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Phase Space Formulation
The phase-space formulation of quantum mechanics places the position ''and'' momentum variables on equal footing in phase space. In contrast, the Schrödinger picture uses the position ''or'' momentum representations (see also position and momentum space). The two key features of the phase-space formulation are that the quantum state is described by a quasiprobability distribution (instead of a wave function, state vector, or density matrix) and operator multiplication is replaced by a star product. The theory was fully developed by Hilbrand Groenewold in 1946 in his PhD thesis, and independently by Joe Moyal, each building on earlier ideas by Hermann Weyl and Eugene Wigner. The chief advantage of the phase-space formulation is that it makes quantum mechanics appear as similar to Hamiltonian mechanics as possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of the Hilbert space". This formulation is statistical in nature and offers l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both particles and waves (wave–particle duality); and there are limits to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic behavior of nature from the behavior of such ensembles. Statistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscopic parameters that fluctuate about average values and are characterized by probability distributions. This established the fields of statistical thermodynamics and statistical physics. The founding of the field of statistical mechanics is generally credited to three physicists: *Ludwig Boltzmann, who developed the fundamental interpretation of entropy in terms of a collection of microstates *James Clerk Maxwell, who developed models of probability distr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Order
In quantum field theory a product of quantum fields, or equivalently their creation and annihilation operators, is usually said to be normal ordered (also called Wick order) when all creation operators are to the left of all annihilation operators in the product. The process of putting a product into normal order is called normal ordering (also called Wick ordering). The terms antinormal order and antinormal ordering are analogously defined, where the annihilation operators are placed to the left of the creation operators. Normal ordering of a product quantum fields or creation and annihilation operators can also be defined in many #Alternative definitions, other ways. Which definition is most appropriate depends on the expectation values needed for a given calculation. Most of this article uses the most common definition of normal ordering as given above, which is appropriate when taking expectation values using the vacuum state of the creation and annihilation operators. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Husimi Q Representation
The Husimi Q representation, introduced by Kôdi Husimi in 1940, is a quasiprobability distribution commonly used in quantum mechanics to represent the phase space distribution of a quantum state such as light in the phase space formulation. It is used in the field of quantum optics and particularly for tomographic purposes. It is also applied in the study of quantum effects in superconductors. Definition and properties The Husimi Q distribution (called Q-function in the context of quantum optics) is one of the simplest distributions of quasiprobability in phase space. It is constructed in such a way that observables written in ''anti''-normal order follow the optical equivalence theorem. This means that it is essentially the density matrix put into normal order. This makes it relatively easy to calculate compared to other quasiprobability distributions through the formula : Q(\alpha)=\frac\langle\alpha, \hat, \alpha\rangle, which is effectively a trace of the densit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wigner Quasiprobability Distribution
The Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution, after Eugene Wigner and Jean-André Ville) is a quasiprobability distribution. It was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. The goal was to link the wavefunction that appears in Schrödinger's equation to a probability distribution in phase space. It is a generating function for all spatial autocorrelation functions of a given quantum-mechanical wavefunction . Thus, it maps on the quantum density matrix in the map between real phase-space functions and Hermitian operators introduced by Hermann Weyl in 1927, in a context related to representation theory in mathematics (see Weyl quantization). In effect, it is the Wigner–Weyl transform of the density matrix, so the realization of that operator in phase space. It was later rederived by Jean Ville in 1948 as a quadratic (in signal) representation of the local ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomas Curtright
