourth power of the absolute temperature T:

Some confusion arose because Stefan supposed that this law would be valid

in general without taking into account the condition of black body.4)

Boltz-mann derived with the method of thermodynamics the relation between the

radiation pressure and the temperature, starting from Bartoli's work

men-tioned 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

5' 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

in terms of the electric and magnetic fields E(E X , E y ,E z ) and H(H X , H y , H z ), respectively

Since the pressure is given by the average over the time p xx of p X x, one makes use of the

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 + udV

dT Inserting this and p = u / 3 into dS given above, and using the condition of complete differ-

ential

_ 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

oo

§5 Wien's displacement 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 ne has

Thermodynamical Investigation of Black Body Radiation

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,

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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10 Quantum of Radiation

(9)

dx c

n = — • — 2(a — x) v The use of this relation for the wavelength after n times of reflection

The integration of this equation gives

Wien applies next the second law of thermodynamics for adiabatic

where U is the total energy in the box when the piston has covered the distance

x, the area of the piston being taken to be unitary The energy density u

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Thermodynamical Investigation of Black Body Radiation 11

increases with the movement of the piston, on account of the decrease in the

volume, on one hand, and of the work done by the piston, on the other hand:

where UQ is the initial energy density

From Eqs (15) and (22), it results that

r = $- (23)

wo A4

Thus, one sees from the description given above, that the distribution law

of the density u of the energy radiated from a body at a temperature T is

identical with that of the energy density of the radiation which is obtained by

compressing the radiation of the energy density UQ at a lower temperature To

Now, according to Stefan-Boltzmann's law one has

12 Quantum of Radiation

The proof given above follows the one-dimensional treatment in Wien's first

paper As was done by Planck, the proof can be given in a general way by

considering reflections of light at a angle to the piston

Wien's displacement law can also be derived more generally to determine

the universal function F(\, T) in a more particular form Between the energy

density u\ considered as a function of wavelength A and the energy density u u

considered as a function of frequency v, there is the relation

The studies of thermal radiation described in the previous section were based

on Maxwell's theory of electromagnetics and the second law of

thermodynam-ics In order to make a further step toward the distribution law, it was

nec-essary in addition to give atomistic consideration This could be developed

on the basis of atomistic viewpoint, which was advanced by Boltzmann and

others to gain victories step by step over the phenomenological and empirical

energetics

As was described in Wien's paper,9) in which his famous distribution law

was presented, any change in wavelength of black body radiation takes place

without consumption and absorption of energy because black body radiation

8' F r o m this one has for the wavelength \ m at which the energy density becomes maximum

the relation

A m T = const

9 )W Wien, Wied Ann 58 (1896), 662

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Atomistic Investigations of Black Body Radiation 13

corresponds to a state of maximum entropy, and the energy distribution in the

spectrum of black body radiation can well be determined from the condition of

maximum entropy, if some process of increasing the entropy is known In the

studies described above, though it is possible to give the entropy of radiation

of given intensity and wavelength, "there is no physical process to bring about

changes in color in a clear-cut way, so that it is impossible to determine the

energy distribution without some assumption" This viewpoint had already

been taken by V A Michelson in Moscow in an attempt made in 1887,10)

before Wien's work on the displacement law Michelson (1) considered that

the continuity of the spectrum of a solid body was caused by the complete

disorder of oscillations of the constituent molecules of the solid body, and

applied Maxwell's law of velocity distribution to the molecules, (2) assumed

that the period r of the oscillation excited by a molecule was related to its

velocity w by the relation r — 4a/w, a being a constant, and (3) supposed

that the intensity of the radiation emitted by molecules was proportional to

the number of the molecules having the same frequency as that of the radiation,

and to an unknown function of the temperature and the kinetic energy of such

a molecule which was assumed to be given by a power of w 2

From these assumptions Michelson obtained for the energy density per unit

wavelength the formula

where B\ and C are constants.n) This formula gives the relation X^T =

const., for the wavelength Am at which the energy density becomes maximum

At that time this relation was considered to be right, because the experimental

result was only that by Langley which turned out incorrect In 1883 H F

Weber obtained with a purely empirical method the experimental formula

ux = tf.l_e.T-i/6W , ( 2 9 )

where C, a and b are constants From this formula he got the relation

X m T = const., showing that Michelson's theoretical formula did not fit the

experimental result This relation is the one that was confirmed theoretically

by Wien in 1893 as the displacement law

10 >V A Michelson, Journ d Phys 6 (1887), 467

11 'Cf also Sec 1, §1, Chap 1, Vol II of the present book

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14 Quantum of Radiation

Though the electromagnetic theory as the theory of emission and tion of radiation was laid its foundation by the experimental proof of electro-magnetic wave done by Hertz in 1888, it could not immediately be applied to the theory of thermal radiation, so that Wien's distribution law should start

