Arising out of the need for Quantum Mechanics (QM) to be part of the common education of mathematics students, this book formulates the mathematical structure of QM in terms of the C*-algebra of observables, which is argued on the basis of the operational definition of measurements and the duality between states and observables.
This book arises out of the need for Quantum Mechanics (QM) to be part of the common education of mathematics students. Rather than starting from the Dirac-Von Neumann axioms, the book offers a short presentation of the mathematical structure of QM using the C--algebraic structure of the observable based on the operational definition of measurements and the duality between states and observables. The description of states and observables as Hilbert space vectors and operators is then derived from the GNS and Gelfand-Naimark Theorems.For finite degrees of freedom, the Weyl algebra codifies the experimental limitations on the measurements of position and momentum (Heisenberg uncertainty relations) and Schroedinger QM follows from the von Neumann uniqueness theorem.The existence problem of the dynamics is related to the self-adjointness of the differential operator describing the Hamiltonian and solved by the Rellich-Kato theorems. Examples are discussed which include the explanation of the discreteness of the atomic spectra.Because of the increasing interest in the relation between QM and stochastic processes, a final chapter is devoted to the functional integral approach (Feynman-Kac formula), the formulation in terms of ground state correlations (Wightman functions) and their analytic continuation to imaginary time (Euclidean QM). The quantum particle on a circle as an example of the interplay between topology and functional integral is also discussed in detail.
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. The numerous exercises at the end of each chapter make the book suitable for both graduate courses and independent study. Most of the text is accessible to graduate students in mathematics who have had a first course in real analysis, covering the basics of L2 spaces and Hilbert spaces. The final chapters introduce readers who are familiar with the theory of manifolds to more advanced topics, including geometric quantization.
The first volume (General Theory) differs from most textbooks as it emphasizes the mathematical structure and mathematical rigor, while being adapted to the teaching the first semester of an advanced course in Quantum Mechanics (the content of the book are the lectures of courses actually delivered.). It differs also from the very few texts in Quantum Mechanics that give emphasis to the mathematical aspects because this book, being written as Lecture Notes, has the structure of lectures delivered in a course, namely introduction of the problem, outline of the relevant points, mathematical tools needed, theorems, proofs. This makes this book particularly useful for self-study and for instructors in the preparation of a second course in Quantum Mechanics (after a first basic course). With some minor additions it can be used also as a basis of a first course in Quantum Mechanics for students in mathematics curricula. The second part (Selected Topics) are lecture notes of a more advanced course aimed at giving the basic notions necessary to do research in several areas of mathematical physics connected with quantum mechanics, from solid state to singular interactions, many body theory, semi-classical analysis, quantum statistical mechanics. The structure of this book is suitable for a second-semester course, in which the lectures are meant to provide, in addition to theorems and proofs, an overview of a more specific subject and hints to the direction of research. In this respect and for the width of subjects this second volume differs from other monographs on Quantum Mechanics. The second volume can be useful for students who want to have a basic preparation for doing research and for instructors who may want to use it as a basis for the presentation of selected topics.
This work covers quantum mechanics by answering questions such as where did the Planck constant and Heisenberg algebra come from, what motivated Feynman to introduce his path integral and why does one distinguish two types of particles, the bosons and fermions. The author addresses all these topics with utter mathematical rigor. The high number of instructive Appendices and numerous Remark sections supply the necessary background knowledge.
Graduate-level text offers unified treatment of mathematics applicable to many branches of physics. Theory of vector spaces, analytic function theory, theory of integral equations, group theory, and more. Many problems. Bibliography.
This marvelous book is aimed at strengthening the mathematical background and sharpening the mathematical tools of students without rigorous training before taking the quantum mechanics course. The abstract construction of quantum postulates in the framework of Hilbert space and Hermitian operators are realized by q-representation in the formulation to demonstrate the conventional approach to quantum theory.Symmetry property is emphasized and extensively explored in this book both in continuous transformations as well as in the discrete ones. The space-time structure is discussed in depth and Dirac equation is formulated by symmetry consideration of Lorentz group.
Presents a comprehensive treatment of quantum mechanics from a mathematics perspective. Including traditional topics, like classical mechanics, mathematical foundations of quantum mechanics, quantization, and the Schrodinger equation, this book gives a mathematical treatment of systems of identical particles with spin.
