The book offers a self-contained description of the theory of finite and continuous groups, and their representations, applied to problems from various areas of quantum mechanics, such as the theory of atoms, quantum chemistry, the theory of solids, and relativistic quantum mechanics.

Special attention is paid to the analysis of symmetries of Shroedinger's wave function, and to an explanation of the "additional" degeneracies in a Coulomb field. These, and some other topics which are examined in detail in the book, are difficult to find elsewhere in the literature.

Compared to the previous edition, the book contains the classification of point groups according to Weyl, and a demonstration of the Wigner-Eckart theorem together with its application to the Zeeman effect.

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].

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.