Quantum Theory

This book is devoted to aspects of the foundations of Quantum Mechanics in which probabilistic and statistical concepts play an essential role. The main part of the book concerns the quantitative statistical theory of quantum measurement, based on the notion of Positive Operator-valued Measures.

The quantum computer is no longer the stuff of science fiction. Pioneering physicists are on the brink of unlocking a new quantum universe which provides a better representation of reality than our everyday experiences and common sense ever could. The birth of quantum computers - which, like Schrödinger's famous "dead and alive" cat, rely on entities like electrons, photons, or atoms existing in two states at the same time - is set to turn the computing world on its head.

For many physicists quantum theory contains strong conceptual difficulties, while for others the apparent conclusions about the reality of our physical world and the ways in which we discover that reality remain philosophically unacceptable.

Although ideas from quantum physics play an important role in many parts of modern mathematics, there are few books about quantum mechanics aimed at mathematicians. This book introduces the main ideas of quantum mechanics in language familiar to mathematicians. Readers with little prior exposure to physics will enjoy the book's conversational tone as they delve into such topics as the Hilbert space approach to quantum theory; the Schrödinger equation in one space dimension; the Spectral Theorem for bounded and unbounded self-adjoint operators; the Stone–von Neumann Theorem; the Wentzel–Kramers–Brillouin approximation; the role of Lie groups and Lie algebras in quantum mechanics; and the path-integral approach to quantum mechanics.

Quantum computing explained in terms of elementary linear algebra, emphasizing computation and algorithms and requiring no background in physics.

Relativistic quantum electrodynamics, which describes the electromagneticinteractions of electrons and atomic nuclei, provides the basis for modeling the electronic structure of atoms, molecules and solids and of their interactions with photons and other projectiles. The theory underlying the widely used GRASP relativistic atomic structure program, the DARC electron-atom scattering code and the new BERTHA relativistic molecular structure program is presented in depth, together with computational aspects relevant to practical calculations. Along with an understanding of the physics and mathematics, the reader will gain some idea of how to use these programs to predict energy levels, ionization energies, electron affinities, transition probabilities, hyperfine effects and other properties of atoms and molecules.