Group Theory and Its Applications

Group Theory and its Applications, Volume II covers the two broad areas of applications of group theory, namely, all atomic and molecular phenomena, as well as all aspects of nuclear structure and elementary particle theory.

This volume contains five chapters and begins with the representation and tensor operators of the unitary groups. The next chapter describes wave equations, both Schrödinger’s and Dirac’s for a wide variety of potentials. These topics are followed by discussions of the applications of dynamical groups in dealing with bound-state problems of atomic and molecular physics. A chapter explores the connection between the physical constants of motion and the unitary group of the Hamiltonian, the symmetry adaptation with respect to arbitrary finite groups, and the Dixon method for computing irreducible characters without the occurrence of numerical errors. The last chapter deals with the study of the extension, representation, and applications of Galilei group.

This book will prove useful to mathematicians, practicing engineers, and physicists.

List of Contributors

Preface

Contents of Volume I

The Representations and Tensor Operators of the Unitary Groups U(n)

I. Introduction: The Connections Between the Representation Theory of S(n) and That of U(n), and Other Preliminaries

II. The Group SU(2) and Its Representations

III. The Matrix Elements for the Generators of U(n)

IV. Tensor Operators and Wigner Coefficients on the Unitary Groups

References

Symmetry and Degeneracy

I. Introduction

II. Symmetry of the Hydrogen Atom

III. Symmetry of the Harmonic Oscillator

IV. Symmetry of Tops and Rotators

V. Bertrand's Theorem

VI. Non-Bertrandian Systems

VII. Cyclotron Motion

VIII. The Magnetic Monopole

IX. Two Coulomb Centers

X. Relativistic Systems

XI. Zitterbewegung

XII. Dirac Equation for the Hydrogen Atom

XIII. Other Possible Systems and Symmetries

XIV. Universal Symmetry Groups

XV. Summary

References

Dynamical Groups in Atomic and Molecular Physics

I. Introduction

II. The Second Vector Constant of Motion in Kepler Systems

III. The Four-Dimensional Orthogonal Group and the Hydrogen Atom

IV. Generalization of Fock's Equation: O(5) as a Dynamical Noninvariance Group

V. Symmetry Breaking in Helium

VI. Symmetry Breaking in First-Row Atoms

VII. The Conformal Group and One-Electron Systems

VIII. Conclusion

References

Symmetry Adaptation of Physical States by Means of Computers

I. Introduction

II. Constants of Motion and the Unitary Group of the Hamiltonian

III. Separation of Hilbert Space with Respect to the Constants of Motion

IV. Dixon's Method for Computing Irreducible Characters

V. Computation of Irreducible Matrix Representatives

VI. Group Theory and Computers

References

Galilei Group and Galilean Invariance

I. Introduction

II. The Galilei Group and Its Lie Algebra

III. The Extended Galilei Group and Lie Algebra

IV. Representations of the Galilei Groups

V. Applications to Classical Physics

VI. Applications to Quantum Physics

References

Author Index

Subject Index