量子科学Paramagnetism顺磁性（Paramagnetism）指的是一种材料的磁性状态。有些材料可以受到外部磁场的影响，产生跟外部磁场同样方向的磁化矢量的特性。这样的物质具有正的磁化率。与顺磁性相反的现象被称为抗磁性。
Paramagnetism is a form of magnetism whereby some materials are weakly attracted by an externally applied magnetic field, and form internal, induced magnetic fields in the direction of the applied magnetic field. In contrast with this behavior, diamagnetic materials are repelled by magnetic fields and form induced magnetic fields in the direction opposite to that of the applied magnetic field. Paramagnetic materials include most chemical elements and some compounds; they have a relative magnetic permeability slightly greater than 1 (i.e., a small positive magnetic susceptibility) and hence are attracted to magnetic fields. The magnetic moment induced by the applied field is linear in the field strength and rather weak. It typically requires a sensitive analytical balance to detect the effect and modern measurements on paramagnetic materials are often conducted with a SQUID magnetometer. Paramagnetism is due to the presence of unpaired electrons in the material, so most atoms with incompletely filled atomic orbitals are paramagnetic, although exceptions such as copper exist.
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查看内容许可 ↗居里点（英语：Curie point），又作居里温度（Curie temperature，Tc）或磁性转变点。是指磁性材料中自发磁化强度降到零时的温度，是铁磁性或亚铁磁性物质转变成顺磁性物质的临界点。低于居里点温度时该物质成为铁磁体，此时和材料有关的磁场很难改变。当温度高于居里点时，该物质成为顺磁体，磁体的磁场很容易随周围磁场的改变而改变。这时的磁敏感度约为10−6。居里点由物质的化学成分和晶体结构决定。居里温度是以皮埃尔·居里命名的，他表明在临界温度以上磁性材料会失去磁性。 居里点的温度可以用平均场理论估计。
In physics and materials science, the Curie temperature (TC), or Curie point, is the temperature above which certain materials lose their permanent magnetic properties, which can (in most cases) be replaced by induced magnetism. The Curie temperature is named after Pierre Curie, who showed that magnetism is lost at a critical temperature. The force of magnetism is determined by the magnetic moment, a dipole moment within an atom that originates from the angular momentum and spin of electrons. Materials have different structures of intrinsic magnetic moments that depend on temperature; the Curie temperature is the critical point at which a material's intrinsic magnetic moments change direction. Permanent magnetism is caused by the alignment of magnetic moments, and induced magnetism is created when disordered magnetic moments are forced to align in an applied magnetic field.
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查看内容许可 ↗光电导效应或光电导（英语：Photoconductivity）是电磁波入射到物体表面导致其电导率变化的现象，是内光电效应的一种。 当光被诸如半导体的材料吸收时，电子受到足够能量时将越过能隙被激发至导带。半导体内自由的电子和空穴增加，进而导致电导增加。
Photoconductivity is an optical and electrical phenomenon in which a material becomes more electrically conductive due to the absorption of electromagnetic radiation such as visible light, ultraviolet light, infrared light, or gamma radiation. When light is absorbed by a material such as a semiconductor, the number of free electrons and holes increases, resulting in increased electrical conductivity. To cause excitation, the light that strikes the semiconductor must have enough energy to raise electrons across the band gap, or to excite the impurities within the band gap. When a bias voltage and a load resistor are used in series with the semiconductor, a voltage drop across the load resistors can be measured when the change in electrical conductivity of the material varies the current through the circuit. Classic examples of photoconductive materials include: photographic film: Kodachrome, Fujifilm, Agfachrome, Ilford, etc., based on silver sulfide and silver bromide.
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查看内容许可 ↗光子晶体是由周期性排列的不同折射率的介质制造的规则光学结构。这种材料因为具有光子带隙而能够阻断特定频率的光子，从而影响光子运动。这种影响类似于半导体晶体对于电子行为的影响。由半导体在电子方面的应用，人们推想可以通过光子晶体制造的器件来控制光子运动，例如制造光子计算机。另外，光子晶体也在自然界中发现。
A photonic crystal is an optical nanostructure in which the refractive index changes periodically. This affects the propagation of light in the same way that the structure of natural crystals gives rise to X-ray diffraction and that the atomic lattices (crystal structure) of semiconductors affect their conductivity of electrons. Photonic crystals occur in nature in the form of structural coloration and animal reflectors, and, as artificially produced, promise to be useful in a range of applications. Photonic crystals can be fabricated for one, two, or three dimensions. One-dimensional photonic crystals can be made of thin film layers deposited on each other. Two-dimensional ones can be made by photolithography, or by drilling holes in a suitable substrate. Fabrication methods for three-dimensional ones include drilling under different angles, stacking multiple 2-D layers on top of each other, direct laser writing, or, for example, instigating self-assembly of spheres in a matrix and dissolving the spheres.