Thomas L. Curtright (born 1948) is a theoretical physicist at the University of Miami. He did undergraduate work in physics at the University of Missouri (B.S., M.S., 1970), and graduate work at Caltech (Ph.D., 1977) under the supervision of Richard Feynman. He has made numerous influential contributions in particle and mathematical physics, notably in supercurrent anomalies, higher-spin fields (Curtright field), quantum Liouville theory, geometrostatic sigma models, quantum algebras, and deformation quantization. Curtright is a Fellow of the American Physical Society, a co-recipient (with Charles Thorn) of the SESAPS Jesse Beams Award, a University of Miami Cooper Fellow, and a recipient of the Distinguished Faculty Scholar Award from the University's Senate. He is also the recipient of Distinguished Alumni Awards from the Department of Physics and Astronomy (2021) and from the College of Arts and Science (2022), University of Missouri at Columbia. He has co-edited ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Fairlie
David B. Fairlie (born in South Queensferry, Scotland, 1935) is a British mathematician and theoretical physicist, Professor Emeritus at the University of Durham (UK). He was educated in mathematical physics at the University of Edinburgh (BSc 1957), and he earned a PhD at the University of Cambridge in 1960, under the supervision of John Polkinghorne. After postdoctoral training at Princeton University and Cambridge, he was lecturer in St. Andrews (1962–64) and at Durham University (1964), retiring as Professor (2000). He has made numerous influential contributions in particle and mathematical physics, notably in the early formulation of string theory, as well as the determination of the weak mixing angle in extra dimensions, infinite-dimensional Lie algebras, classical solutions of gauge theories, higher-dimensional gauge theories, and deformation quantization. He has co-authored several volumes, notablyThomas L Curtright, David B Fairlie, Cosmas K Zachos, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosmas Zachos
Cosmas K. Zachos ( el, Κοσμάς Ζάχος; born 1951) is a theoretical physicist. He was educated in physics (undergraduate A.B. 1974) at Princeton University, and did graduate work in theoretical physics at the California Institute of Technology (Ph.D. 1979 ) under the supervision of John Henry Schwarz. Zachos is an emeritus staff member in the theory group of the High Energy Physics Division of Argonne National Laboratory. He is considered an authority on the subject of phase-space quantization. His early research involved, jointly, the introduction of renormalization geometrostasis, and the so-called FFZ Lie algebra of noncommutative geometry. His thesis work revealed a balancing repulsive gravitational force present in extended supergravity. He is co-author of treatises on quantum mechanics in phase space, Thomas L. Curtright, David B. Fairlie, Cosmas K. Zachos, ''A Concise Treatise on Quantum Mechanics in Phase Space'', (World Scientific, Singapore, 2014) . a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noncommutative Geometry
Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of ''spaces'' that are locally presented by noncommutative algebras of functions (possibly in some generalized sense). A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which xy does not always equal yx; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions. An approach giving deep insight about noncommutative spaces is through operator algebras (i.e. algebras of bounded linear operators on a Hilbert space). Perhaps one of the typical examples of a noncommutative space is the " noncommutative tori", which played a key role in the early development of this field in 1980s and lead to noncommutativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kontsevich Quantization Formula
In mathematics, the Kontsevich quantization formula describes how to construct a generalized ★-product operator algebra from a given arbitrary finite-dimensional Poisson manifold. This operator algebra amounts to the deformation quantization of the corresponding Poisson algebra. It is due to Maxim Kontsevich. Deformation quantization of a Poisson algebra Given a Poisson algebra , a deformation quantization is an associative unital product \star on the algebra of formal power series in , subject to the following two axioms, :\begin f\star g &=fg+\mathcal(\hbar)\\ ,g&=f\star g-g\star f=i\hbar\+\mathcal(\hbar^2) \end If one were given a Poisson manifold , one could ask, in addition, that :f\star g=fg+\sum_^\infty \hbar^kB_k(f\otimes g), where the are linear bidifferential operators of degree at most . Two deformations are said to be equivalent iff they are related by a gauge transformation of the type, :\begin D: A \hbar\to A \hbar \\ \sum_^\infty \hbar^k f_k \mapsto \sum_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decoherence
Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics. If a quantum system were perfectly isolated, it would maintain coherence indefinitely, but it would be impossible to manipulate or investigate it. If it is not perfectly isolated, for example during a measurement, coherence is shared with the environment and appears to be lost with time; a process called quantum decoherence. As a result of this process, quantum behavior is apparently lost, just as e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
