Foreword |

1. | Introduction |

2. | Abstract Groups |

3. | Representations of Finite Groups |

4. | Composition of Representations and the Direct Products of Groups |

5. | Wigner's Theorem |

6. | Point Groups |

7. | Decomposition of a Reducible Representation into an Irreducible Representation |

8. | Space Groups and their Irreducible Representations |

9. | Classification of the Vibrational and Electronic States of a Crystal |

10. | Continuous Groups |

11. | Irreducible Representations of the Three-Dimensional Rotation Group |

12. | The Properties of Irreducible Representations of the Rotation Group |

13. | Some Applications of the Theory of Representation of the Rotation Group in Quantum Mechanics |

14. | Additional Degeneracy in a Spherically Symmetric Field |

15. | Permutation Groups |

16. | Symmetrized Powers of Representations |

17. | Symmetry Properties of Multi-Electron Wave Functions |

18. | Symmetry Properties of Wave Functions for a System of Identical Particles with Arbitrary Spins |

19. | Classification of the States of a Multi-Electron Atom |

20. | Applications of Group Theory to Problems Connected with the Perturbation Theory |

21. | Selection Rules |

22. | The Lorentz Group and its Irreducible Representations |

23. | The Dirac Equation |

Appendix to Chapter 7 |

Bibliography |

Preface for the Second Edition

I saw Maria Ivanovna Petrashen for the first time in 1952,
when she gave us, the students of the second year of the Faculty
of Physics of St.Petersburg (at that time the Leningrad) State
University, lectures on Linear
Algebra and Mathematical Physics.
In addition to this general course of mathematics she later gave
a special course on the application of group theory
to quantum mechanics.

As is well known, the main principles of the application of group theory
formalism to quantum mechanics arose at almost the birth of this field of
physics, at the beginning of the 1930s. At this time the famous books by
Hermann Weyl and E.P.Wigner were published. However, even at the middle
of 1950s this method had not circulated widely amongst our theorists, though
it had found an important place in the *Quantum Mechanics * by L.Landau
and E.Lifshitz which had appeared in 1948. The textbook by
G.Ya.Lubarskii "The application of group theory in physics" was
published in 1957.

After graduating M.I.Petrashen was a mathematician, although
her scientific interest always lay in theoretical physics. The first
calculation of atomic structure by the Hartree--Fock method was performed by
V.Fock and M.Petrashen. V.A.Fock, who was her tutor, never used group
theory in his investigations. Nevertheless, just his fundamental works about
the dynamical symmetry of the hydrogen atom and of many-electron wave
functions promoted applications of the group theoretical method in physics.
Apparently M.I.'s interest in group theory were connected with her transition
from atomic to solid state physics.
M.I.'s lectures were interesting to me,
but I must confess that I did not find everything clear, and I needed to
study group theory by reading the third volume of a *Course of Higher
Mathematics * by V.I.Smirnov. M.I. was also a supervisor of the student
seminar, where we each had to make a report on given topics. I was obliged
to read Wigner's paper about space groups and Bethe's paper on splitting
atomic levels in crystals. My increasing interest in the application of
group theory to quantum mechanics was also excited by my aspirant entrance
examination. Professor M.G.Veselov gave me a question on Fock's paper
about symmetry of many-electron wave functions. He had told us about this
work in his lectures, but to reproduce Fock's complicated logic without a
preliminary reading of the paper was too much for me. M.G. gave me
permission to answer at the next day, and I went far into Fock's paper
intently... Later I succeeded in stating the connection between Fock's
properties of a many-electron function and those which follow from an
irreducible representation of the permutation group. This was done in my
first paper in JETP(1959). I was also engaged in `translating' Fock's
paper about the symmetry of the hydrogen atom in four-dimensional space into
group-theoretical language. It was connected with the thesis of my friend of
student days, Yurii Dobronravov, who had died out of time in 1955. I helped
to prepare his work for publication, and it was published in 1956 in *Vestnik Leningrad University*. My thesis, also published in that journal in
1956, was devoted to the irreducible representation of the rotation group in
four-dimensional space. At that time I also made acquaintance with the work
by Yu.N.Demkov on the dynamical symmetry of an isotropic harmonic oscillator
relative to the unitary group in phase space. Later I used this symmetry to
state the analogy between the theory of the M\"ossbauer effect and the theory
of luminescence in impurity crystals (1961), and thus to explain the
existence of zero-phonon lines in optical spectra in solids.

M.I. encouraged me in these studies and suggested I read certain
lectures in her course for students. As a result the idea arose
of writing a textbook reflecting this broader course.
We thought that such a book could be published by the university's
publishing house, but did not make any enquiries -- we decided
to write the text first. The book was finished in 1966.
The first part was written by M.I., the second -- by me.
We continually exchanged our manuscripts and discussed their contents.
On some occasions contradictory points of view arose, and agreement
was reached through the help of M.N.Adamov, who later became
an editor of the book.

