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Quantum Science维尔纳州维尔纳态是一个 d 2 {\displaystyle d^{2}} × d 2 {\displaystyle d^{2}} 维二分量子态密度矩阵，在 U ⊗ U {\displaystyle U\otimes U} 形式的所有酉算子下保持不变。
A Werner state is a d 2 {\displaystyle d^{2}} × d 2 {\displaystyle d^{2}} -dimensional bipartite quantum state density matrix that is invariant under all unitary operators of the form U ⊗ U {\displaystyle U\otimes U} .
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View content license ↗ 共形场论（CFT）是一种在共形变换下不变的量子场论。在二维中，存在局部共形变换的无限维代数，共形场论有时可以被精确求解或分类。共形场论在凝聚态物理、统计力学、量子统计力学和弦理论中具有重要的应用。统计和凝聚态物质系统确实在热力学或量子临界点通常具有共形不变性。
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified. Conformal field theory has important applications to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points.
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View content license ↗ 在微分几何（数学领域）中，达布定理是为特殊类别的微分 1-形式提供范式的定理，部分推广了 Frobenius 积分定理。它以 Jean Gaston Darboux 的名字命名，他将其作为普法夫问题的解决方案。它是多个领域的基础性成果，其中最主要的是辛几何。事实上，它的众多后果之一是任何两个相同维度的辛流形彼此局部辛同胚。也就是说，每一个 2 n {\displaystyle 2n} 维辛流形都可以局部看起来像线性辛空间 C n {\displaystyle \mathbb {C} ^{n}} 及其规范辛形式。该定理应用于接触几何也有类似的结果。
In differential geometry, a field in mathematics, Darboux's theorem is a theorem providing a normal form for special classes of differential 1-forms, partially generalizing the Frobenius integration theorem. It is named after Jean Gaston Darboux who established it as the solution of the Pfaff problem. It is a foundational result in several fields, the chief among them being symplectic geometry. Indeed, one of its many consequences is that any two symplectic manifolds of the same dimension are locally symplectomorphic to one another. That is, every 2 n {\displaystyle 2n} -dimensional symplectic manifold can be made to look locally like the linear symplectic space C n {\displaystyle \mathbb {C} ^{n}} with its canonical symplectic form. There is also an analogous consequence of the theorem applied to contact geometry.
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View content license ↗ 在数学和物理学中，变形量化大致相当于寻找一个（量子）代数，其经典极限是给定的（经典）代数，例如李代数或泊松代数。
In mathematics and physics, deformation quantization roughly amounts to finding a (quantum) algebra whose classical limit is a given (classical) algebra such as a Lie algebra or a Poisson algebra.
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View content license ↗ 经典电磁学的协变公式是指以在洛伦兹变换下明显不变的形式，在使用直线惯性坐标系的狭义相对论形式中书写经典电磁学定律（特别是麦克斯韦方程组和洛伦兹力）的方法。这些表达式不仅可以轻松证明经典电磁定律在任何惯性坐标系中都采用相同的形式，而且还提供了一种将场和力从一个坐标系转换到另一个坐标系的方法。然而，这并不像弯曲时空或非直线坐标系中的麦克斯韦方程那么普遍。
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems. These expressions both make it simple to prove that the laws of classical electromagnetism take the same form in any inertial coordinate system, and also provide a way to translate the fields and forces from one frame to another. However, this is not as general as Maxwell's equations in curved spacetime or non-rectilinear coordinate systems.
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View content license ↗ 在数学中，达朗贝尔方程，有时也称为拉格朗日方程，是一阶非线性常微分方程，以法国数学家让·勒隆德·达朗贝尔命名。该方程为 y = x f ( d y d x ) + g ( d y d x ) 。
In mathematics, d'Alembert's equation, sometimes also known as Lagrange's equation, is a first order nonlinear ordinary differential equation, named after the French mathematician Jean le Rond d'Alembert. The equation reads as y = x f ( d y d x ) + g ( d y d x ) .
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View content license ↗ 在数学物理中，Degasperis-Procesi 方程 u t − u x x t + 2 κ u x + 4 u u x = 3 u x u x x + u u x x x {\displaystyle \displaystyle u_{t}-u_{xxt}+2\kappa u_{x}+4uu_{x}=3u_{x}u_{xx}+uu_{xxx}} 是仅有的两个完全可解的方程之一以下三阶、非线性、色散偏微分方程组中的方程：...