absorp-by being given rather unfamiliar assumptions as those in Michelson's theory mentioned above In other words, the development of the theory of thermal radiation was accomplished by building theories on suitable assumptions from the atomistic viewpoint, before the results derived from the classical theory

in strictly logical ways were shown to be incompatible with experiments It was made clear in later studies done by physicists, from Rayleigh to Einstein and Jeans, that the classical theory was not applicable to the phenomena of thermal radiation This may be one of the reasons, that the quantum theory came to be accepted as revolutionary by physicists in general for the first time,

at such a time as 1911 when the first Solvay conference was held

§2 Wien's distribution law

As we have described above, Wien derived in 1896, the distribution law of black body radiation from an atomistic viewpoint In this work, Wien makes use of Maxwell's law of velocity distribution as Michelson Wien considers, however, restricting the excitation of radiation by Stefan-Boltzmann's law and Wien's displacement law, which are obtained with the use of the electromag-netic theory and thermodynamical method, because about it there is so little knowledge as to make any reliable assumption Wien says that "the still re-maining hypothesis can as usual be allowed to return to uncertainty in the theoretical foundation", but its results can be extensively compared with ex-perience, and "the confirmation or refusal by experience will decide the cor-rectness or incorrectness of the hypothesis and in this respect it will be useful for further improvement of the molecular theory" These words of Wien are very noticeable

Because black body radiation is realized in a cavity enclosed by perfectly reflecting walls, when the radiation in it is emitted by a gas having a finite ab-sorptive power for all wavelengths, Wien applies Maxwell's law of velocity dis-tribution to the gas molecules, according to which the number of the molecules

having a velocity between v and v + dv\s proportional to v 2 e~ u l a du, a being

a constant related to the mean velocity, or the absolute temperature Wien says that, though it is entirely unknown how the frequency of the radiation

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Atomistic Investigations of Black Body Radiation 15

emitted by a molecule of velocity v is related to the state of the molecule, "the

view that the electric charge of a molecule excites electromagnetic waves may

be assumed in general at present"

Wien considers first, as an assumption, that every molecule emits radiation

of a certain wavelength and intensity, both of which depend only on its velocity

Though a variety of arbitrary suppositions can be made, he chooses to make

a simple and general supposition as far as possible Because the wavelength

A of the radiation emitted by a molecule of velocity v is a function of v, v is

inversely a function of A Wien assumes thus that the intensity <p\d\ of the

radiation of the wavelengths lying between A and A + dX is proportional to

1 the number of the molecules emitting the radiation of these

wave-lengths,

2 a function of the velocity v, or of the wavelength A

ip\ is therefore given by

where g{\) and /(A) are unknown functions We see that v and hence the

molecules are eliminated in this equation by the supposition of v being a

func-tion of A

Now, from the comparison of Eq (30) with the universal function

given in Eq (27), which is derived from Wien's displacement law, it turns

out that

where C is a constant On the other hand, because of Stefan-Boltzmann's law,

there holds the relation

I g{\)e- c ' XT d\ = const • T4 , (32)

o

from which g(\) can be determined by means of the method of undetermined

coefficients In other words, by inserting the power series expansion of g(X)

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16 Quantum of Radiation

in A into Eq (32), and comparing the both hand sides with the use of the

dimension-less variable y = C/XT, one gets

This distribution formula takes the same form as that found experimentally

at that time by Paschen However, Paschen's experiment was not determinate

because it did not satisfy the condition of black body

§3 Planck's effort

From nearly that time experiments on black body radiation became

prosper-ous in connection with demands by German industry, getting higher accuracy,

as is shown by Amano in his book cited above In particular, in 1895 Wien

and O Lummer showed from Kirchhoff 's law that the cavity radiation realizes

the ideal condition of black body radiation, and that the experiments done in

earlier times were defective in respect to the condition of black body After

their studies experiments on black body radiation came to have a reliable

foun-dation Thus, in 1898 O Lummer and F Kurlbaum accomplished the black

body electric furnace In the same year Kurlbaum published his measurement

of the absolute intensity of black body radiation, and also the study of black

body radiation with the use of remnant infrared rays In 1899 the famous

spec-troscopic measurements of black body radiation by Lummer and Pringsheim

and by Pascen were published

Planck started his efforts at that time In order to find the universal

func-tion proposed by Kirchhoff, Planck considered instead of making an unfamiliar

