Quantum field theory, statistical mechanics: Perspectives in statistical mechanics by M. Aizenman Coherent infrared representations in non-relativistic QED by T. Chen and J. Frohlich Quantum spin systems after DLS 1978 by B. Nachtergaele Barry Simon's contributions to quantum field theory by L. Rosen Nonrelativistic two-body and N-body quantum systems, resonances: Isoperimetric inequalities for eigenvalues of the Laplacian by M. S. Ashbaugh and R. D. Benguria Non-self-adjoint operators and pseudospectra by E. B. Davies Barry Simon's contribution to non-relativistic quantum mechanics: Two-body and N-body Schrodinger operators and resonances by R. Froese N-body quantum scattering and quantum resonances: An overview by C. Gerard Mathematical analysis of Born-Oppenheimer approximations by G. A. Hagedorn and A. Joye Perturbation theory and atomic resonances since Schrodinger's time by E. M. Harrell II On a theorem for quantum mechanical scattering theory by A. Lenard Analytic criteria in the qualitative spectral analysis of the Schrodinger operator by V. Maz'ya The spectral geometry of geometrically finite hyperbolic manifolds by P. Perry Topics in the theory of positive solutions of second-order elliptic and parabolic partial differential equations by Y. Pinchover Complex scaling in atomic physics: A staging ground for experimental mathematics and for extracting physics from otherwise impossible computations by W. P. Reinhardt Recent results on the Bethe-Sommerfeld conjecture by A. V. Sobolev Electric and magnetic fields, semiclassical limit: Recent developments in quantum mechanics with magnetic fields by L. Erdos Aspects of the integer quantum Hall effect by G. M. Graf Barry Simon's work on electric and magnetic fields and the semi-classical limit by I. W. Herbst Some bound state problems in quantum mechanics by D. Hundertmark.
We prove that any nonrelativistic classical system must obey a statistical wave equation that is exactly the same as the Schrödinger equation for the system, including the usual “canonical quantization” and Hamiltonian operator, provided an unknown constant is set equal to . We show why the two equations must have exactly the same sets of solutions, whereby this classical statistical theory (CST) and nonrelativistic quantum mechanics may differ only in their interpretations of the same quantitative results. We identify some of the different interpretations. We show that the results also imply nonrelativistic Lagrangian classical mechanics and the associated Newtonian laws of motion. We prove that the CST applied to a nonrelativistic rigid rotator yields spin angular momentum operators that obey the quantum commutation rules and allow both integer and half-odd-integer spin. We also note that the CST applied to systems of identical massive particles is mathematically equivalent to nonrelativistic quantum field theory for those particles.
Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century. States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space. This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators. Part 1 of the book is a concise introduction to the spectral theory of unbounded operators. Only those topics that will be needed for later applications are covered. The spectral theorem is a central topic in this approach and is introduced at an early stage. Part 2 starts with the free Schrodinger equation and computes the free resolvent and time evolution. Position, momentum, and angular momentum are discussed via algebraic methods. Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom. Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory. This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required. In particular, no functional analysis and no Lebesgue integration theory are assumed. It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics. Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature. This new edition has additions and improvements throughout the book to make the presentation more student friendly.