Topics in Quantum Mechanics (Progress in Mathematical Physics, 27)

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.

The theories of quantum fields and strings have had a fruitful impact on certain exciting developments in mathematics and have sparked mathematicians' interest in further understanding some of the basic elements of these grand physical theories. This self-contained text presents quantum mechanics from the point of view of some computational examples with a mixture of mathematical clarity often not found in texts offering only a purely physical point of view. Emphasis is placed on the systematic application of the Nikiforov—Uvarov theory of generalized hypergeometric differential equations to solve the Schrödinger equation and to obtain the quantization of energies from a single unified point of view. This theory is developed and is also used to give a uniform approach to the theory of special functions.
Additional key features:* Considerable material is devoted to the foundations of classical mechanics using conventional mathematical terminology
• The first 10 chapters of Part I cover Planck and Schrödinger quantization, Pauli's spin functions, and an introduction to multielectron atoms
• Part II treats such topics as Feynman path integrals, quantum statistical partition functions, high and low temperature asymptotics of quantum fields of over a negatively curved space-time
• Selected special topics involve some applications of the theory of automorphic forms, zeta functions, the Jacobi inversion formula, spherical harmonic analysis and the Selberg trace formula
• Excellent bibliography and index.

Communication between physicists and mathematicians requires continual bridges to eliminate the divide. This monograph furthers that goal in presenting some new and exciting applications of so-called pure mathematics, including number theory, to various problems arising in physics. An excellent resource for classroom or self-study.

The theories of quantum fields and strings have had a fruitful impact on certain exciting developments in mathematics and have sparked mathematicians' interest in further understanding some of the basic elements of these grand physical theories. This self-contained text presents quantum mechanics from the point of view of some computational examples with a mixture of mathematical clarity often not found in texts offering only a purely physical point of view. Emphasis is placed on the systematic application of the Nikiforov-- Uvarov theory of generalized hypergeometric differential equations to solve the Schr"dinger equation and to obtain the quantization of energies from a single unified point of view. This theory is developed and is also used to give a uniform approach to the theory of special functions. Additional key features:* Considerable material is devoted to the foundations of classical mechanics using conventional mathematical terminology * The first 10 chapters of Part I cover Planck and Schr"dinger quantization, Pauli's spin functions, and an introduction to multielectron atoms * Part II treats such topics as Feynman path integrals, quantum statistical partition functions, high and low temperature asymptotics of quantum fields of over a negatively curved space-time * Selected special topics involve some applications of the theory of automorphic forms, zeta functions, the Jacobi inversion formula, spherical harmonic analysis and the Selberg trace formula * Excellent bibliography and index. Communication between physicists and mathematicians requires continual bridges to eliminate the divide. This monograph furthers that goal in presenting some new and exciting applications of so-called pure mathematics, including number theory, to various problems arising in Front Matter....Pages i-xv Front Matter....Pages 1-1 Units of Measurement....Pages 3-6 Quantum Mechanics: Some Remarks and Themes....Pages 7-23 Equations of Motion in Classical Mechanics....Pages 25-49 Quantization and the Schrödinger Equation....Pages 51-79 Hypergeometric Equations and Special Functions....Pages 81-121 Hydrogen-like Atoms....Pages 123-155 Heisenberg’s Uncertainty Principle....Pages 157-170 Group Representations and Selection Rules....Pages 171-216 The Quantized Hamiltonian for a Charged Particle in an Electromagnetic Field....Pages 217-231 Spin Wave Functions....Pages 233-252 Introduction to Multi-Electron Atoms....Pages 253-267 Front Matter....Pages 269-269 Fresnel Integrals and Feynman Integrals....Pages 271-290 Path Integral for the Harmonic Oscillator....Pages 291-297 Euclidean Path Integrals....Pages 299-305 The Density Matrix and Partition Function in Quantum Statistical Mechanics....Pages 307-315 Zeta Regularization....Pages 317-320 Helmholtz Free Energy for Certain Negatively Curved Space-Times, and the Selberg Trace Formula....Pages 321-332 The Zeta Function of a Product of Laplace Operators and the Multiplicative Anomaly for X Γ d ....Pages 333-339 Schrödinger’s Equation and Gauge Theory....Pages 341-357 Back Matter....Pages 359-398 This book is largely targeted toward a mathematical audience, whose grasp and recollection of principles of physics may vary from small to great.