In mathematical physics, the Degasperis–Procesi equation u t − u x x t + 2 κ u x + 4 u u x = 3 u x u x x + u u x x x {\displaystyle \displaystyle u_{t}-u_{xxt}+2\kappa u_{x}+4uu_{x}=3u_{x}u_{xx}+uu_{xxx}} is one of only two exactly solvable equations in the following family of third-order, non-linear, dispersive PDEs:...
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View content license ↗ 量子场论中的常见积分是一组公式，可用于量子场论中各种类型的计算，例如配分函数、环图积分等。
Common integrals in quantum field theory are set of formulas that are useful for computation of various types in quantum field theory such as partition function, integrals of loop diagrams, etc.
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View content license ↗ 在数学物理学中，德唐德-韦尔理论是变分法中哈密顿形式主义和时空经典场论的推广，它平等地对待空间和时间坐标。在这个框架中，力学中的哈密顿形式主义被推广到场论，场被表示为在空间和时间上变化的系统。这种概括不同于场论中的经典哈密顿形式主义，后者以不同的方式对待空间和时间变量，并将经典场描述为随时间演化的无限维系统。
In mathematical physics, the De Donder–Weyl theory is a generalization of the Hamiltonian formalism in the calculus of variations and classical field theory over spacetime which treats the space and time coordinates on equal footing. In this framework, the Hamiltonian formalism in mechanics is generalized to field theory in the way that a field is represented as a system that varies both in space and in time. This generalization is different from the canonical Hamiltonian formalism in field theory which treats space and time variables differently and describes classical fields as infinite-dimensional systems evolving in time.
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View content license ↗ 在李群理论中，指数映射是从李群G的李代数g到G的映射。如果G是矩阵李群，则指数映射简化为矩阵指数。指数映射表示为 exp:g → G，是解析映射，并且具有导数 d/dtexp(X(t)):Tg → TG，其中 X(t) 是李代数中的 C1 路径，以及密切相关的微分 dexp:Tg → TG。 dexp 的公式首先由 Friedrich Schur (1891) 证明。后来由 Henri Poincaré (1899) 在使用李代数项表达李群乘法问题的背景下详细阐述。它有时也称为杜哈梅尔公式。该公式在纯数学和应用数学中都很重要。
In the theory of Lie groups, the exponential map is a map from the Lie algebra g of a Lie group G into G. In case G is a matrix Lie group, the exponential map reduces to the matrix exponential. The exponential map, denoted exp:g → G, is analytic and has as such a derivative d/dtexp(X(t)):Tg → TG, where X(t) is a C1 path in the Lie algebra, and a closely related differential dexp:Tg → TG. The formula for dexp was first proved by Friedrich Schur (1891). It was later elaborated by Henri Poincaré (1899) in the context of the problem of expressing Lie group multiplication using Lie algebraic terms. It is also sometimes known as Duhamel's formula. The formula is important both in pure and applied mathematics.
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View content license ↗ Victor Batyrev 使用 d {\displaystyle d} 维凸多面体的极对偶性提出了一种镜面对称的纯粹组合方法。极性对偶性的最著名的例子提供了柏拉图立体：例如，立方体是八面体的对偶，十二面体是二十面体的对偶。
A purely combinatorial approach to mirror symmetry was suggested by Victor Batyrev using the polar duality for d {\displaystyle d} -dimensional convex polyhedra. The most famous examples of the polar duality provide Platonic solids: e.g., the cube is dual to octahedron, the dodecahedron is dual to icosahedron.
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View content license ↗ 在物理学中，哈密顿力学是 1833 年出现的拉格朗日力学的重新表述。由 William Rowan Hamilton 爵士提出，哈密顿力学用（广义）动量取代了拉格朗日力学中使用的（广义）速度 q ˙ i {\displaystyle {\dot {q}}^{i}}。两种理论都提供了经典力学的解释并描述了相同的物理现象。哈密顿力学与几何学（特别是辛几何学和泊松结构）有着密切的关系，并且是经典力学和量子力学之间的纽带。
In physics, Hamiltonian mechanics is a reformulation of Lagrangian mechanics that emerged in 1833. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities q ˙ i {\displaystyle {\dot {q}}^{i}} used in Lagrangian mechanics with (generalized) momenta. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics.
