Topic
Mathematical formulation of quantum mechanics
About: Mathematical formulation of quantum mechanics is a research topic. Over the lifetime, 1216 publications have been published within this topic receiving 37821 citations.
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Book•
01 Jan 1955
TL;DR: The Mathematical Foundations of Quantum Mechanics as discussed by the authors is a seminal work in theoretical physics that introduced the theory of Hermitean operators and Hilbert spaces and provided a mathematical framework for quantum mechanics.
Abstract: Mathematical Foundations of Quantum Mechanics was a revolutionary book that caused a sea change in theoretical physics. Here, John von Neumann, one of the leading mathematicians of the twentieth century, shows that great insights in quantum physics can be obtained by exploring the mathematical structure of quantum mechanics. He begins by presenting the theory of Hermitean operators and Hilbert spaces. These provide the framework for transformation theory, which von Neumann regards as the definitive form of quantum mechanics. Using this theory, he attacks with mathematical rigor some of the general problems of quantum theory, such as quantum statistical mechanics as well as measurement processes. Regarded as a tour de force at the time of publication, this book is still indispensable for those interested in the fundamental issues of quantum mechanics.
4,908 citations
Book•
01 Jan 1964
TL;DR: R. Shankar has introduced major additions and updated key presentations in this second edition of "Principles of Quantum Mechanics", including an entirely rewritten mathematical introduction, a discussion of Time-reversal invariance, and extensive coverage of a variety of path integrals and their applications.
Abstract: Reviews from the First Edition: 'An excellent text? The postulates of quantum mechanics and the mathematical underpinnings are discussed in a clear, succinct manner' - ("American Scientist"). 'No matter how gently one introduces students to the concept of Dirac's bras and kets, many are turned off. Shankar attacks the problem head-on in the first chapter, and in a very informal style suggests that there is nothing to be frightened of' - ("Physics Bulletin").Reviews of the Second Edition: 'This massive text of 700 and odd pages has indeed an excellent get-up, is very verbal and expressive, and has extensively worked out calculational details - all just right for a first course. The style is conversational, more like a corridor talk or lecture notes, though arranged as a text. It would be particularly useful to beginning students and those in allied areas like quantum chemistry' - ("Mathematical Reviews").R. Shankar has introduced major additions and updated key presentations in this second edition of "Principles of Quantum Mechanics". New features of this innovative text include an entirely rewritten mathematical introduction, a discussion of Time-reversal invariance, and extensive coverage of a variety of path integrals and their applications. Additional highlights include: clear, accessible treatment of underlying mathematics; a review of Newtonian, Lagrangian, and Hamiltonian mechanics; student understanding of quantum theory is enhanced by separate treatment of mathematical theorems and physical postulates; and, unsurpassed coverage of path integrals and their relevance in contemporary physics.The requisite text for advanced undergraduate- and graduate-level students, "Principles of Quantum Mechanics, Second Edition" is fully referenced and is supported by many exercises and solutions. The book's self-contained chapters also make it suitable for independent study as well as for courses in applied disciplines.
1,739 citations
25 Oct 1993
TL;DR: In this article, Abdus Salam argued that the observable degrees of freedom can best be described as if they were Boolean variables defined on a two-dimensional lattice, evolving with time.
Abstract: The requirement that physical phenomena associated with gravitational collapse should be duly reconciled with the postulates of quantum mechanics implies that at a Planckian scale our world is not 3+1 dimensional. Rather, the observable degrees of freedom can best be described as if they were Boolean variables defined on a two-dimensional lattice, evolving with time. This observation, deduced from not much more than unitarity, entropy and counting arguments, implies severe restrictions on possible models of quantum gravity. Using cellular automata as an example it is argued that this dimensional reduction implies more constraints than the freedom we have in constructing models. This is the main reason why so-far no completely consistent mathematical models of quantum black holes have been found. Essay dedicated to Abdus Salam.
1,191 citations
01 Apr 1994
TL;DR: P-adic numbers padic analysis non-Archimedean geometry distribution theory pseudo differential operators and spectral theory p-adic quantum mechanics and representation theory quantum field theory padic strings as discussed by the authors.
Abstract: P-adic numbers p-adic analysis non-Archimedean geometry distribution theory pseudo differential operators and spectral theory p-adic quantum mechanics and representation theory quantum field theory p-adic strings.
1,147 citations
TL;DR: In this article, the general definition of quantization is proposed and two classical systems are considered: the phase space is a Lobachevskii plane and the two-dimensional sphere.
Abstract: The general definition of quantization is proposed. As an example two classical systems are considered. For the first of them the phase space is a Lobachevskii plane, for the second one the two-dimensional sphere.
931 citations