M.I. decided to seek the advice of V.I.Smirnov
(who was a head of her Department) about publishing
the book, and he suggested sending the manuscript to the Moscow
publishing house "Nauka".
He read the text and signed a letter of recommendation,
which was also signed by Fock. That was enough for the book
to be accepted by "Nauka" for printing, and it was published in 1967.
Over the next two years we learned that it had been translated
and published in England, France, Germany, and the USA.
This happened without any effort by us, and even with
out our agreement (at that time the Soviet Union had
not accepted the International Copyright Convention).
Soon a positive review appeared in *Physics Today*.
However, we were disappointed that the foreign publishing
houses had not approached us about translating our book,
for, as usual, in the first printing there were some
misprints and inaccuracies, which could have been removed
but unfortunately most of them were preserved in the translations.
For this reason we had thought to prepare a new edition
in Russian, in which we planned to add some new sections.
But a long and difficult illness and subsequent death
of M.I.Petrashen, in 1977, led to forget about the idea
for a long time.

However, it all became possible again when the Moscow
publishing house "URSS"
proposed translating the book into Spanish and agreed to
publish a revised and accomplished version in Russian as well.
The general plan of the book (aimed at a reader's first acquaintance
with the subject) is completely retained. Only a few topics
of major significance have been added; in particular, a section
devoted to the classification of point groups,
using the method of H. Weyl in which the problem is reduced
to the solution of a simple algebraic equation in integers.
A section missing from the first edition is supplied --
the Wigner--Eckart theorem, which is important for many applications.
We illustrate it in the theory of the Zeeman effect.

Many misprints have been removed. In
this connection I thank very much A.V.Tulub, who brought
my attention to misprints in the Appendix, which is now improved.
Inevitably some errors will occur in this second edition,
and I shall be indebted to the attentive reader for criticisms
and remarks. I hope that this new edition will be of interest
not only to those who are just beginning to study the subject
but also to those who have been acquainted with its first edition.

As was pointed out in the Preface to the first edition, the book can be
considered as an introduction to the subject.
For more detailed study of the application of group theory
to different branches of quantum physics, apart from the books
mentioned in the first edition, we recommend the literature
appeared backward [13--28].

*E.D.Trifonov*

This monograph is based on a course of lectures on the
applications of group theory to problems in quantum mechanzics,
given by the authors to undergraduates at the Physics Department
of Leningrad University.

Following a period of scepticism about the value of
group theory as a means of investigating physical systems,
this mathematical theory eventually won a very general
acceptance by physicists. The group-theory formalism is
now widely used in various branches of quantum physics,
including the theory of the atom, the theory of the solid
state, quantum chemistry, and so on. Recent achievements
in the theory of elementary particles, which are intimately
connected with the application of group theory, have intensified
general interest in the possibility of using group-theoretical
methods in physics, and have shown once again the
importance and eminent suitability of such methods in
quantum theory.

A relatively large number of textbooks and monographs
on applications of group theory in physics is already available.
A bibliography is given at the end of the book.

The range of applications of the methods of group
theory to physics is continually expanding, and it is hardly
possible at the present time to produce a monograph which
would cover all these applications. The best course to
adopt, therefore, is to include the relevant applications in
monographs or textbooks devoted to special topics in physics.
This is done, for example, in the well-known course on
theoretical physics by Landau and Lifshits. It is likely that
this tendency will continue in the future.

At the same time, a theoretical physicist should have
a general knowledge of the leading ideas and methods of
group theory as used in physics. Our aim in this course
was to satisfy this need. Moreover, we thought it would be
useful to include in the book a number of problems which
have not been discussed in existing monographs, or treated
in sufficient detail. We refer, above all, to studies of the
symmetry properties of the Schroedinger wave function, to
the explanation of `Additional' degeneracy in the Coulomb
field, and to certain problems in solid-state physics.

In our course, we have restricted our attention to
applications of group theory to quantum mechanics. It follows
that the book can be regarded as the first part of a broader
course, the second part of which should be devoted to applications
of group-theoretical methods to quantum field theory.
We conclude our book with an account of related problems
concerned with the conditions for relativistic invariance in
quantum theory.

We are grateful to M.N.Adamov, who read this monograph
in manuscript and made a number of valuable suggestions,
and to A.G.Zhilich and I.B.Levinson, who reviewed
individual chapters. In the preparation of the manuscript
for press we made use of the kind assistance of A.A.Kiselev,
B.Ya. Frezinskii, R.A.Evarestov, A.A.Berezin and
G.A.Natanzon.