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View content license ↗ 在数学中，海森堡群 H {\displaystyle H} 以维尔纳·海森堡 (Werner Heisenberg) 的名字命名，是 ( 1 a c 0 1 b 0 0 1 ) {\displaystyle {\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\\\end{pmatrix}}} 在矩阵乘法运算下的 3×3 上三角矩阵的群。
In mathematics, the Heisenberg group H {\displaystyle H} , named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ( 1 a c 0 1 b 0 0 1 ) {\displaystyle {\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\\\end{pmatrix}}} under the operation of matrix multiplication.
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View content license ↗ 在数学物理中，Hunter–Saxton 方程 ( u t + u u x ) x = 1 2 u x 2 {\displaystyle (u_{t}+uu_{x})_{x}={\frac {1}{2}}\,u_{x}^{2}} 是向列液晶理论研究中出现的可积偏微分方程。如果液晶中的分子最初全部排列，然后其中一些分子轻微摆动，则这种取向扰动将通过晶体传播，亨特-萨克斯顿方程描述了这种取向波的某些方面。
In mathematical physics, the Hunter–Saxton equation ( u t + u u x ) x = 1 2 u x 2 {\displaystyle (u_{t}+uu_{x})_{x}={\frac {1}{2}}\,u_{x}^{2}} is an integrable PDE that arises in the theoretical study of nematic liquid crystals. If the molecules in the liquid crystal are initially all aligned, and some of them are then wiggled slightly, this disturbance in orientation will propagate through the crystal, and the Hunter–Saxton equation describes certain aspects of such orientation waves.
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View content license ↗ 在数学和数学物理中，可微流形 M 的坐标基或完整基是一组基向量场 {e1, ..., en}，在流形区域的每个点 P 处定义为 e α = lim δ x α → 0 δ s δ x α , {\displaystyle \mathbf {e} _{\alpha }=\lim _{\delta x^{\alpha }\to 0}{\frac {\delta \mathbf {s} }{\delta x^{\alpha }}},} 其中 δs 是点 P 和附近点 Q 之间的位移矢量，该点与 P 的坐标沿坐标曲线 xα 的间距为 δxα （即
In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold M is a set of basis vector fields {e1, ..., en} defined at every point P of a region of the manifold as e α = lim δ x α → 0 δ s δ x α , {\displaystyle \mathbf {e} _{\alpha }=\lim _{\delta x^{\alpha }\to 0}{\frac {\delta \mathbf {s} }{\delta x^{\alpha }}},} where δs is the displacement vector between the point P and a nearby point Q whose coordinate separation from P is δxα along the coordinate curve xα (i.e.
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View content license ↗ 在理论物理学中，乔丹图（通常也称为乔丹-施温格图）是从矩阵 Mij 到量子振子双线性表达式的映射，可加快物理学中李代数表示的计算。它由 Pascual Jordan 于 1935 年提出，并于 1952 年被 Julian Schwinger 用来有效地重新制定量子角动量理论，因为该图很容易在福克空间中组织 su(2) 的（对称）表示。该图利用了量子场论和多体问题中常规使用的几个创建和湮灭算子 a i † {\displaystyle a_{i}^{\dagger }} 和 a i {\displaystyle a_{i}^{\,}}，每一对代表一个量子谐振子。
In theoretical physics, the Jordan map, often also called the Jordan–Schwinger map is a map from matrices Mij to bilinear expressions of quantum oscillators which expedites computation of representations of Lie algebras occurring in physics. It was introduced by Pascual Jordan in 1935 and was utilized by Julian Schwinger in 1952 to re-work out the theory of quantum angular momentum efficiently, given that map’s ease of organizing the (symmetric) representations of su(2) in Fock space. The map utilizes several creation and annihilation operators a i † {\displaystyle a_{i}^{\dagger }} and a i {\displaystyle a_{i}^{\,}} of routine use in quantum field theories and many-body problems, each pair representing a quantum harmonic oscillator.
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View content license ↗ 在理论物理学中，哈密顿场论是经典哈密顿力学的场论模拟。它是经典场论与拉格朗日场论的形式主义。它在量子场论中也有应用。
In theoretical physics, Hamiltonian field theory is the field-theoretic analogue to classical Hamiltonian mechanics. It is a formalism in classical field theory alongside Lagrangian field theory. It also has applications in quantum field theory.
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View content license ↗ 在数学中，埃尔米特变换是一种以数学家 Charles Hermite 命名的积分变换，它使用埃尔米特多项式 H n ( x ) {\displaystyle H_{n}(x)} 作为变换的核。
In mathematics, the Hermite transform is an integral transform named after the mathematician Charles Hermite that uses Hermite polynomials H n ( x ) {\displaystyle H_{n}(x)} as kernels of the transform.
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View content license ↗ 在数学中，雅可比变换是以数学家 Carl Gustav Jacob Jacobi 命名的积分变换，它使用雅可比多项式 P n α , β ( x ) {\displaystyle P_{n}^{\alpha ,\beta }(x)} 作为变换的核。
In mathematics, the Jacobi transform is an integral transform named after the mathematician Carl Gustav Jacob Jacobi, which uses Jacobi polynomials P n α , β ( x ) {\displaystyle P_{n}^{\alpha ,\beta }(x)} as kernels of the transform .
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View content license ↗ 在凝聚态物理学中，系统的态密度 (DOS) 描述了每单位能量范围允许的模式或状态的数量。态密度定义为 D ( E ) = N ( E ) / V {\displaystyle D(E)=N(E)/V} ，其中 N ( E ) δ E {\displaystyle N(E)\delta E} 是体积 V {\displaystyle V} 系统中的态数，其能量范围为 E {\displaystyle E} 到 E + δ E {\displaystyle E+\delta E} 。它在数学上表示为概率密度函数的分布，通常是系统所占据的各种状态的空间和时间域上的平均值。
In condensed matter physics, the density of states (DOS) of a system describes the number of allowed modes or states per unit energy range. The density of states is defined as D ( E ) = N ( E ) / V {\displaystyle D(E)=N(E)/V} , where N ( E ) δ E {\displaystyle N(E)\delta E} is the number of states in the system of volume V {\displaystyle V} whose energies lie in the range from E {\displaystyle E} to E + δ E {\displaystyle E+\delta E} . It is mathematically represented as a distribution by a probability density function, and it is generally an average over the space and time domains of the various states occupied by the system.
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View content license ↗ 固态体的费米能级是向该体添加一个电子所需的热力学功。为简洁起见，它是一个热力学量，通常用 μ 或 EF 表示。费米能级不包括将电子从其来源处移走所需的功。费米能级的概念是用于确定电子特性的电子能带结构模型的重要组成部分，特别是因为它与电子电路中的电压和电荷流动有关。在固态物理学中用于分析固体能级的能带结构理论中，费米能级可以被认为是电子的假设能级，因此在热力学平衡时，该能级在任何给定时间都有 50% 的概率被占据。
The Fermi level of a solid-state body is the thermodynamic work required to add one electron to the body. It is a thermodynamic quantity usually denoted by μ or EF for brevity. The Fermi level does not include the work required to remove the electron from wherever it came from. The concept of the Fermi level is an important component of the electronic band structure model for determining electronic properties, especially as it relates to the voltage and flow of charge in electronic circuits. In band structure theory, used in solid state physics to analyze the energy levels in a solid, the Fermi level can be considered to be a hypothetical energy level of an electron, such that at thermodynamic equilibrium this energy level would have a 50% probability of being occupied at any given time.
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View content license ↗ 声子是凝聚态物质（特别是固体和某些液体）中原子或分子周期性弹性排列的准粒子集体激发。在光学捕获物体的背景下，只要振荡的模态波长小于物体的尺寸，量子化振动模式就可以定义为声子。声子是物理学中的一种准粒子，是相互作用粒子的弹性结构的振动模式的量子力学量子化中的激发态。声子可以被认为是量子化的声波，类似于光子被认为是量子化的光波。声子的研究是凝聚态物理的重要组成部分。
A phonon is a quasiparticle, collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. In the context of optically trapped objects, the quantized vibration mode can be defined as phonons as long as the modal wavelength of the oscillation is smaller than the size of the object. A type of quasiparticle in physics, a phonon is an excited state in the quantum mechanical quantization of the modes of vibrations for elastic structures of interacting particles. Phonons can be thought of as quantized sound waves, similar to photons as quantized light waves. The study of phonons is an important part of condensed matter physics.
