Topics in Quantum Mechanics

Quantum mechanics and quantum field theory are highly successful physical theo ries that have numerous practical applications. Largely mathematical in character, these theories continue to stimulate the imaginations of applied mathematicians and purists as weIl. In recent years, in particular, as a new array of tools have emerged, including a representative amount from the domain of so-called pure mathematics, interest in both the conceptual and physical aspects of these beau tiful subjects has especially blossomed. Given the emergence of newer and of ten spectacular applications of mathematics to quantum theory, and to theoretical physics in general, one notes that certain communication gaps between physicists and mathematicians continue to be bridged. This text on quantum mechanics, designed primarily for mathematics students and researchers, is an attempt to bridge further gaps. Although the mathematical style presented is generally precise, it is counterbalanced at some points by a re laxation of precision, as our overall purpose is to capture the basic fiavor of the subject both formally and intuitively. The approach is one in which we attempt to maintain sensitivity with respect to diverse backgrounds of the readers, including those with modest backgrounds in physics. Thus we have included several con crete computational examples to fortify stated principles, several appendices, and certain basic physical concepts that help to provide for a reasonably self-contained account of the material, especially in the first 11 chapters.

I Introductory Concepts in Quantum Theory. - 0 Units of Measurement. - 1 Quantum Mechanics: Some Remarks and Themes. - 2 Equations of Motion in Classical Mechanics. - 3 Quantization and the Schrödinger Equation. - 4 Hypergeometric Equations and Special Functions. - 5 Hydrogen-like Atoms. - 6 Heisenberg s Uncertainty Principle. - 7 Group Representations and Selection Rules. - 8 The Quantized Hamiltonian for a Charged Particle in an Electromagnetic Field. - 9 Spin Wave Functions. - 10 Introduction to Multi-Electron Atoms. - II Some Selected Topics. - 11 Fresnel Integrals and Feynman Integrals. - 12 Path Integral for the Harmonic Oscillator. - 13 Euclidean Path Integrals. - 14 The Density Matrix and Partition Function in Quantum Statistical Mechanics. - 15 Zeta Regularization. - 16 Helmholtz Free Energy for Certain Negatively Curved Space-Times, and the Selberg Trace Formula. - 17 The Zeta Function of a Product of Laplace Operators and the Multiplicative Anomaly for X? d. - 18 Schrödinger s Equation and Gauge Theory. - About the Author. - General Appendices. - Appendix A: Some Further Electron Configurations. - Appendix B: Mendeléev Periodic Table. - Appendix C: Determinants for String World-Sheets That Are Tori: Another Example of Zeta Regularization. - Appendix E: Some Informal Comments on QFT. - References.