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- ...somorphic]] to its [[dual space|dual]] and is actually its own dual if the Hilbert space is real. ...ties or "observables" are postulated as [[self-adjoint operator]]s on that Hilbert space. States serve to assign statistical properties to observables of the system2 KB (258 words) - 12:33, 4 January 2009
- 12 bytes (1 word) - 08:14, 13 October 2007
- 67 bytes (8 words) - 12:32, 4 January 2009
- 955 bytes (150 words) - 15:15, 28 July 2009
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- ...somorphic]] to its [[dual space|dual]] and is actually its own dual if the Hilbert space is real. ...ties or "observables" are postulated as [[self-adjoint operator]]s on that Hilbert space. States serve to assign statistical properties to observables of the system2 KB (258 words) - 12:33, 4 January 2009
- {{r|Hilbert space}}423 bytes (60 words) - 15:14, 28 July 2009
- {{r|Hilbert space}}359 bytes (48 words) - 15:04, 28 July 2009
- {{r|Hilbert space}}297 bytes (43 words) - 12:20, 4 January 2009
- {{r|Hilbert space}}347 bytes (48 words) - 14:08, 26 July 2008
- * [[Hilbert space]]982 bytes (148 words) - 07:17, 3 December 2007
- ...2 is special since it is also a [[Hilbert space]] and is in fact the only Hilbert space among the <math>\scriptstyle L^p(\mathbb{T})</math> spaces, <math> \scripts2 KB (317 words) - 13:13, 14 July 2008
- {{r|Hilbert space}}497 bytes (65 words) - 16:06, 11 January 2010
- ...t class of reflexive Banach spaces are the [[Hilbert space]]s, i.e., every Hilbert space is a reflexive Banach space. This follows from an important result known as4 KB (605 words) - 17:25, 20 November 2008
- {{r|Hilbert space}}940 bytes (149 words) - 15:13, 28 July 2009
- {{r|Hilbert space}}1 KB (172 words) - 15:25, 15 May 2011
- ...is defined. A [[completeness|complete]] inner product space is called a [[Hilbert space]].1 KB (204 words) - 14:38, 4 January 2009
- {{r|Hilbert space}}296 bytes (36 words) - 18:51, 12 July 2008
- * [[Hilbert space]]3 KB (441 words) - 12:23, 4 January 2009
- ...tes of a quantum (mechanical) system. The state of a quantum system on a [[Hilbert space]] <math>\scriptstyle \mathcal{H}</math> is represented by a non-negative de ...(\mathcal{H})</math>. Let <math>\scriptstyle \mathcal{K}</math> be another Hilbert space (can be the same as <math>\scriptstyle \mathcal{H}</math>). A quantum opera4 KB (681 words) - 12:41, 22 February 2010
- ...f a complex [[matrix]] to linear operators on [[complex number|complex]] [[Hilbert space|Hilbert spaces]]. In this article the adjoint of a linear operator ''M'' wi ...matics, the ''adjoint'' of a linear operator ''T'' on an arbitrary complex Hilbert space ''H'', with inner product ⟨ ⋅, ⋅ ⟩<sub>''H''</sub>, c5 KB (914 words) - 08:41, 17 October 2009
- ...omain <math>\scriptstyle H_0</math> which is a dense subspace of a complex Hilbert space ''H'' then it is self-adjoint if <math>\scriptstyle A=A^*</math>, where <ma On an infinite dimensional Hilbert space, a self-adjoint operator can be thought of as the analogy of a real symme4 KB (709 words) - 06:58, 23 December 2008
- {{r|Hilbert space}}905 bytes (145 words) - 15:27, 28 November 2008
- {{r|Hilbert space}}598 bytes (75 words) - 20:17, 11 January 2010
- ...of quantum mechanics the state of a system corresponds to a vector in a [[Hilbert space]], so the state <math>|\psi\rangle</math> is analogous to the [[wave functi Let <math>\mathcal{H}</math> be a Hilbert space and <math>\mathcal{H}^*</math> its [[dual space]] (which is [[isomorphic]]4 KB (690 words) - 12:51, 26 March 2011
- .../math> denotes the [[inner product]] between two vectors in the associated Hilbert space), and immediately after measurement the state of the system becomes (or, in ...so a density operator). Density operators are trace class operators on the Hilbert space of the system with unity trace. The density operator corresponding to a pur5 KB (726 words) - 01:57, 30 September 2009
- ...lues is some functional space, usually a [[Banach space|Banach]] or even [[Hilbert space|Hilbert]] space; the function <math> F </math> is then an [[operator]] on t6 KB (951 words) - 05:01, 8 December 2009
