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Wigner's Theorem
Wigner's theorem, proved by Eugene Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT transformations are represented on the Hilbert space of states. The physical states in a quantum theory are represented by unit vectors in Hilbert space up to a phase factor, i.e. by the complex line or ''ray'' the vector spans. In addition, by the Born rule the absolute value of the unit vector's inner product with a unit eigenvector, or equivalently the cosine squared of the angle between the lines the vectors span, corresponds to the transition probability. Ray space, in mathematics known as projective Hilbert space, is the space of all unit vectors in Hilbert space up to the equivalence relation of differing by a phase factor. By Wigner's theorem, any transformation of ray space that preserves the absolute value of the inner products can be represented by a unitary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wigner
Eugene Paul Wigner (, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles". A graduate of the Technical Hochschule Berlin (now Technische Universität Berlin), Wigner worked as an assistant to Karl Weissenberg and Richard Becker at the Kaiser Wilhelm Institute in Berlin, and David Hilbert at the University of Göttingen. Wigner and Hermann Weyl were responsible for introducing group theory into physics, particularly the theory of symmetry in physics. Along the way he performed ground-breaking work in pure mathematics, in which he authored a number of mathematical theorems. In particular, Wigner's theorem is a cornerstone in the mathematical formulation of quantum mechanics. He is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Representation
In the field of representation theory in mathematics, a projective representation of a group ''G'' on a vector space ''V'' over a field ''F'' is a group homomorphism from ''G'' to the projective linear group \mathrm(V) = \mathrm(V) / F^*, where GL(''V'') is the general linear group of invertible linear transformations of ''V'' over ''F'', and ''F''∗ is the normal subgroup consisting of nonzero scalar multiples of the identity transformation (see Scalar transformation). In more concrete terms, a projective representation of G is a collection of operators \rho(g)\in\mathrm(V),\, g\in G satisfying the homomorphism property up to a constant: :\rho(g)\rho(h) = c(g, h)\rho(gh), for some constant c(g, h)\in F. Equivalently, a projective representation of G is a collection of operators \tilde\rho(g)\subset\mathrm(V), g\in G, such that \tilde\rho(gh)=\tilde\rho(g)\tilde\rho(h). Note that, in this notation, \tilde\rho(g) is a ''set'' of linear operators related by multiplication with som ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (, ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term " Fréchet space". Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete nor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Matrices
In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed states. These arise in quantum mechanics in two different situations: # when the preparation of a system can randomly produce different pure states, and thus one must deal with the statistics of possible preparations, and # when one wants to describe a physical system that is entangled with another, without describing their combined state. This case is typical for a system interacting with some environment (e.g. decoherence). In this case, the density matrix of an entangled system differs from that of an ensemble of pure states that, combined, would give the same statistical results upon measurement. Density matrices are thus crucial tools in areas of quantum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Set
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \sim on S, the of an element a in S is denoted /math> or, equivalently, to emphasize its equivalence relation \sim, and is defined as the set of all elements in S with which a is \sim-related. The definition of equivalence relations implies that the equivalence classes form a partition of S, meaning, that every element of the set belongs to exactly one equivalence class. The set of the equivalence classes is sometimes called the quotient set or the quotient space of S by \sim, and is denoted by S /. When the set S has some structure (such as a group operation or a topology) and the equivalence re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric, and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. A simpler example is equality. Any number a is equal to itself (reflexive). If a = b, then b = a (symmetric). If a = b and b = c, then a = c (transitive). Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definitions A binary relation \,\si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Hilbert Space
