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Mathematical Foundations Of Quantum Mechanics
''Mathematical Foundations of Quantum Mechanics'' () is a quantum mechanics book written by John von Neumann in 1932. It is an important early work in the development of the mathematical formulation of quantum mechanics. The book mainly summarizes results that von Neumann had published in earlier papers. Von Neumman formalized quantum mechanics using the concept of Hilbert spaces and linear operators. He acknowledged the previous work by Paul Dirac on the mathematical formalization of quantum mechanics, but was skeptical of Dirac's use of delta functions. He wrote the book in an attempt to be even more mathematically rigorous than Dirac. It was von Neumann's last book in German, afterwards he started publishing in English. Publication history The book was originally published in German in 1932 by Springer. It was translated into French by Alexandru Proca in 1946, and into Spanish in 1949. An English translation by Robert T. Beyer was published in 1955 by Princeton Universit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Von Neumann
John von Neumann ( ; ; December 28, 1903 – February 8, 1957) was a Hungarian and American mathematician, physicist, computer scientist and engineer. Von Neumann had perhaps the widest coverage of any mathematician of his time, integrating Basic research, pure and Applied science#Applied research, applied sciences and making major contributions to many fields, including mathematics, physics, economics, computing, and statistics. He was a pioneer in building the mathematical framework of quantum physics, in the development of functional analysis, and in game theory, introducing or codifying concepts including Cellular automaton, cellular automata, the Von Neumann universal constructor, universal constructor and the Computer, digital computer. His analysis of the structure of self-replication preceded the discovery of the structure of DNA. During World War II, von Neumann worked on the Manhattan Project. He developed the mathematical models behind the explosive lense ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Function Collapse
In various interpretations of quantum mechanics, wave function collapse, also called reduction of the state vector, occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an ''observation'' and is the essence of a measurement in quantum mechanics, which connects the wave function with classical observables such as position and momentum. Collapse is one of the two processes by which quantum systems evolve in time; the other is the continuous evolution governed by the Schrödinger equation. : In the Copenhagen interpretation, wave function collapse connects quantum to classical models, with a special role for the observer. By contrast, objective-collapse proposes an origin in physical processes. In the many-worlds interpretation, collapse does not exist; all wave function outcomes occur while quantum decoherence accounts for the appearance of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grete Hermann
Grete Hermann (2 March 1901 – 15 April 1984) was a German mathematician and philosopher noted for her work in mathematics, physics, philosophy and education. She is noted for her early philosophical work on the foundations of quantum mechanics, and is now known most of all for an early, but long-ignored critique of the no hidden variables proof by John von Neumann. Mathematics Hermann studied mathematics at Göttingen under Emmy Noether and Edmund Landau, where she achieved her PhD in 1926. Her doctoral thesis, ''The Question of Finitely Many Steps in Polynomial Ideal Theory'' (), published in '' Mathematische Annalen'', is the foundational paper for modern computer algebra. It first established the existence of algorithms (including complexity bounds) for many of the basic problems of abstract algebra, such as ideal membership for polynomial rings. Hermann's algorithm for primary decomposition is still in contemporary use. Assistant to Leonard Nelson From 1925 to 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Positive Operator
In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator A acting on an inner product space is called positive-semidefinite (or ''non-negative'') if, for every x \in \operatorname(A), \langle Ax, x\rangle \in \mathbb and \langle Ax, x\rangle \geq 0, where \operatorname(A) is the domain of A. Positive-semidefinite operators are denoted as A\ge 0. The operator is said to be positive-definite, and written A>0, if \langle Ax,x\rangle>0, for all x\in\mathop(A) \setminus \. Many authors define a positive operator A to be a self-adjoint (or at least symmetric) non-negative operator. We show below that for a complex Hilbert space the self adjointness follows automatically from non-negativity. For a real Hilbert space non-negativity does not imply self adjointness. In physics (specifically quantum mechanics), such operators represent quantum states, via the density matrix formalism. Cauchy–Schwarz inequality Take ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, and , of a group , is the element : . This element is equal to the group's identity if and only if and commute (that is, if and only if ). The set of all commutators of a group is not in general closed under the group operation, but the subgroup of ''G'' generated by all commutators is closed and is called the ''derived group'' or the '' commutator subgroup'' of ''G''. Commutators are used to define nilpotent and solvable groups and the largest abelian quotient group. The definition of the commutator above is used throughout this article, but many group theorists define the commutator as : . Using the first definition, this can be expressed as . Identities (group theory) Commutator identities are an important tool in group th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Observable
In physics, an observable is a physical property or physical quantity that can be measured. In classical mechanics, an observable is a real-valued "function" on the set of all possible system states, e.g., position and momentum. In quantum mechanics, an observable is an operator, or gauge, where the property of the quantum state can be determined by some sequence of operations. For example, these operations might involve submitting the system to various electromagnetic fields and eventually reading a value. Physically meaningful observables must also satisfy transformation laws that relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations that preserve certain mathematical properties of the space in question. Quantum mechanics In quantum mechanics, observables manifest as self-adjoint operators on a separable complex Hilbert space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-adjoint Operator
