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Stone–von Neumann Theorem
In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. It is named after Marshall Stone and John von Neumann. Representation issues of the commutation relations In quantum mechanics, physical observables are represented mathematically by linear operators on Hilbert spaces. For a single particle moving on the real line \mathbb, there are two important observables: position and momentum. In the Schrödinger representation quantum description of such a particle, the position operator and momentum operator p are respectively given by \begin[] [x \psi](x_0) &= x_0 \psi(x_0) \\[] [p \psi](x_0) &= - i \hbar \frac(x_0) \end on the domain V of infinitely differentiable functions of compact support on \mathbb. Assume \hbar to be a fixed ''non-zero'' real number—in quantum theory \hbar is the reduced Planck's const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Position Operator
In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. When the position operator is considered with a wide enough domain (e.g. the space of tempered distributions), its eigenvalues are the possible position vectors of the particle. In one dimension, if by the symbol , x \rangle we denote the unitary eigenvector of the position operator corresponding to the eigenvalue x, then, , x \rangle represents the state of the particle in which we know with certainty to find the particle itself at position x. Therefore, denoting the position operator by the symbol X in the literature we find also other symbols for the position operator, for instance Q (from Lagrangian mechanics), \hat \mathrm x and so on we can write X, x\rangle = x , x\rangle, for every real position x. One possible realization of the unitary state with position x is the Dirac delta (function) distribution centered at the position x, often denoted by \d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Calculus
In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral theory. (Historically, the term was also used synonymously with calculus of variations; this usage is obsolete, except for functional derivative. Sometimes it is used in relation to types of functional equations, or in logic for systems of predicate calculus.) If f is a function, say a numerical function of a real number, and M is an operator, there is no particular reason why the expression f(M) should make sense. If it does, then we are no longer using f on its original function domain. In the tradition of operational calculus, algebraic expressions in operators are handled irrespective of their meaning. This passes nearly unnoticed if we talk about 'squaring a matrix', though, which is the case of f(x) = x^2 and M an n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone's Theorem On One-parameter Unitary Groups
In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space \mathcal and one-parameter families :(U_)_ of unitary operators that are strongly continuous, i.e., :\forall t_0 \in \R, \psi \in \mathcal: \qquad \lim_ U_t(\psi) = U_(\psi), and are homomorphisms, i.e., :\forall s,t \in \R : \qquad U_ = U_t U_s. Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups. The theorem was proved by , and showed that the requirement that (U_t)_ be strongly continuous can be relaxed to say that it is merely weakly measurable, at least when the Hilbert space is separable. This is an impressive result, as it allows one to define the derivative of the mapping t \mapsto U_t, which is only supposed to be continuous. It is also related to the theory of Lie groups and Lie algebras. Formal statem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Clock And Shift Matrices
''The'' () is a grammatical article in English, denoting persons or things that are already or about to be mentioned, under discussion, implied or otherwise presumed familiar to listeners, readers, or speakers. It is the definite article in English. ''The'' is the most frequently used word in the English language; studies and analyses of texts have found it to account for seven percent of all printed English-language words. It is derived from gendered articles in Old English which combined in Middle English and now has a single form used with nouns of any gender. The word can be used with both singular and plural nouns, and with a noun that starts with any letter. This is different from many other languages, which have different forms of the definite article for different genders or numbers. Pronunciation In most dialects, "the" is pronounced as (with the voiced dental fricative followed by a schwa) when followed by a consonant sound, and as (homophone of the archaic pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Joseph Sylvester