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查看内容许可 ↗阴极射线发光（Cathodoluminescence，CL）或阴极发光、阴极射线致发光，是一种冷发光现象，指的是磷光体之类的材料受电子照射时发射出可见光的现象。阴极射线发光常见于老式电视的显像管；其利用电子束在电视屏幕内侧的磷光体上来回扫描，通过控制屏幕上不同区域的发光强度生成图像。
Cathodoluminescence is an optical and electromagnetic phenomenon in which electrons impacting on a luminescent material such as a phosphor, cause the emission of photons which may have wavelengths in the visible spectrum. A familiar example is the generation of light by an electron beam scanning the phosphor-coated inner surface of the screen of a television that uses a cathode-ray tube. Cathodoluminescence is the inverse of the photoelectric effect, in which electron emission is induced by irradiation with photons.
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查看内容许可 ↗卢瑟福背散射分析或卢瑟福背散射谱学（Rutherford Backscattering Spectrometry，RBS），有时候被称为高能离子散射谱学（High-Energy Ion Scattering，HEIS），是一种离子束分析技术，被用在材料科学中，用以分析、测量材料的结构和组成。通过将一束确定能量的高能离子束（通常是质子或α粒子）打到待分析材料上，检测背向反射的离子的能量，即可确定靶原子的种类、浓度和深度分布。 卢瑟福背散射分析的基本原理详见卢瑟福散射。
Rutherford backscattering spectrometry (RBS) is an analytical technique used in materials science. Sometimes referred to as high-energy ion scattering (HEIS) spectrometry, RBS is used to determine the structure and composition of materials by measuring the backscattering of a beam of high energy ions (typically protons or alpha particles) impinging on a sample.
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查看内容许可 ↗穆斯堡尔效应（Mössbauer effect），即原子核辐射的无反冲共振吸收。这个效应首先是由德国物理学家穆斯堡尔于1958年首次在实验中实现的，因此被命名为穆斯堡尔效应。其主要应用是穆斯堡尔谱学。 理论上，当一个原子核由激发态跃迁到基态，发出一个γ射线光子。当这个光子遇到另一个同样的原子核时，就能够被共振吸收。但是实际情况中，处于自由状态的原子核要实现上述过程是困难的。因为原子核在放出一个光子的时候，自身也具有了一个反冲动量，这个反冲动量会使光子的能量减少。同样原理，吸收光子的原子核光子由于反冲效应，吸收的光子能量会有所增大。这样造成相同原子核的发射谱和吸收谱有一定差异，所以自由的原子核很难实现共振吸收。迄今为止，人们还没有在气体和不太粘稠的液体中观察到穆斯堡尔效应。
The Mössbauer effect, or recoilless nuclear resonance fluorescence, is a physical phenomenon, named after Rudolf Mössbauer who investigated it in 1958. It involves the resonant and recoil-free emission and absorption of gamma radiation by atomic nuclei bound in a solid. Its main application is in Mössbauer spectroscopy. In the Mössbauer effect, a narrow resonance for nuclear gamma emission and absorption results from the momentum of recoil being delivered to a surrounding crystal lattice rather than to the emitting or absorbing nucleus alone. When this occurs, no gamma energy is lost to the kinetic energy of recoiling nuclei at either the emitting or absorbing end of a gamma transition: emission and absorption occur at the same energy, resulting in strong, resonant absorption.