assumption as Wien did, that black body radiation was realized through the

equilibrium, which was arrived at in a cavity by electromagnetic waves

emit-ted and absorbed by linear oscillators, because that time was just after Hertz's

experiment which made clear that a linear oscillator emits and absorbs

elec-tromagnetic waves Planck thought that for this purpose it was sufficient to

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Atomistic Investigations of Black Body Radiation 17

set up a theory on the basis of electrodynamics etc as they were at that time without the need of being restricted particularly In 1895 Planck started his study of the problem of emission and absorption of electromagnetic waves In this way, he succeeded in getting a general relation between the energy of an oscillator and the energy of radiation, that was independent of the oscillator His theory could not take into account the entropy, however, because it was a theory of reversible processes as long as it was a pure electromagnetic theory When this point was indicated by Boltzmann, Planck introduced the idea of natural radiation, in the same way as Boltzmann introduced the molecular disorder into the second law of thermodynamics That is, Planck supposed that the energy of radiation was distributed completely in disorder among the individual partial oscillations constituting the radiation On this supposition, however, he could not achieve his purpose yet Then, he conversely began to treat the problem from the side of thermodynamics, and became aware of re-lating the entropy of an oscillator to its energy (about in 1899) But, he made

no more progress, because "Planck inclined so much toward phenomenological thinking that he did not made any detailed consideration of the relation be-

tween entropy and probability" , 12 > paying attention only to experimental data

at that time

As Amano shows in his book cited above, in 1891 Planck was contrary together with Ostwald, an advocate of phenomenological energetics, to Boltz-mann, an advocate of atomism, but in 1895 Planck conversely refuted with Boltzmann, a materialist, Ostwald's opinion of "overcoming scientific materi-alism" Planck attacked the energetics as in reality a baseless assertion in spite

of its apparent universality, for the reason that "it could not bring forth at all any new result to be verified by experience" though it would give birth to no contradiction.13)

§4 Foundation by Planck of Wien's distribution formula

About in 1899 Planck was collecting experimental data for the distribution law, but the experiments at that time were all to support in a certain degree Wien's distribution formula Only the experiments on remnant infrared rays showed that Wien's distribution formula would not be valid This possibility was not

1 2' K Amano, ibid., p 50

1 3' K Amano, ibid., p 53 Amano's description of these circumstances is in particular

excellent As a materialist, he gives really a sharp critique of Machism

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18 Quantum of Radiation

affirmed with certainty, however, because these experiments were difficult ones

Because of the fact, in addition to the prevailing belief in Wien's distribution

formula, that Planck succeeded in deriving Wien's formula from Wien's

dis-placement law together with a simple consideration, according to which the

inverse of the second derivative of the entropy of an oscillator with respect to

its energy was proportional to the energy, Planck thought that Wien's

distri-bution formula was a correct law Even in 1900, when it was finally made

clear through the experiments done step by step by Lummer-Pringsheim and

others that Wien's formula did not fit the experimental data, Planck persisted

in Wien's formula

In other words, in a paper in 1900, Planck14) takes the following method

Though the law of energy distribution can be immediately determined, if the

entropy 5 of a linear resonator oscillating in response to radiation is known

as a function of its energy of oscillation, the law of increase of entropy does

not suffice alone to determine this function completely Planck derives it by

calculating in two ways the infinitesimal increase in the entropy of a system of

n identical resonators in a stationary radiation field

Now, suppose that a resonator in a stationary radiation field has the energy

which is different from its stationary value UQ by AC/ If its energy changes by

dU during the time dt, the change in its entropy is given by

where s is the density of the entropy of the radiation field, whose integration

over the whole space gives the entropy of the radiation field Consider that St

14 >M Planck, Ann d Phys 1 (1900), 719

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Atomistic Investigations of Black Body Radiation 19

increases by

if the resonator changes its energy by dU when its energy U has the

infinites-imal increase AU over its stationary value UQ With the use of the equations

of the interaction between the resonator and the radiation field, that is, the

electromagnetic field, Planck obtains the relation

% d 2e !

