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Runge–Gross Theorem
In quantum mechanics, specifically time-dependent density functional theory, the Runge–Gross theorem (RG theorem) shows that for a many-body system evolving from a given initial wavefunction, there exists a one-to-one mapping between the potential (or potentials) in which the system evolves and the density (or densities) of the system. The potentials under which the theorem holds are defined up to an additive purely time-dependent function: such functions only change the phase of the wavefunction and leave the density invariant. Most often the RG theorem is applied to molecular systems where the electronic density, ''ρ''(r,''t'') changes in response to an external scalar potential, ''v''(r,''t''), such as a time-varying electric field. The Runge–Gross theorem provides the formal foundation of time-dependent density functional theory. It shows that the density can be used as the fundamental variable in describing quantum many-body systems in place of the wavefunction, and that al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is the fundamental physical Scientific theory, theory that describes the behavior of matter and of light; its unusual characteristics typically occur at and below the scale of atoms. Reprinted, Addison-Wesley, 1989, It is the foundation of all quantum physics, which includes quantum chemistry, quantum field theory, quantum technology, and quantum information science. Quantum mechanics can describe many systems that classical physics cannot. Classical physics can describe many aspects of nature at an ordinary (macroscopic and Microscopic scale, (optical) microscopic) scale, but is not sufficient for describing them at very small submicroscopic (atomic and subatomic) scales. Classical mechanics can be derived from quantum mechanics as an approximation that is valid at ordinary scales. Quantum systems have Bound state, bound states that are Quantization (physics), quantized to Discrete mathematics, discrete values of energy, momentum, angular momentum, and ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Van Leeuwen Theorem
A van is a type of road vehicle used for transporting goods or people. There is some variation in the scope of the word across the different English-speaking countries. The smallest vans, microvans, are used for transporting either goods or people in tiny quantities. Mini MPVs, compact MPVs, and MPVs are all small vans usually used for transporting people in small quantities. Larger vans with passenger seats are used for institutional purposes, such as transporting students. Larger vans with only front seats are often used for business purposes, to carry goods and equipment. Specially equipped vans are used by television stations as mobile studios. Postal services and courier companies use large step vans to deliver packages. Word origin and usage Van meaning a type of vehicle arose as a contraction of the word caravan. The earliest records of a van as a vehicle in English are in the mid-19th century, meaning a covered wagon for transporting goods; the earliest reported rec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anomalous Density
Anomaly or The Anomaly may refer to: Science Natural *Anomaly (natural sciences) ** Atmospheric anomaly **Geophysical anomaly Medical *Congenital anomaly (birth defect), a disorder present at birth ** Physical anomaly, a deformation of an anatomical structure ***Congenital vertebral anomaly, any of several malformations of the spine **Collie eye anomaly, eye disease of dogs **Coronary artery anomaly, a congenital abnormality in the heart **Ebstein's anomaly, a congenital heart defect ** Uhl anomaly, a congenital heart disease affecting the myocardial muscle **Vaginal anomalies Biology * Anomalous, a species of moth in the Noctuid family *Chromosome anomaly, a disorder caused by a structural error in a chromosome or an atypical number of chromosomes * Genetic anomaly, a disorder caused by mutation *Teratology, the study of developmental anomalies Physics *Anomalous diffusion, the movement of molecules from a region of lower concentration to a region of higher concentration *Ano ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pairing Potential
In mathematics, a pairing is an ''R''-bilinear map from the Cartesian product of two ''R''- modules, where the underlying ring ''R'' is commutative. Definition Let ''R'' be a commutative ring with unit, and let ''M'', ''N'' and ''L'' be ''R''-modules. A pairing is any ''R''-bilinear map e:M \times N \to L. That is, it satisfies :e(r\cdot m,n)=e(m,r \cdot n)=r\cdot e(m,n), :e(m_1+m_2,n)=e(m_1,n)+e(m_2,n) and e(m,n_1+n_2)=e(m,n_1)+e(m,n_2) for any r \in R and any m,m_1,m_2 \in M and any n,n_1,n_2 \in N . Equivalently, a pairing is an ''R''-linear map :M \otimes_R N \to L where M \otimes_R N denotes the tensor product of ''M'' and ''N''. A pairing can also be considered as an ''R''-linear map \Phi : M \to \operatorname_ (N, L) , which matches the first definition by setting \Phi (m) (n) := e(m,n) . A pairing is called perfect if the above map \Phi is an isomorphism of ''R''-modules and the other evaluation map \Phi'\colon N\to \operatorname_(M,L) is an isomorphism also ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superconductivity
Superconductivity is a set of physical properties observed in superconductors: materials where Electrical resistance and conductance, electrical resistance vanishes and Magnetic field, magnetic fields are expelled from the material. Unlike an ordinary metallic Electrical conductor, conductor, whose resistance decreases gradually as its temperature is lowered, even down to near absolute zero, a superconductor has a characteristic Phase transition, critical temperature below which the resistance drops abruptly to zero. An electric current through a loop of superconducting wire can persist indefinitely with no power source. The superconductivity phenomenon was discovered in 1911 by Dutch physicist Heike Kamerlingh Onnes. Like ferromagnetism and Atomic spectral line, atomic spectral lines, superconductivity is a phenomenon which can only be explained by quantum mechanics. It is characterized by the Meissner effect, the complete cancellation of the magnetic field in the interior of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Potential