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查看内容许可 ↗原子核磁矩是质子和中子固有的磁矩，分别记为μp和μn。原子核由质子和中子组成，这两种核子都表现出类似微小磁体的特性。中子的磁矩以及质子磁矩的巨大数值表明，核子并非基本粒子。它们的磁性强弱通过磁矩来度量。核子通过核力或其磁矩与常规物质发生相互作用，而带正电的质子还会额外通过库仑力参与相互作用。 质子的磁矩于1933年由汉堡大学的奥托·施特恩团队测得。虽然中子的磁矩在20世纪30年代中期通过间接方法被确定，但路易斯·阿尔瓦雷茨和费利克斯·布洛赫在1940年首次直接测得中子的磁矩。
The nucleon magnetic moments are the intrinsic magnetic dipole moments of the proton and neutron, symbols μp and μn . The nucleus of an atom comprises protons and neutrons, both nucleons that behave as small magnets. Their magnetic strengths are measured by their magnetic moments. The nucleons interact with normal matter through either the nuclear force or their magnetic moments, with the charged proton also interacting by the Coulomb force. The proton's magnetic moment was directly measured in 1933 by Otto Stern team in University of Hamburg. While the neutron was determined to have a magnetic moment by indirect methods in the mid-1930s, Luis Alvarez and Felix Bloch made the first accurate, direct measurement of the neutron's magnetic moment in 1940. The proton's magnetic moment is exploited to make measurements of molecules by proton nuclear magnetic resonance. The neutron's magnetic moment is exploited to probe the atomic structure of materials using scattering methods and to manipulate the properties of neutron beams in particle accelerators.
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查看内容许可 ↗Bra–ket 表示法或 Dirac 表示法是有限维和无限维情况下复向量空间及其对偶空间上线性代数和线性算子的数学表示法。它是专门为简化量子力学中经常出现的计算类型而设计的。现在它在该主题中已普遍使用。括号表示法是由保罗·狄拉克（Paul Dirac）在其1930年牛津大学出版的专着《量子力学原理》中创建的。名称来源于英文单词“bracket”。
Bra–ket notation or Dirac notation is a mathematical notation for linear algebra and linear operators on complex vector spaces together with their dual spaces both in the finite- and infinite-dimensional cases. It is specifically designed to ease the types of calculations that frequently arise in quantum mechanics. It is now of ubiquitous usage in that subject. Bra–ket notation was created by Paul Dirac in his monograph, "The Principles of Quantum Mechanics" published by Oxford University 1930. The name comes from the English word bracket.
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查看内容许可 ↗在数学中，特别是在泛函分析中，投影值测度是一种映射，其将给定集合的特定子集映射为给定的希尔伯特空间上的一个自伴投影算子。 投影值测度 (projection-valued measure, PVM) 在形式上类似于实值测度，不过其值是自伴投影而不是实数。与普通测度一样，也可以关于PVM进行复值函数的积分；这种积分的结果是给定希尔伯特空间上的线性算子。 投影值测度用于表达谱理论中的结果，例如自伴算子的谱定理，在这种情况下 PVM 有时被称为谱测度。自伴算子的博雷尔函数演算是通过关于 PVM 的积分构造的。在量子力学中，PVM 提供了投影测量的数学表述，它们可推广为正算子值测度（POVM），正如混合态或密度矩阵推广了纯态的概念一样。
In mathematics, particularly in functional analysis, a projection-valued measure, or spectral measure, is a function defined on certain subsets of a fixed set and whose values are self-adjoint projections on a fixed Hilbert space. A projection-valued measure (PVM) is formally similar to a real-valued measure, except that its values are self-adjoint projections rather than real numbers. As in the case of ordinary measures, it is possible to integrate complex-valued functions with respect to a PVM; the result of such an integration is a linear operator on the given Hilbert space. Projection-valued measures are used to express results in spectral theory, such as the important spectral theorem for self-adjoint operators, in which case the PVM is sometimes referred to as the spectral measure. The Borel functional calculus for self-adjoint operators is constructed using integrals with respect to PVMs. In quantum mechanics, PVMs are the mathematical description of projective measurements.
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查看内容许可 ↗在数学中，完全正映射的 Choi 定理是对有限维（矩阵）C* 代数之间的完全正映射进行分类的结果。这个 1975 年定理是由 Man-Duen Choi 提出的。 Choi 定理的无限维代数推广被称为完全正映射的 Belavkin 的“Radon-Nikodym”定理。
In mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional (matrix) C*-algebras. This 1975 theorem is due to Man-Duen Choi. An infinite-dimensional algebraic generalization of Choi's theorem is known as Belavkin's "Radon–Nikodym" theorem for completely positive maps.