Thus, dS t depends only on dU, AU and U Moreover, it is proportional to dU

and AU Because dU and AU are shown to be opposite each other in their sign

from the relation between the resonator and the radiation field, Planck puts

~^A = -f(U), (41) 5dU 2 M h K '

where f(U) is a positive function of U dSt is accordingly expressed as

Planck considers the case of n resonators which make independently of

each other the same kind of process Their total energy U n = nU is supposed

to differ by an infinitesimal quantity AU n = nAU from its stationary value

Uo n = nUo, so that

Their total entropy is the sum of their respective entropies, and is given by

S n = nS U n is then supposed to change by dU n = ndU during the time dt

Now, Planck discusses as follows "One could for the moment tend towards

the supposition that S n depends on U n in the same way as S does on U In

that case, it would be possible to get the value of S n by putting U n instead of

U in the expression of S taken as a function of U However, this supposition

could not be founded by any consideration of a physical kind, since no physical

meaning must actually at all be attached to the quantity of entropy, just as

it is the case with the absolute quantity of a potential of force, for example

Rather, only the increase in entropy of the whole system coming in during

the time element dt possesses a definite physical meaning, as this forms the

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Quantum of Radiation

numerical measure for the irreversibility of the process or for the

uncompensa-table transformation of work to heat, and some corresponding conclusion can

actually be applied to or made on this quantity" He then puts forth his basic

idea — "The necessary physical connection between the change in energy and

the increase in entropy would hardly be likely to appear understandably, if

one does not want to assume that the increase in entropy in the case of the

radiation under consideration of the n resonators is completely determined by

their energy U n , its deviation AU n from the stationary value, and the change

in energy dU n occurring in the time dt, and that as a result of this the quantity

of the increase in entropy can then be obtained, if one puts U n instead of U

everywhere in the expression" of dSt given by Eq (42)

In this way, Planck extends Eq (42) to the case of n resonators The new

dSt should be, on the other hand, n times the old dSt, because the n resonators

make identical processes at the same time and independently of each other

Planck thus gives the equation

{-dUAUf(U)} Un = -ndU • AU • f(U), (44)

or

dU n AU n f(U n ) = ndUAUfiU) (45) Putting all U n 's in this equation to be nU, he gets

Atomistic Investigations of Black Body Radiation 21

from which Planck obtains with the use of the second law of thermodynamics

dS/dU = l/T

a being a coefficient Wien's displacement law shows that

a = Au, b = Bu, (51) where A and B are numerical constants Planck arrives thus at Wien's distri-

bution formula This theory seemed to Planck to be natural, because in it any

special assumption such as Wien's one was included and only the resonators

were taken into consideration Wien's distribution formula seemed to Planck

accordingly to be a necessary and natural one

§5 Rayleigh's critique based on the theory of vibration

The year 1900 was eventful to the theory of thermal radiation On one hand,

many experiments were carried out on thermal radiation, and the experiments

on long wavelength rays, especially those on infrared rays, made the deviation

of Wien's formula undeniable On the other hand, Rayleigh15) presented a

formula different from Wien's one, by calculating the number of the proper

vibrations of the radiation field with frequencies between v and v + dv by

analogy with those of air in a cubic box, and by applying the equipartition

law of energy to them, and criticized Wien's theory Rayleigh's formula was

derived by Jeans16) later in 1905 strictly from the classical electromagnetic

theory

In the paper cited above, Rayleigh says that "Paschen finds that his

ob-servations are well represented" by Wien's distribution formula; "nevertheless,

the law seems rather difficult of acceptance, especially the implication that

as the temperature is raised, the radiation of given wave-length approaches

a limit", indicating just the point to be examined Rayleigh thus "ventures

to suggest a modification of Wien's formula" on the basis of the theory of

vibration, which is his specialty He considers that although for some reason

not yet explained the Boltzmann-Maxwell doctrine of the partition of energy,

according to which every mode of vibration should be alike favoured, fails in

general, it seems possible that it may apply to the graver modes In the case of

1 5 )Lord Rayleigh, Phil Mag 4 9 (1900), 539

16 >J H Jeans, Phil Mag 10 (1905), 91

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22 Quantum of Radiation

a string vibrating transversely in one dimensional way, the Boltzmann-Maxwell

law states that the energy should be equally divided among all modes, whose

frequencies are as 1, 2, 3 , (in units of the fundamental frequency) Hence,

if k = 1/A, representing the frequency, the energy between k and k + dk is

represented by dk when k is large enough

Passing to the case of three dimensions, and considering for example the

vibrations of a cubical mass of air, one has the equation

k 2 = p 2 + q 2 + r 2 , (52)