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View content license ↗ 密度泛函理论 (DFT) 是一种计算量子力学建模方法，用于物理、化学和材料科学，用于研究多体系统（特别是原子、分子和凝聚相）的电子结构（或核结构）（主要是基态）。利用这一理论，多电子系统的属性可以通过使用泛函来确定，即接受函数作为输入并输出单个实数的函数。就 DFT 而言，这些是空间相关电子密度的函数。 DFT 是凝聚态物理、计算物理和计算化学领域最流行、最通用的方法之一。自 20 世纪 70 年代以来，DFT 在固体物理计算中一直非常流行。
Density functional theory (DFT) is a computational quantum mechanical modeling method used in physics, chemistry and materials science to investigate the electronic structure (or nuclear structure) (principally the ground state) of many-body systems, in particular atoms, molecules, and the condensed phases. Using this theory, the properties of a many-electron system can be determined by using functionals – that is, functions that accept a function as input and output a single real number. In the case of DFT, these are functionals of the spatially dependent electron density. DFT is among the most popular and versatile methods available in condensed-matter physics, computational physics, and computational chemistry. DFT has been very popular for calculations in solid-state physics since the 1970s.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 凝聚态物理学（condensed matter physics）又称凝态物理学、凝（聚）体物理学等，是研究物质凝聚态（固态和液态）或凝聚相的物理本质、结构和性质的一门学科，为固体物理学的延拓。该领域的研究者力图通过物理学定律来解释凝聚相物质的行为。其中，量子力学、电磁学以及统计力学的相关定律对于该领域尤为重要。 凝（聚）态（condensed state）或凝（聚）相（condensed phase）是由大量粒子组成，且粒子间有很强相互作用的系统。固相以及液相是人们最为熟悉的凝聚相，由于两者通常体积固定，不似气相具有可压缩性，故称之为凝（聚）相。除了这两种相之外，凝聚相还包括一些特定的物质在低温条件下的超导相、自旋有关的铁磁相及反铁磁相、超低温原子系统的玻色-爱因斯坦凝聚相等等。对于凝聚态的研究包括通过实验手段测定物质的各种性质，以及利用理论方法发展数学模型以深入理解这些物质的物理行为。 由于尚有大量的系统及现象亟待研究，凝聚态物理学成为了目前物理学最为活跃的领域之一。仅在美国，该领域的研究者就占到该国物理学者整体的近三分之一，凝聚态物理学部也是美国物理学会最大的部门。此外，该领域还与化学，材料科学以及纳米技术等学科领域交叉，并与原子物理学以及生物物理学等物理学分支紧密相关。该领域研究者在理论研究中所采用的一些概念与方法也适用于粒子物理学及核物理学等领域。 晶体学、冶金学、弹性力学以及磁学等等起初是各自独立的学科领域。这些学科在二十世纪四十年代被物理学家统合为固体物理学。时间进入二十世纪六十年代后，有关液体物理性质的研究也被纳入其中，形成凝聚态物理学这一新学科。据物理学家菲利普·安德森所述，术语“凝聚态物理学”是他和福尔克尔·海涅首创。1967年，他们把位于卡文迪许实验室的研究组名称由“固体理论”改为“凝聚态理论”。二人觉得原来的名称并没有涵盖液体及核物质等方面研究。但是，“凝聚态”这一术语此前已在欧洲学界出现，只是由他们普及而已。较为著名的例子是施普林格公司于1963年创建的期刊《凝聚态物理学》（英语：Physics of Condensed Matter）。二十世纪六、七十年代的资金环境以及各国政府采取的冷战政策促使相关领域物理学家接纳了“凝聚态物理学”这一术语。他们认为这一术语相对于“固体物理学”而言更为突出了固体、液体、等离子体以及其他复杂物质研究之间的共通性。这些研究与金属和半导体在工业上的应用息息相关。贝尔实验室是最早开展凝聚态物理学研究项目的研究机构之一。
Condensed matter physics is the field of physics that deals with the macroscopic and microscopic physical properties of matter, especially the solid and liquid phases, that arise from electromagnetic forces between atoms and electrons. More generally, the subject deals with condensed phases of matter: systems of many constituents with strong interactions among them. More exotic condensed phases include the superconducting phase exhibited by certain materials at extremely low cryogenic temperatures, the ferromagnetic and antiferromagnetic phases of spins on crystal lattices of atoms, the Bose–Einstein condensates found in ultracold atomic systems, and liquid crystals. Condensed matter physicists seek to understand the behavior of these phases by experiments to measure various material properties, and by applying the physical laws of quantum mechanics, electromagnetism, statistical mechanics, and other physics theories to develop mathematical models and predict the properties of very large groups of atoms.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 固体物理学是凝聚态物理学中最大的分支。它研究的对象是固体，特别是原子排列具有周期性结构的晶体。固体物理学的基本任务是从微观上解释固体材料的宏观物理性质，主要理论基础是非相对论性的量子力学，还会使用到电动力学、统计物理中的理论。主要方法是应用薛定谔方程来描述固体物质的电子态，并使用布洛赫波函数表达晶体周期性势场中的电子态。在此基础上，发展了固体的能带论，预言了半导体的存在，并且为晶体管的制造提供理论基础。
Solid-state physics is the study of rigid matter, or solids, through methods such as solid-state chemistry, quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the large-scale properties of solid materials result from their atomic-scale properties. Thus, solid-state physics forms a theoretical basis of materials science. Along with solid-state chemistry, it also has direct applications in the technology of transistors and semiconductors.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 分子动力学（MD）是一种分析原子和分子物理运动的计算机模拟方法。原子和分子可以在固定的时间内相互作用，从而可以看到系统的动态“演化”。在最常见的版本中，原子和分子的轨迹是通过数值求解相互作用粒子系统的牛顿运动方程来确定的，其中粒子之间的力及其势能通常使用原子间势或分子机械力场来计算。 MD模拟广泛应用于化学物理、材料科学和生物物理学。由于分子系统通常由大量粒子组成，因此不可能通过分析确定此类复杂系统的性质； MD模拟通过使用数值方法来规避这个问题。
Molecular dynamics (MD) is a computer simulation method for analyzing the physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a fixed period of time, giving a view of the dynamic "evolution" of the system. In the most common version, the trajectories of atoms and molecules are determined by numerically solving Newton's equations of motion for a system of interacting particles, where forces between the particles and their potential energies are often calculated using interatomic potentials or molecular mechanical force fields. MD simulations are widely applied in chemical physics, materials science, and biophysics. Because molecular systems typically consist of a vast number of particles, it is impossible to determine the properties of such complex systems analytically; MD simulation circumvents this problem by using numerical methods.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements. The Chinese definition is a machine-assisted translation of the cited English introduction; check technical terminology against the original.
View content license ↗ 在电磁学里，介电质响应外电场的施加而电极化的衡量，称为电容率。在非真空中由于介电质被电极化，在物质内部的总电场会减小。电容率关系到介电质传输（或容许）电场的能力。电容率衡量电场怎样影响介电质，怎样被介电质影响。电容率又称为“绝对电容率”。 在国际单位制中，电容率的测量单位是法拉每米（F/m）。真空的电容率，称为真空电容率，或“真空介电常数”，标记为 ε 0 {\displaystyle \varepsilon _{0}} 。 ε 0 {\displaystyle \varepsilon _{0}} ≈8.854187817…×10⁻¹² F/m。
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ε (epsilon), is a measure of the electric polarizability of a dielectric material. A material with high permittivity polarizes more in response to an applied electric field than a material with low permittivity, thereby storing more energy in the material. In electrostatics, the permittivity plays an important role in determining the capacitance of a capacitor. In the simplest case, the electric displacement field D resulting from an applied electric field E is D = ε E . {\displaystyle \mathbf {D} =\varepsilon \ \mathbf {E} ~.} More generally, the permittivity is a thermodynamic function of state. It can depend on the frequency, magnitude, and direction of the applied field. The SI unit for permittivity is farad per meter (F/m).