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查看内容许可 ↗张量（英语：Tensor）在数学中是一个代数对象，描述了与矢量空间相关的代数对象集之间的多重线性映射。张量可以作为不同的对象之间的映射，例如矢量、标量以及其他张量。张量有很多种类型，包括标量和矢量、对偶矢量、矢量空间之间的多重线性映射，甚至还有一些运算，例如点积。张量的定义独立于任何基，尽管它们通常由与特定坐标系相关的基中的分量来表示；这些分量形成一个数组，可以将其视为高维矩阵。 n {\displaystyle n} 维空间上的 r {\displaystyle r} 阶张量有 n r {\displaystyle n^{r}} 个分量， r {\displaystyle r} 也称为该张量的秩（与矩阵的秩和阶均无关系）。
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix.
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查看内容许可 ↗线性映射（英语：linear map）是向量空间之间，保持向量加法和标量乘法的函数。线性映射也是向量空间作为模的同态。 线性算子（英语：linear operator）与线性变换（英语：linear transformation，又称线性变换）是与线性映射相关的惯用名词，但其实际意义存在许多分歧，详见相关名词一节。
In mathematics, and more specifically in linear algebra, a linear map, linear mapping, or linear operator is a particular kind of function between vector spaces, which respects the basic operations of vector addition and scalar multiplication. A standard example of a linear map is an m × n {\displaystyle m\times n} matrix, which takes vectors in n {\displaystyle n} -dimensions into vectors in m {\displaystyle m} -dimensions in a way that is compatible with addition of vectors, and multiplication of vectors by scalars. When the two vector spaces are the same, a linear map is also called a linear transformation or linear endomorphism. A linear map is a homomorphism of vector spaces.
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查看内容许可 ↗矩阵范数（matrix norm）亦译矩阵模是数学中矩阵论、线性代数、泛函分析等领域中常见的基本概念，是将一定的矩阵空间建立为赋范向量空间时为矩阵装备的范数。应用中常将有限维赋范向量空间之间的映射以矩阵的形式表现，这时映射空间上装备的范数也可以通过矩阵范数的形式表达。
In mathematics, a norm is a function from a vector space to non-negative real numbers that satisfies certain axioms. A matrix norm is a norm defined on a vector space of matrices. As for every norm, a matrix norm defines a distance, the distance between two matrices being the norm of their difference. The specificity of matrix norms is that they may often be defined either directly as for every norm, or by mean of their properties as linear operators. There are many specific matrix norms. Most arise from the following three perspectives, though different perspectives may sometimes yield the same norm. Consider the matrix as a linear operator; then a matrix norm may describe how much the operator can stretch vectors. Such matrix norms induced by vector norms are called operator norms. Consider the matrix as a rectangular array of numbers; then a matrix norm may be defined as a function of the entries of the matrix. Such matrix norms are sometimes called "entry-wise" norms. The singular value decomposition is useful in analyzing matrices.
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查看内容许可 ↗数学中，特别是泛函分析中，作用于希尔伯特空间X、Y之间的紧算子 T : X → Y {\displaystyle T:X\rightarrow Y} 的奇异值是自伴算子 T ∗ T {\displaystyle T^{*}T} （ T ∗ {\displaystyle T^{*}} 表示T的伴随）的非负特征值的平方根。 奇异值是非负实数，一般按递减顺序排列（ σ 1 ( T ) , σ 2 ( T ) , … {\displaystyle \sigma _{1}(T),\ \sigma _{2}(T),\dots } ）。最大的奇异值 σ 1 ( T ) {\displaystyle \sigma _{1}(T)} 等于T的算子范数（见极小-极大定理）。
In mathematics, in particular in functional analysis, the singular values of a compact operator T : X → Y {\displaystyle \,T\!:X\rightarrow Y} acting between Hilbert spaces X {\displaystyle X} and Y {\displaystyle Y} , are the square roots of the (necessarily non-negative) eigenvalues of the self-adjoint operator T ∗ T {\displaystyle T^{*}T} (where T ∗ {\displaystyle T^{*}} denotes the adjoint of T {\displaystyle T} ).