James Joseph Sylvester (3 September 1814 – 15 March 1897) was an English mathematician. He made fundamental contributions to matrix theory, invariant theory, number theory, partition theory, and combinatorics. He played a leadership role in American mathematics in the later half of the 19th century as a professor at the Johns Hopkins University and as founder of the ''American Journal of Mathematics''. At his death, he was a professor at Oxford University. Biography James Joseph was born in London on 3 September 1814, the son of Abraham Joseph, a Jewish merchant. James later adopted the surname Sylvester when his older brother did so upon emigration to the United States—a country which at that time required all immigrants to have a given name, a middle name, and a surname. At the age of 14, Sylvester was a student of Augustus de Morgan at the University of London. His family withdrew him from the University after he was accused of stabbing a fellow student wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normed Algebra
In mathematics, a normed algebra ''A'' is an algebra over a field which has a sub-multiplicative norm: : \forall x,y\in A\qquad \, xy\, \le\, x\, \, y\, . Some authors require it to have a multiplicative identity 1 such that ║1║ = 1. See also * Banach algebra * Composition algebra * Division algebra * Gelfand–Mazur theorem * Hurwitz's theorem (composition algebras) In mathematics, Hurwitz's theorem is a theorem of Adolf Hurwitz (1859–1919), published posthumously in 1923, solving the Hurwitz problem for finite-dimensional unital real non-associative algebras endowed with a positive-definite quadratic fo ... External reading {{algebra-stub Algebras ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmut Wielandt
__NOTOC__ Helmut Wielandt (19 December 1910 – 14 February 2001) was a German mathematician who worked on permutation groups. He was born in Niedereggenen, Lörrach, Germany. He gave a plenary lecture ''Entwicklungslinien in der Strukturtheorie der endlichen Gruppen'' (Lines of Development in the Structure Theory of Finite Groups) at the International Congress of Mathematicians (ICM) in 1958 at EdinburghWielandt, H"Entwicklungslinien in der Strukturtheorie der endlichen Gruppen." In ''Proc. Intern. Congress Math.'', Edinburgh, pp. 268-278. 1958. and was an Invited Speaker with talk ''Bedingungen für die Konjugiertheit von Untergruppen endlicher Gruppen'' (Conditions for the Conjugacy of Finite Groups) at the ICM in 1962 in Stockholm. See also * Collatz–Wielandt formula * Wielandt theorem In mathematics, the Wielandt theorem characterizes the gamma function, defined for all complex numbers z for which \mathrm\,z > 0 by :\Gamma(z)=\int_0^ t^ \mathrm e^\,\mathrm dt, as the o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y. If X and Y are normed vector spaces (a special type of TVS), then L is bounded if and only if there exists some M > 0 such that for all x \in X, \, Lx\, _Y \leq M \, x\, _X. The smallest such M is called the operator norm of L and denoted by \, L\, . A bounded operator between normed spaces is continuous and vice versa. The concept of a bounded linear operator has been extended from normed spaces to all topological vector spaces. Outside of functional analysis, when a function f : X \to Y is called "bounded" then this usually means that its image f(X) is a bounded subset of its codomain. A linear map has this property if and only if it is identically 0. Consequently, in functional analysis, when a linear operator is called "bounded" then it is never meant in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace (linear Algebra)
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proved that for any two matrices and . This implies that similar matrices have the same trace. As a consequence one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). Definition The trace of an square matrix is defined as \operatorname(\mathbf) = \sum_^n a_ = a_ + a_ + \dots + a_ where denotes the entry on the th row and th column of . The entries of can be real numbers or (more generally) complex numbers. The trace is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to distinguish it from other types of dimension. For every vector space there exists a basis, and all bases of a vector space have equal cardinality; as a result, the dimension of a vector space is uniquely defined. We say V is if the dimension of V is finite, and if its dimension is infinite. The dimension of the vector space V over the field F can be written as \dim_F(V) or as : F read "dimension of V over F". When F can be inferred from context, \dim(V) is typically written. Examples The vector space \R^3 has \left\ as a standard basis, and therefore \dim_(\R^3) = 3. More generally, \dim_(\R^n) = n, and even more generally, \dim_(F^n) = n for any field F. The complex numbers \Complex are both a real and complex vector space; we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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