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Unitary (physics)
In quantum physics, unitarity is (or a unitary process has) the condition that the time evolution of a quantum state according to the Schrödinger equation is mathematically represented by a unitary operator. This is typically taken as an axiom or basic postulate of quantum mechanics, while generalizations of or departures from unitarity are part of speculations about theories that may go beyond quantum mechanics. A unitarity bound is any inequality that follows from the unitarity of the evolution operator, i.e. from the statement that time evolution preserves inner products in Hilbert space. Hamiltonian evolution Time evolution described by a time-independent Hamiltonian is represented by a one-parameter family of unitary operators, for which the Hamiltonian is a generator: U(t) = e^. In the Schrödinger picture, the unitary operators are taken to act upon the system's quantum state, whereas in the Heisenberg picture, the time dependence is incorporated into the observables i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Physics
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] |
Hermitian Matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and : A \text \quad \iff \quad a_ = \overline or in matrix form: A \text \quad \iff \quad A = \overline . Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A is denoted by A^\mathsf, then the Hermitian property can be written concisely as A \text \quad \iff \quad A = A^\mathsf Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are A^\mathsf = A^\dagger = A^\ast, although in quantum mechanics, A^\ast typically means the complex conjugate onl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Antiunitary Operator
In mathematics, an antiunitary transformation is a bijective antilinear map :U: H_1 \to H_2\, between two complex Hilbert spaces such that :\langle Ux, Uy \rangle = \overline for all x and y in H_1, where the horizontal bar represents the complex conjugate. If additionally one has H_1 = H_2 then U is called an antiunitary operator. Antiunitary operators are important in quantum mechanics because they are used to represent certain symmetries, such as time reversal. Their fundamental importance in quantum physics is further demonstrated by Wigner's theorem. Invariance transformations In quantum mechanics, the invariance transformations of complex Hilbert space H leave the absolute value of scalar product invariant: : , \langle Tx, Ty \rangle, = , \langle x, y \rangle, for all x and y in H. Due to Wigner's theorem these transformations can either be unitary or antiunitary. Geometric Interpretation Congruences of the plane form two distinct classes. The first co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mandelstam Variables
In theoretical physics, the Mandelstam variables are numerical quantities that encode the energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ..., momentum, and angles of particles in a scattering process in a Lorentz-invariant fashion. They are used for scattering processes of two particles to two particles. The Mandelstam variables were first introduced by physicist Stanley Mandelstam in 1958. If the Minkowski metric is chosen to be \mathrm(1, -1,-1,-1), the Mandelstam variables s,t,u are then defined by :*s=(p_1+p_2)^2 c^2 =(p_3+p_4)^2 c^2 :*t=(p_1-p_3)^2 c^2 =(p_4-p_2)^2 c^2 :*u=(p_1-p_4)^2 c^2 =(p_3-p_2)^2 c^2, where ''p''1 and ''p''2 are the four-momenta of the incoming particles and ''p''3 and ''p''4 are the four-momenta of the outgoing particles. s is also known as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Froissart Bound
In particle physics the Froissart bound, or Froissart limit, is a generic constraint that the total scattering cross section of two colliding high-energy particles cannot increase faster than c \ln^2(s) , with ''c'' a normalization constant and ''s'' the square of the center-of-mass energy (''s'' is one of the three Mandelstam variables In theoretical physics, the Mandelstam variables are numerical quantities that encode the energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical ...). See also * * ''S''-matrix theory * Regge theory Further readingThe Froissart bound on scholarpedia, by M. Froissart References Scattering theory {{Particle-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Faddeev–Popov Ghost
In physics, Faddeev–Popov ghosts (also called Faddeev–Popov gauge ghosts or Faddeev–Popov ghost fields) are extraneous fields which are introduced into gauge quantum field theories to maintain the consistency of the path integral formulation. They are named after Ludvig Faddeev and Victor Popov. A more general meaning of the word "ghost" in theoretical physics is discussed in Ghost (physics). