Wave-Mechanics, Supplement to Atomic Structure And Spectral Lines Sommerfeld 3rd. Ed.

319 Pages·1936·18.298 MB·English

by  Arnold Sommerfeld

WAV E-MECHANICS

rrHE AUTHOR'S PREFACE TO THE
GERMAN EDITION
T
HE author believes that he is fulfilling a general wish in
attempting in the present supplementary volume to depict
the recent developments which since 1924 to 1926 (thanks
to the work of L. de Broglie, Heisenberg, Schrodinger) have
transformed the external aspect of atomic physics. The fact that
the inner content of the theory, that is, the quantitative assertions
that can be tested by experiment, has for the most part survived
this process of regeneration is manifest to those acquainted with
the subject. The new development does not signify a radical
change but a welcome evolution of the existing theory, while many
fundamental points are classified and made more precise.
I have called this volume a " wave-mechanical" supplement,
because for practical manipulation Schrodinger's methods are
obviously superior to the specifically " quantum-mechanical"
methods. On the other hand, however, I have left no doubt
that the general ideas that have led Heisenberg to enunciate
quantum-mechanics are also indispensable for the elaboration of
wave-mechanics. The original standpoint of Schrodinger, that
transitions are to occur only between co-existing states, is clearly
too narrow and does not accurately fit the facts. I have there
fore taken over into wave-mechanics the equal treatment of states
and transitions,-as is done by Heisenberg from the very beginning,
-in particular in deriving the frequency-condition and the rules
of polarisation and intensity in § 5, Chap. I. This, of course,
denotes that I am renouncing the more definite wave-kinematic
objective, set up by Schrodinger and de Broglie, and am sacrificing
pictorial representation to formalism. In wave-mechanics the
electron still remains a point-charge ultimately, and the light
quantum a point-like centre of energy. But the dualism between
the light-quantum and the light-wave extends into the corpuscular
v
vi Wave-Mechanics
region; beside the electron-corpuscle we have the electron-wave
with all the accumulative experimental evidence to confirm it.
During several semesters I have endeavoured in my university
lectures to make clear to my class and to myself the chief results
of wave-mechanics in as simple a form as possible. We found
that in all cases that permitted of complete integration the
"polynomial method" is adequate and leads to the final analytical
form. The "method of generating functions," although_ often
very elegant, is here replaced by directly applying the conditions
of orthogonality. My purpose in the present volume was to bring
out such points of view clearly. A more significant simplification
is that which we have been able to achieve in Dirac's theory of.
the electron.
The form in which this supplementary volume is presented
follows directly on the 4th German edition of Atombau und
Spektrallinien. Chapter I contains the fundamental conceptions
and their application to elementary problems, whereas Chapter II
deals with the more difficult calculations of perturbations and dif
fraction and-the most difficult problem of all-the relativistic
theory of the electron. The present volume can of course also
be regarded as a supplement to the English edition, Atomic
Structure and Spectral Lines, to which references are made
where possible.
As in the previous editions of the main volume I have kept in
view both the theoretical and the' experimental physicist as readers.
I have therefore restricted myself to such problems as may claim
direct physical interest. The general speculations of the Theory
of Transformations of Probabilities receive far too small a share
of space, as do the fundamental questions of Indeterminacy and
Observability. Concerning these more general subjects, I under
stand that other accounts by authoritative writers are about to
appear. My wish was to preserve the _original charaqter of my
book, and I therefore kept my attention as much as possible
on concrete questions. I would gladly have treated . afresh the
systematic arrangement of spectra frem the point of ~iew of
wave-mechanics. But there was neither sufficient space nor time
for this; moreover, the theory of the relativisti-0 electron will have
to be further investigated before these things can become sufficiently
clear.
In §§ 4 and 7 of Chapter II on the photo-electric and Compton
effects I have had the pleasure of the collaboration of my colleague,
Mr. F. Kirchner. I am indebted to Dr. S. Bochner for many
Preface vii
mathematical references and suggestions. But my special thanks
are due to my faithful associate, Dr. K. Bechert, who has given
me his untiring help not only in all the technical questions of the
printing and in all the details of the manuscript, but has also
worked out many points of the text independently, particularly in
the final paragraph on electron spin, so that if I had not had his
devoted help it would have scarcely been possible to· bring the
book to completion at the appointed time.
A. SOMMERFELD
MUNICH
Augmt, 1928
ERRATA
P. 172. Second line from bottom, read
qlcj = YkJ (that is, omit Zn)·
P. 173. The equation above (a) should read-
~ fkJ = - 2 ~ Aki qkJ = 1.
J ;
In the following equations, as far as the middle
of p. 17 4, read dn-for dt and -r for t everywhere.
In the last line of p. 173 we must add d-r under the
integral sign.
P. 174. The third equation from the top must be called
(d).

1'RANSLATOR'S PREFACE
S
INCE the appearance of the last edition of the author's
Atombau und Spektrallinien, which is generally recognised
as the standard work on the subject, many striking advances
have been made owing to the birth and growth of quantum- and
wave-mechanics. What is the attitude of the author towards
these new theories? Professor Sommerfeld has endeavoured in
the present supplementary volume to answer this question and to
bring his treatment of the subject completely up to date. The
success of his attempt is clearly demonstrated by the highly
appreciative way in which the German original has been received.
The great number of original researches which have contributed
to these recent advances often appear little connected outwardly;
by-paths which are ventured upon and later abandoned tend to
confuse the average reader whose mathematical attainments may
be insufficient to allow him to form a just estimate of such contri
butions. The masterly and brilliant way in which Professor
Sommerfeld has sifted the highly abstract material and brought
into prominence what is of practical importance to the physicist
will be recognised by all who wish to follow the thread of progress
of atomic physics under his inspiring guidance. He has described
in a particularly convenient form the mathematical machinery
that is necessary for an understanding of the theory.
The present English edition has had the advantage of being
accurately checked and revised by the author. Many additions
and alterations have been made; for example, the last section of
Chapter I has a different form from that which it was given in
the German edition; several oversights in the German edition
have also been corrected. A particularly welcome feature is that
Professor Sommerfeld has corrected and supplemented the section
on the Photo-electric Effect, on which much interest is focussed at
the present time, by adding three Notes, rich in content, which
b ~