where p, q, r are integers representing the number of subdivisions in the three

dimensions If p, q, r are regarded as the coordinates of points forming a

cubic array, k is the distance of any point from the origin Hence, the number

of points for which k likes between k and k + dk, being proportional to the

volume of the corresponding spherical shell, is represented by k 2 dk, and "this

expresses the distribution of energy according to the Boltzmann-Maxwell law,

so far as regards the wave-length or frequency" Rayleigh concludes thus that

"if we apply this result to radiation, we shall have, since the energy in each

mode is proportional to T,

or, if we prefer it,

Rayleigh makes the suggestion that this may be the proper form than

\~ 5 dA, which results from Wien's formula when AT is great Now, he says

without any reasoning that "if we introduce the exponential factor, the

com-plete expression will be

and that "if, as is probably to be preferred, we make k the independent variable,

this becomes

His introduction of this exponential factor might be made groundlessly, on

account of his idea, described at the beginning of the present subsection, of

suggesting a modification of Wien's formula

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Atomistic Investigations of Black Body Radiation

§6 Breakdown of the classical theory

Rayleigh's idea consisted in taking the condition of the oscillation of radiation

in space into consideration In the theory of thermal radiation, radiation as

an electromagnetic wave was applied for the first time to the calculation of

radiation pressure in the derivation of the Stefan-Boltzmann law, and then to

the Doppler effect in the derivation of Wien's displacement law No application

of the other laws of radiation field was made, until the calculation of the

proper oscillations of radiation in a cavity was made first by Rayleigh, and then

in 1905 by Jeans strictly from the viewpoint of the classical electromagnetic

theory, that is, a theory of continuous wave Thus, Rayleigh-Jean's formula is

to give the spectral distribution of black body radiation according purely to

the classical theory, and expresses the energy density u v dv in the frequency

interval dv as

c 6

Rayleigh-Jean's formula contains a great difficulty In other words, this

formula implies that black body radiation has a great deal of energy at high

frequencies even if the temperature is very low, so that any body of low

tem-perature should appear brilliant, as is seen from the equation of the total

energy17'

— kTdX = oo (58)

This difficulty is called the ultraviolet catastrophe Since Rayleigh made his

study from the standpoint of modifying Wien's formula by taking into

consider-ation the vibrconsider-ations in a cavity, he introduced the exponential factor probably

from a molecular theoretical idea, in order to settle the ultraviolet catastrophe

and make his formula fit the fact that Wien's formula was in accord with the

experimental data for short wavelengths.18) It seems that Rayleigh thought

that the exponential factor could not be deduced from the classical theory

In this way, about in 1900 it was not thought clearly that the classical

elec-tromagnetic theory was in contradiction with black body radiation, even after

1 7 ' T h i s point was pointed out in particular by Einstein, as we shall describe later

experiment

L

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Quantum of Radiation

the proposal of Planck's theory of energy quantum This contradiction became decisively recognized in the studies by Einstein and by Jeans in particular.19)

§7 Planck's new distribution formula

Because it was made clear that Wien's formula was off the experimental data for long wavelengths by Lummer-Pringsheim's work, and in particular

by Rubens-Kurlbaum's result20) of the measurement of remnant infrared rays from rock salt (51/J, in wavelength), Planck21) proceeded to reexamine his for-mer thermodynamical consideration In the new study under consideration, Planck admits that Wien's distribution formula does not possess a general meaning but has at most the character of a limiting law, and accepts now the opinion that "its extremely simple form owes its origin just to a restriction to short wavelengths or low temperatures", contrary to the thought in his former study that Wien's formula should be right because of the simplicity of the thermodynamical equations used for it About Eq (45), that is,

dU n AU n f(U n ) = ndUAUf(U),

which was taken as the starting point in the former study, Planck says "I would consider it well possible, even so not easily understandable and anyway hardly provable, that the expression does not possess generally the meaning ascribed formerly by me to it"

Then, about the mistake22) of having simply applied in a formal way the

expression dU AU f (U) derived for a single resonator to the case of n resonators

in Eq (45), considering that "the values of U n , dU n and U n are not sufficient to

determine the change in entropy in question, but at the same time U must also

be known", he states "Following this thought I have finally come to construct quite arbitrary expressions of the entropy, which, though more complicated than Wien's expression, nevertheless seem to satisfy all the demands of the thermodynamical and electromagnetic theories just as much as that" Planck starts therefore this time by putting semiempirical equations, instead of basing

on theoretical necessities as in the former study

19>Cf K Amano, ibid., p 85

20 >H Rubens und F Kurlbaum, Berlin Ber 1900, 4 1

21 >M Planck, Verhand d Deut Phys Gesel 2 (1900), 202

22)Cf K Amano, ibid., p 169

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