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查看内容许可 ↗柯西-施瓦茨不等式（也称为柯西-布尼亚科夫斯基-施瓦茨不等式）是内积空间中两个向量之间的内积绝对值的上限，以向量范数的乘积表示。它被认为是数学中最重要和最广泛使用的不等式之一。向量的内积可以描述有限和（通过有限维向量空间）、无限级数（通过序列空间中的向量）和积分（通过希尔伯特空间中的向量）。总和不等式由 Augustin-Louis Cauchy (1821) 发表。相应的积分不等式由 Viktor Bunyakovsky (1859) 和 Hermann Schwarz (1888) 发表。施瓦茨给出了积分版本的现代证明。
The Cauchy–Schwarz inequality (also called Cauchy–Bunyakovsky–Schwarz inequality) is an upper bound on the absolute value of the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is considered one of the most important and widely used inequalities in mathematics. Inner products of vectors can describe finite sums (via finite-dimensional vector spaces), infinite series (via vectors in sequence spaces), and integrals (via vectors in Hilbert spaces). The inequality for sums was published by Augustin-Louis Cauchy (1821). The corresponding inequality for integrals was published by Viktor Bunyakovsky (1859) and Hermann Schwarz (1888). Schwarz gave the modern proof of the integral version.
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维基百科条目作者 · 获取于 2026-10-04 · CC BY-SA 4.0。简介经过纯文本提取与截取；两个语言版本的内容侧重可能不同。用于概念速查，不替代标准原文。 本条中文为英文百科简介的机器辅助翻译，请结合英文原文核对专业术语。
查看内容许可 ↗在数学中，正交性是垂直于双线性形式的线性代数的几何概念的推广。当 B ( u , v ) = 0 {\displaystyle B(\mathbf {u} ,\mathbf {v} )=0} 时，具有双线性形式 B {\displaystyle B} 的向量空间的两个元素 u 和 v 是正交的。根据双线性形式，向量空间可能包含零向量、非零自正交向量，在这种情况下，垂直性被双曲正交性取代。在函数空间的情况下，函数族用于形成正交基，例如在正交多项式、正交函数和组合数学的背景下。
In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity to linear algebra of bilinear forms. Two elements u and v of a vector space with bilinear form B {\displaystyle B} are orthogonal when B ( u , v ) = 0 {\displaystyle B(\mathbf {u} ,\mathbf {v} )=0} . Depending on the bilinear form, the vector space may contain null vectors, non-zero self-orthogonal vectors, in which case perpendicularity is replaced with hyperbolic orthogonality. In the case of function spaces, families of functions are used to form an orthogonal basis, such as in the contexts of orthogonal polynomials, orthogonal functions, and combinatorics.
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维基百科条目作者 · 获取于 2026-10-04 · CC BY-SA 4.0。简介经过纯文本提取与截取；两个语言版本的内容侧重可能不同。用于概念速查，不替代标准原文。 本条中文为英文百科简介的机器辅助翻译，请结合英文原文核对专业术语。
查看内容许可 ↗正交基（英语：orthogonal basis）在线性代数中是指内积空间的一些特定基底，其元素（称作基向量）两两正交。若一组正交基的元素均为单位向量（其范数皆为1），则称这组正交基为标准正交基（英语：orthonormal basis，亦称规范正交基）。 无论在有限维还是无限维空间中，正交基的概念都是很重要的。在无限维希尔伯特空间中，正交基不再是哈默尔基，也就是说不是每个元素都可以写成有限个基中元素的线性组合。因此在无限维空间中，正交基应该被更严格地定义为由线性无关而且两两正交的元素组成、张成的空间是原空间的一个稠密子空间（而不是整个空间）的集合。 注意，在没有定义内积的空间中，“正交基”一词是没有意义的。因此，一个具有正交基的巴拿赫空间，就是一个希尔伯特空间。
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V {\displaystyle V} with finite dimension is a basis for V {\displaystyle V} whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space R n {\displaystyle \mathbb {R} ^{n}} is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for R n {\displaystyle \mathbb {R} ^{n}} arises in this fashion. An orthonormal basis can be derived from an orthogonal basis via normalization.
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维基百科条目作者 · 获取于 2026-10-04 · CC BY-SA 4.0。简介经过纯文本提取与截取；两个语言版本的内容侧重可能不同。用于概念速查，不替代标准原文。
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