In mathematics and the foundations of quantum mechanics, the projective Hilbert space or ray space \mathbf(H) of a complex Hilbert space H is the set of equivalence classes /math> of non-zero vectors v \in H, for the equivalence relation \sim on H given by :w \sim v if and only if v = \lambda w for some non-zero complex number \lambda. This is the usual construction of projectivization, applied to a complex Hilbert space. In quantum mechanics, the equivalence classes /math> are also referred to as rays or projective rays. Each such projective ray is a copy of the nonzero complex numbers, which is topologically a two-dimensional plane after one point has been removed. Overview The physical significance of the projective Hilbert space is that in quantum theory, the wave functions \psi and \lambda \psi represent the same ''physical state'', for any \lambda \ne 0. The Born rule demands that if the system is physical and measurable, its wave function has unit norm, \langle\psi, \p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superselection Rule
In quantum mechanics, superselection extends the concept of selection rules. Superselection rules are postulated rules forbidding the preparation of quantum states that exhibit coherence (physics), coherence between eigenstates of certain observables. It was originally introduced by Gian Carlo Wick, Arthur Wightman, and Eugene Wigner to impose additional restrictions to quantum theory beyond those of selection rules. Mathematically speaking, two quantum states \psi_1 and \psi_2 are separated by a selection rule if \langle \psi_1 , H , \psi_2 \rangle = 0 for the given Hamiltonian H , while they are separated by a superselection rule if \langle \psi_1 , A , \psi_2 \rangle = 0 for ''all ''physical observables A . Because no observable connects \langle \psi_1 , and , \psi_2 \rangle they cannot be put into a quantum superposition \alpha , \psi_1 \rangle + \beta , \psi_2 \rangle , and/or a quantum superposition cannot be distinguished from a classical mixture of the two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it were applied once (i.e. P is idempotent). It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object. Definitions A projection on a vector space V is a linear operator P\colon V \to V such that P^2 = P. When V has an inner product and is complete, i.e. when V is a Hilbert space, the concept of orthogonality can be used. A projection P on a Hilbert space V is called an orthogonal projection if it satisfies \langle P \mathbf x, \mathbf y \rangle = \langle \mathbf x, P \mathbf y \rangle for all \mathbf x, \mathbf y \in V. A projection on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ray (quantum Theory)
In mathematics and the foundations of quantum mechanics, the projective Hilbert space or ray space \mathbf(H) of a complex Hilbert space H is the set of equivalence classes /math> of non-zero vectors v \in H, for the equivalence relation \sim on H given by :w \sim v if and only if v = \lambda w for some non-zero complex number \lambda. This is the usual construction of projectivization, applied to a complex Hilbert space. In quantum mechanics, the equivalence classes /math> are also referred to as rays or projective rays. Each such projective ray is a copy of the nonzero complex numbers, which is topologically a two-dimensional plane after one point has been removed. Overview The physical significance of the projective Hilbert space is that in quantum theory, the wave functions \psi and \lambda \psi represent the same ''physical state'', for any \lambda \ne 0. The Born rule demands that if the system is physical and measurable, its wave function has unit norm, \langle\psi, \ps ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase Factor
For any complex number written in polar form (such as ), the phase factor is the complex exponential (), where the variable is the ''phase'' of a wave or other periodic function. The phase factor is a unit complex number, i.e. a complex number of absolute value 1. It is commonly used in quantum mechanics and optics. It is a special case of ''phasors'', which may have arbitrary magnitude (i.e. not necessarily on the unit circle in the complex plane). Multiplying the equation of a plane wave by a phase factor shifts the phase of the wave by : e^ A\,e^ = A\,e^. In quantum mechanics, a phase factor is a complex coefficient that multiplies a ket , \psi\rangle or bra \langle\phi, . It does not, in itself, have any physical meaning, since the introduction of a phase factor does not change the expectation values of a Hermitian operator. That is, the values of \langle\phi, A , \phi\rangle and \langle\phi, A e^ , \phi\rangle are the same. However, ''differences'' in phas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum State
In quantum physics, a quantum state is a mathematical entity that embodies the knowledge of a quantum system. Quantum mechanics specifies the construction, evolution, and measurement of a quantum state. The result is a prediction for the system represented by the state. Knowledge of the quantum state, and the rules for the system's evolution in time, exhausts all that can be known about a quantum system. Quantum states may be defined differently for different kinds of systems or problems. Two broad categories are * wave functions describing quantum systems using position or momentum variables and * the more abstract vector quantum states. Historical, educational, and application-focused problems typically feature wave functions; modern professional physics uses the abstract vector states. In both categories, quantum states divide into pure versus mixed states, or into coherent states and incoherent states. Categories with special properties include stationary states for tim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
