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- ...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
- ...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 ...fined operator) ''X'' of a quantum system <math>\scriptstyle Q</math> with Hilbert space <math>\scriptstyle \mathbb{C}^n</math>, and suppose that ''X'' has a finit8 KB (1,259 words) - 03:56, 22 November 2023
- ...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
- :''An impure state of a quantum system is represented on a [[Hilbert space]] <math>\scriptstyle \mathcal{H}</math> by a non-negative definite [[Trace2 KB (317 words) - 11:30, 12 April 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
- ...ons which form an orthonormal basis follows from the [[compact operator on Hilbert space|spectral theorem for compact operators]]." So, on the surface, it seems the :We consider a Hilbert space ''H'' of complex functions with an inner product32 KB (5,383 words) - 17:15, 20 October 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