05230 - Group Theory - The Application to Quantum Mechanics [Meijer-Bauer].pdf

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GROUP THEORY
THB APPLICATION TO QUANTUM MECHANICS
PAUL H. E. MEIJER
17Ie
Clltllollc
UniNriity
0/
AIIterlt:G,
Wash'IIgton D.C.
EDMOND BAUBR
Laboratoire
de
Chi",. Physlqw,
Pari3
1962
NOR.TH-HOLLAND PUBLISHING COMPANY -
AMSTERDAM
No part
0/
this book
1IItly
be reproduced In
any
form by print. plwioprint,
microfilm or any othe, means
wlthollt
written
permission
from the publisher
PUBLISHERS :
NORTH-HOLLAND PUBLISHING CO. - AMSTERDAM
SOLE DISTlUBUTORS FOR U.S.A.:
1NIEUCmNCE PUB USHERS,
JOHN WILEY
&
SONS~
a division
of
INC. - NEW YORK
PRINT.ED IN THE NE.THERLANDS
BY N.V. DIJKSTRA'S DRUKKERU
v/H
BOEKDRUKKERU GEBROEDERS HOITSHMA
GJtQN1N9EN
PREFACE
Seldom has an application of so-called pure mathematics to mathematical
physics had more appeal than the use of group theory to quantum mecha-
nics. Almost every student in this subject, after going through the necessary
theorems, felt the satisfaction of overlooking a broad
field,)
Inastering
it
in
its
complete generality, as an award to his efforts.
In recent years the availability of tables of coefficients has increased the
applicability
of
many
ideas introduced one or
two decades ago and the num-
ber of papers applying the results of representation theory has been steadily
increasing.
The application of group theory to problems in Physics can
be
classified
in two types. As an example of the first type we mention the considerati<;lns
based on
the symmetry of
a
crystal
used to
reduce
the 6
by
6 stress-
strain.:matrix (the generalization of Hooke's
law)~
An example of the same
type
is the heat conduction tensor in a crystal. Instead of nine components the
number of different elements is reduced by the
symmetry
of the crystal and
still further reduced by the Onsager relations (which are based on micros-
copiq time reversal). Still another example of the first type is the set of piezo-
electric constants .
T~e
second type of problems are those eigenvalue problems where the
differential equation or the boundary
are
of such a geometric nature that
certain rotations or translations leave the. problem unchanged. In this case
it
may
happen that the eigenvalue connected with the solution of the problem
is degenerated; that is, more than one eigenfunction belongs to the same
eigenvalue.
The central problem of the book is the study of this second type through the
transformation properties of these eigenfunctions. In the first cases the ap-
plication of "group theory" is hardly more than
the
application of
symmetry
considerations.
In
the second case the application of group theory, or ac-
tually the application of representation theory, is a much
tn<?re
essential
matter. The
general idea of the transformation induced in function space
by
a
rotat.ion
(or translation) in configuration space as explained in Chap-
ter 4 is not only useful in
Qlla.nt.wn.
mechanics but in
any
other
eigenvalue
\
vi
PREFACE
problem s.uch as
vibrating
systems (molecules or lattices) or waveguides as
wen.
The crucial point in the developing of representation theory is Schur's
Lemma. The proof has been illustrated with a symbolic diagram
and
in
subsequent sections the theory is developed on the basis of this lemma.
Great value is attached to represent the ideas in a geometrical fashion:
For instance
the similarity transformation is
~escribed
as a rotation in mul-
tidimensional function space and the reduction is described in terms of
mutual orth0ional spaces..
Although there
are
many books
written
on
group
theory
as
w~n
as on the-
connection between physics and group theory, the number of books of
in.
troductory
nature are
only very
few. The general references
are
chosen
with
'emphasis on
clarity and
readability
and
are mainly mentioned
for further
study
in
this
field.
The
book is based on a French monograph entitled "Introduction
it.
la
theorie des groupes et ses
applications
en physique
quantique
U
which
appear-
ed
in 1933 in the Annales de I'Institut Henri Poincare. Chapters 1 through
S
are a
translation
with addition of the monograph. The subjects treated in
the additional
chapters
deal
mainly
with
developments since then. The
bas.i~
ideas.
of
the
application of group
theory
to quantum mechanics have not
~onsiderably
changed and hence a fairly literal translation of the material
of the first edition is still valuable today.
the
I would like to quote from
the
introduction
to
tl;le
first edition the
follow-
ing s"4&tement: 'The pages
that
follow are
simple introduction.
having
aa
main~8oal
to familiarize the reader
with-
some
new
concepts
and
to permit
him
to
read
the
oriainal papers
orWell
and Wigner
with
less
difficulty."
I ,_ouId like to
thank
Mrs.
Peretti,
Drs. Morrison and
Barry
and Mrs.
Mielczarek for their cooperation.
J
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