In mathematics, a self-adjoint operator on a complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. That is, \langle Ax,y \rangle = \langle x,Ay \rangle for all x, y ∊ ''V''. If ''V'' is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of ''A'' is a Hermitian matrix, i.e., equal to its conjugate transpose ''A''. By the finite-dimensional spectral theorem, ''V'' has an orthonormal basis such that the matrix of ''A'' relative to this basis is a diagonal matrix with entries in the real numbers. This article deals with applying generalizations of this concept to operators on Hilbert spaces of arbitrary dimension. Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as position, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hidden Variable Theory
In physics, a hidden-variable theory is a deterministic model which seeks to explain the probabilistic nature of quantum mechanics by introducing additional, possibly inaccessible, variables. The mathematical formulation of quantum mechanics assumes that the state of a system prior to measurement is indeterminate; quantitative bounds on this indeterminacy are expressed by the Heisenberg uncertainty principle. Most hidden-variable theories are attempts to avoid this indeterminacy, but possibly at the expense of requiring that nonlocal interactions be allowed. One notable hidden-variable theory is the de Broglie–Bohm theory. In their 1935 EPR paper, Albert Einstein, Boris Podolsky, and Nathan Rosen argued that quantum entanglement might imply that quantum mechanics is an incomplete description of reality. John Stewart Bell in 1964, in his eponymous theorem proved that correlations between particles under any local hidden variable theory must obey certain constraints. Subse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Many-worlds Interpretation
The many-worlds interpretation (MWI) is an interpretation of quantum mechanics that asserts that the universal wavefunction is Philosophical realism, objectively real, and that there is no wave function collapse. This implies that all Possible world, possible outcomes of quantum measurements are physically realized in different "worlds". The evolution of reality as a whole in MWI is rigidly Determinism, deterministic and principle of locality, local. Many-worlds is also called the relative state formulation or the Everett interpretation, after physicist Hugh Everett III, Hugh Everett, who first proposed it in 1957.Hugh Everett]Theory of the Universal Wavefunction Thesis, Princeton University, (1956, 1973), pp. 1–140. Bryce DeWitt popularized the formulation and named it ''many-worlds'' in the 1970s. See also Cécile DeWitt-Morette, Cecile M. DeWitt, John A. Wheeler (eds,) The Everett–Wheeler Interpretation of Quantum Mechanics, ''Battelle Rencontres: 1967 Lectures in Mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hugh Everett III
Hugh Everett III (; November 11, 1930 – July 19, 1982) was an American physicist who proposed the relative state interpretation of quantum mechanics. This influential approach later became the basis of the many-worlds interpretation (MWI). Everett's theory dropped the wave function collapse postulate of quantum measurement theory, incorporating the observer in the same quantum state as the observation result. The quantum statistic becomes a measure of the branching of the universal wave function. Everett also helped found small companies specializing in contracts with the US government. Although largely disregarded until near the end of his life, Everett's work received more credibility with the discovery of quantum decoherence in the 1970s and has received increased attention in recent decades, with MWI becoming one of the important interpretations of quantum mechanics. Early life and education Hugh Everett III was born in 1930 and raised in the Washington, D.C. area. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Decoherence
Quantum decoherence is the loss of quantum coherence. It involves generally a loss of information of a system to its environment. Quantum decoherence has been studied to understand how quantum systems convert to systems that can be explained by classical mechanics. Beginning out of attempts to extend the understanding of quantum mechanics, the theory has developed in several directions and experimental studies have confirmed some of the key issues. Quantum computing relies on quantum coherence and is one of the primary practical applications of the concept. Concept In quantum mechanics, physical systems are described by a mathematical representation called a quantum state. Probabilities for the outcomes of experiments upon a system are calculated by applying the Born rule to the quantum state describing that system. Quantum states are either ''pure'' or ''mixed''; pure states are also known as ''wavefunctions''. Assigning a pure state to a quantum system implies certai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Consciousness Causes Collapse
The postulate that consciousness causes collapse is an interpretation of quantum mechanics in which consciousness is postulated to be the main mechanism behind the process of measurement in quantum mechanics. It is a historical interpretation of quantum mechanics that is largely discarded by modern physicists. The idea is attributed to Eugene Wigner who wrote about it in the 1960s, but traces of the idea appear as early as the 1930s. Wigner later rejected this interpretation in the 1970s and 1980s. History Earlier work According to Werner Heisenberg recollections in ''Physics and Beyond'', Niels Bohr is said to have rejected the necessity of a conscious observer in quantum mechanics as early as 1927. In his 1932 book '' Mathematical Foundations of Quantum Mechanics'', John von Neumann argued that the mathematics of quantum mechanics allows the collapse of the wave function to be placed at any position in the causal chain from the measurement device to the "subjective perce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