- ...in other branches of mathematics (e.g. [[Banach space]]s, in particular [[Hilbert space]]s).6 KB (1,068 words) - 07:30, 4 January 2009
- ...rm an orthogonal basis (along with the [[identity matrix]]) for the real [[Hilbert space]] of 2 × 2 Hermitian matrices and for the complex Hilbert s7 KB (1,096 words) - 05:49, 17 October 2013
- We consider an infinite-dimensional space with an inner product (a [[Hilbert space]]). Let <font style="vertical-align: top"><math>\hat{T}</math></font> be a12 KB (1,903 words) - 10:57, 2 February 2009
- ...whole vector space that they belong to; they do not form a basis of the [[Hilbert space]] that they belong to.8 KB (1,273 words) - 11:29, 9 July 2009
- ...d]], [[Banach space|Banach]], [[Inner product space|inner product]], and [[Hilbert space|Hilbert]] spaces==== ...n ellipsoid. Angles between vectors are defined in inner product spaces. A Hilbert space is defined as a complete inner product space. (Some authors insist that it28 KB (4,311 words) - 08:36, 14 October 2010
- :in the [[Hilbert space]] [[Lebesgue space|''L''<sup>2</sup>([''a'', ''b''],''w''(''x'') ...>, ... of the ''L'' operator. The proper setting for this problem is the [[Hilbert space]] [[Lp space#Weighted Lp spaces|''L''<sup>2</sup>([''a'', ''b''],''w''15 KB (2,332 words) - 04:52, 18 October 2009
- ...cally, using finite-dimensional approximations to the infinite-dimensional Hilbert space. However, the Hilbert space approach treats <math>g_2</math> as an equivalence class of functions rathe32 KB (5,149 words) - 15:48, 29 June 2009
- ...he eigenvectors are orthonormal and complete (form an orthonormal basis of Hilbert space),12 KB (1,893 words) - 04:51, 25 March 2010
- ...roperties of the real numbers. [[Hermitian|Self-adjoint operator]]s on a [[Hilbert space]] (for example, self-adjoint square complex [[matrix |matrices]]) generaliz19 KB (2,948 words) - 10:07, 28 February 2024
- ...A measure that takes values in the set of self-adjoint projections on a [[Hilbert space]] is called a ''[[projection-valued measure]]''; these are used mainly in [14 KB (2,350 words) - 17:37, 10 November 2007
- ...'M'' is usually infinite, the space ''V''<sub>''M''</sub> is a one-boson [[Hilbert space]]. When the temperature ''T'' is fairly high, not many states are occupied13 KB (2,014 words) - 04:59, 1 November 2013
- ...Hilbert space. Since no power of the step up operator maps a ket outside Hilbert space, there must exist a maximum value ''k''<sub>max</sub> of the integer ''k'',16 KB (2,632 words) - 04:33, 23 September 2021
- ...he basis set is as close as possible to a complete basis of one-electron [[Hilbert space]], <math>{\scriptstyle L^2[\mathbb{R}^3]}</math>, but computer time is a pr The AOs and MOs spanning the very same orbital subspace of one-electron Hilbert space, it would be conceivable to skip the Hartree-Fock calculation. However, it14 KB (2,265 words) - 05:37, 6 March 2024
- ...tive space|projectivization]] of a Hilbert space. The exact nature of this Hilbert space is dependent on the system; for example, the state space for position and m ...les are Hermitian operators acting on that space, but do not tell us which Hilbert space or which operators. These must be chosen appropriately in order to obtain a37 KB (5,578 words) - 04:54, 21 March 2024
- ...(\theta,\phi)</math>, form an orthogonal and complete set (a basis of a [[Hilbert space]]) of functions of the spherical polar angles, θ and φ, with ''� It is known from [[Hilbert space]] theory that the expansion (Fourier) coefficients are given by34 KB (5,282 words) - 14:21, 1 January 2011
- *[[Hilbert space]]s18 KB (2,669 words) - 08:38, 17 April 2024
- 36 KB (5,928 words) - 10:21, 8 July 2019
- ...st be included to obtain a complete set, i.e., to span all of one-electron Hilbert space.<ref>This was observed as early as 1929 by E. A. Hylleraas, Z. f. Physik vo19 KB (2,981 words) - 18:31, 3 November 2021
- ...t operator—in its form of a resolution in terms of basis elements of Hilbert space (see [[closure relation]]).56 KB (8,720 words) - 07:31, 20 April 2024
- ...(among other matters) atomic interactions using a mathematical model of [[Hilbert space]], is commonly called incomplete, despite its experimental success, as it i44 KB (6,711 words) - 20:01, 11 October 2013