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 在电磁学中，磁导率是一种材料对一个外加磁场线性反应的磁化程度。磁导率通常用希腊字母μ来表示。该形式由奥利弗·赫维赛德于1885年9月创造使用。 在国际单位制单位中，磁导率的单位是亨利每米（H m-1），或牛顿每安培的平方（N A-2）。常数值 μ 0 {\displaystyle \mu _{0}} 为磁场常数或真空磁导率，并有明确定义 值 μ 0 {\displaystyle \mu _{0}} = 4π×10−7 N·A−2 (≈ 1.2566371×10−6 N·A−2)。
In electromagnetism, permeability is the measure of magnetization produced in a material in response to an applied magnetic field. Permeability is typically represented by the (italicized) Greek letter μ. It is the ratio of the magnetic induction B {\displaystyle B} to the magnetizing field H {\displaystyle H} in a material. The term was coined by Lord Kelvin in 1872, and is used alongside its electrostatic equivalent, permittivity, coined by Oliver Heaviside in 1885. The reciprocal of permeability is magnetic reluctivity. In SI units, permeability is measured in henries per meter (H/m), or equivalently in newtons per square ampere (N/A2). The permeability constant μ0, also known as the magnetic constant or the permeability of free space, is the proportionality between magnetic induction and magnetizing force when forming a magnetic field in a classical vacuum.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 在物理学里，感受到外电场的作用，中性原子或分子会改变其正常电子云形状，衡量这改变的物理量称为极化性（polarizability）。以方程表达， p = α E {\displaystyle \mathbf {p} =\alpha \mathbf {E} } ； 其中， p {\displaystyle \mathbf {p} } 是由于电子云形状的改变而产生的电偶极矩， α {\displaystyle \alpha } 是极化性， E {\displaystyle \mathbf {E} } 是外电场。
Polarizability usually refers to the tendency of matter, when subjected to an electric field, to acquire an electric dipole moment in proportion to that applied field. It is a property of particles with an electric charge. When subject to an electric field, the negatively charged electrons and positively charged atomic nuclei are subject to opposite forces and undergo charge separation. Polarizability is responsible for a material's dielectric constant and, at high (optical) frequencies, its refractive index. The polarizability of an atom or molecule is defined as the ratio of its induced dipole moment to the local electric field; in a crystalline solid, one considers the dipole moment per unit cell. Note that the local electric field seen by a molecule is generally different from the macroscopic electric field that would be measured externally.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
View content license ↗ 在经典电磁学里，当给电介质施加一个电场时，由于电介质内部正负电荷的相对位移，会产生电偶极子，这现象称为电极化（英语：electric polarization）。施加的电场可能是外电场，也可能是嵌入电介质内部的自由电荷所产生的电场。因为电极化而产生的电偶极子称为“感应电偶极子”，其电偶极矩称为“感应电偶极矩”。 电极化强度（英语：polarization density），又称为电极化矢量，定义为电介质内的电偶极矩密度，也就是单位体积的电偶极矩。这定义所指的电偶极矩包括永久电偶极矩和感应电偶极矩。它的国际单位制度量单位是库仑每平方米（coulomb/m2），表示为矢量 P。
In classical electromagnetism, polarization density (or electric polarization, or simply polarization) is the vector field that expresses the volumetric density of permanent or induced electric dipole moments in a dielectric material. When a dielectric is placed in an external electric field, its atoms or molecules gain electric dipole moment and the dielectric is said to be polarized. Electric polarization of a given dielectric material sample is defined as the quotient of electric dipole moment (a vector quantity, expressed as coulombs-meters (C⋅m) in SI units) to volume (in meters cubed). Polarization density is denoted mathematically by P; in SI units, it is expressed in coulombs per square meter (C/m2). Polarization density also describes how a material responds to an applied electric field as well as the way the material changes the electric field, and can be used to calculate the forces that result from those interactions. It can be compared to magnetization, which is the measure of the corresponding response of a material to a magnetic field in magnetism.
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Wikipedia contributors · Retrieved2026-10-04 · CC BY-SA 4.0. Introductions were extracted as plain text and shortened. Language versions may emphasize different aspects.For concept reference; consult the original standards for authoritative requirements.
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