In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field. Formally, given a vector field \mathbf, a ''vector potential'' is a C^2 vector field \mathbf such that \mathbf = \nabla \times \mathbf. Consequence If a vector field \mathbf admits a vector potential \mathbf, then from the equality \nabla \cdot (\nabla \times \mathbf) = 0 (divergence of the curl is zero) one obtains \nabla \cdot \mathbf = \nabla \cdot (\nabla \times \mathbf) = 0, which implies that \mathbf must be a solenoidal vector field. Theorem Let \mathbf : \R^3 \to \R^3 be a solenoidal vector field which is twice continuously differentiable. Assume that \mathbf(\mathbf) decreases at least as fast as 1/\, \mathbf\, for \, \mathbf\, \to \infty . Define \mathbf (\mathbf) = \frac \int_ \frac \, d^3\mathbf where \nabla_y \times denotes curl with respect to variab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Density Matrix
In quantum mechanics, a density matrix (or density operator) is a matrix used in calculating the probabilities of the outcomes of measurements performed on physical systems. It is a generalization of the state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent mixed states. These arise in quantum mechanics in two different situations: # when the preparation of a system can randomly produce different pure states, and thus one must deal with the statistics of possible preparations, and # when one wants to describe a physical system that is entangled with another, without describing their combined state. This case is typical for a system interacting with some environment (e.g. decoherence). In this case, the density matrix of an entangled system differs from that of an ensemble of pure states that, combined, would give the same statistical results upon measurement. Density matrices are thus crucial tools in areas of quantum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville Equation
: ''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).'' : ''For Liouville's equation in quantum mechanics, see Von Neumann equation.'' : ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelfand equation.'' In differential geometry, Liouville's equation, named after Joseph Liouville, is the nonlinear partial differential equation satisfied by the conformal factor of a metric on a surface of constant Gaussian curvature : :\Delta_0\log f = -K f^2, where is the flat Laplace operator :\Delta_0 = \frac +\frac = 4 \frac \frac. Liouville's equation appears in the study of isothermal coordinates in differential geometry: the independent variables are the coordinates, while can be described as the conformal factor with respect to the flat metric. Occasionally it is the square that is referred to as the conformal factor, instead of itself. Liouville's equation was also taken as an example by David Hilbert in the formulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistical Ensemble (mathematical Physics)
In physics, specifically statistical mechanics, an ensemble (also statistical ensemble) is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a system, considered all at once, each of which represents a possible state that the real system might be in. In other words, a statistical ensemble is a set of systems of particles used in statistical mechanics to describe a single system. The concept of an ensemble was introduced by J. Willard Gibbs in 1902. A thermodynamic ensemble is a specific variety of statistical ensemble that, among other properties, is in statistical equilibrium (defined below), and is used to derive the properties of thermodynamic systems from the laws of classical or quantum mechanics. Physical considerations The ensemble formalises the notion that an experimenter repeating an experiment again and again under the same macroscopic conditions, but unable to control the microscopic details, may expect to observe a ran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reductio Ad Absurdum
In logic, (Latin for "reduction to absurdity"), also known as (Latin for "argument to absurdity") or ''apagogical argument'', is the form of argument that attempts to establish a claim by showing that the opposite scenario would lead to absurdity or contradiction. This argument form traces back to Ancient Greek philosophy and has been used throughout history in both formal mathematical and philosophical reasoning, as well as in debate. In mathematics, the technique is called ''proof by contradiction''. In formal logic, this technique is captured by an axiom for "Reductio ad Absurdum", normally given the abbreviation RAA, which is expressible in propositional logic. This axiom is the introduction rule for negation (see ''negation introduction''). Examples The "absurd" conclusion of a ''reductio ad absurdum'' argument can take a range of forms, as these examples show: * The Earth cannot be flat; otherwise, since the Earth is assumed to be finite in extent, we would find peo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuity Equation
A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. Since mass, energy, momentum, electric charge and other natural quantities are conserved under their respective appropriate conditions, a variety of physical phenomena may be described using continuity equations. Continuity equations are a stronger, local form of conservation laws. For example, a weak version of the law of conservation of energy states that energy can neither be created nor destroyed—i.e., the total amount of energy in the universe is fixed. This statement does not rule out the possibility that a quantity of energy could disappear from one point while simultaneously appearing at another point. A stronger statement is that energy is ''locally'' conserved: energy can neither be created nor destroyed, ''nor'' can it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Current
In quantum mechanics, the probability current (sometimes called probability flux) is a mathematical quantity describing the flow of probability. Specifically, if one thinks of probability as a heterogeneous fluid, then the probability current is the rate of flow of this fluid. It is a real vector that changes with space and time. Probability currents are analogous to mass currents in hydrodynamics and electric currents in electromagnetism. As in those fields, the probability current (i.e. the probability current density) is related to the probability density function via a continuity equation. The probability current is invariant under gauge transformation. The concept of probability current is also used outside of quantum mechanics, when dealing with probability density functions that change over time, for instance in Brownian motion and the Fokker–Planck equation. The relativistic equivalent of the probability current is known as the probability four-current. Definition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
