Group theory and quantum mechanics (Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Bercksichtigung der Anwendungsgebiete)

The German edition of this book appeared in 1932 under the title "Die gruppentheoretische Methode in der Quantenmechanik." Its aim was, to explain the fundamental notions of the Theory of Groups and their Representations, and the application of this theory to the Quantum Mechanics of Atoms and Molecules. The book was mainly written for the benefit of physicists who were supposed to be familiar with Quantum Mechanics. However, it turned out that it was also used by. mathematicians who wanted to learn Quantum Mechanics from it. Naturally, the physical parts were too difficult for mathematicians, whereas the mathematical parts were sometimes too difficult for physicists. The German language created an additional difficulty for many readers. In order to make the book more readable for physicists and mathe maticians alike, I have rewritten the whole volume. The changes are most notable in Chapters 1 and 6. In Chapter t, I have tried to give a mathematically rigorous exposition of the principles of Quantum Mechanics. This was possible because recent investigations in the theory of self-adjoint linear operators have made the mathematical foundation of Quantum Mechanics much clearer than it was in t 932. Chapter 6, on Molecule Spectra, was too much condensed in the German edition. I hope it is now easier to understand. In Chapter 2-5 too, numerous changes were made in order to make the book more readable and more useful." Table of Contents......Page 5 Preface......Page 4 1. Wave Functions......Page 7 2. Hilbert Spaces......Page 10 3. Linear Operators......Page 15 4. Hypermaximal Operators......Page 18 5. Separation of Variables......Page 22 6. One Electron in a Central Field......Page 25 7. Perturbation Theory......Page 31 8. Angular Momentum and Infinitesimal Rotations......Page 33 9. Linear Transformations......Page 38 10. Groups......Page 46 11. Equivalence and Reducibility of Representations......Page 52 12. Representations of Abelian Groups. Examples......Page 59 13. Uniqueness Theorems......Page 65 14. Kronecker's Product Transformation......Page 67 15. The Operators Commuting with all Operators of a Given Representation......Page 72 16. Representations of Finite Groups......Page 77 17. Group Characters......Page 84 A. Lie Groups......Page 88 B. One-dimensional Lie Groups and Semi-Groups......Page 89 C. Causality and Translations in Time......Page 92 D. The Lie Algebra of a Lie Group......Page 93 E. Representations of Lie Groups......Page 95 19. The Unitary Groups SU(2) and the Rotation Group O_3......Page 96 20. Representations of the Rotation Group O_3......Page 102 A. The Product Representation \rho_j \times \rho_{j'}......Page 107 B. The Clebsch-Gordan Series......Page 108 C. Applications of (21.1)......Page 113 D. The Reflection Character......Page 115 22. Selection and Intensity Rules......Page 116 A. The Group SL(2) and the Restricted Lorentz Group......Page 120 B. Infinitesimal Transformations......Page 123 C. The Relation between World Vectors and Spinors......Page 126 24. The Spin......Page 129 A. Pauli's Pair of Functions (\psi_1, \psi_2)......Page 131 B. Transformation of the Pair (\psi_1, \psi_2)......Page 132 C. Infinitesimal Rotations......Page 134 D. The Angular Momenta......Page 135 E. The Doublet Splitting of the Alkali Terms......Page 137 26. Dirac's Wave Equation......Page 138 A. Dirac's Equation Rewritten......Page 143 B. Weyl's Equation......Page 146 28. The Several Electron Problem. Multiplet Structure. Zeeman Effect......Page 147 29. The Resonance of Equal Particles......Page 154 30. The Exclusion Principle and the Periodical System......Page 163 31. The Eigenfunctions of the Atom......Page 167 32. The Calculation of the Energy Values......Page 177 33. Pure Spin Functions and their Transformation under Rotations and Permutations......Page 180 34. Representations of the Symmetric Group S_n......Page 188 35. The Quantum Numbers of the Molecule......Page 194 36. The Rotation Levels......Page 201 37. The Case of Two Equal Nuclei......Page 208 Index......Page 210 [by] B. L. Van Der Waerden. Translation Of Die Gruppentheoretische Methode In Der Quantenmechanik.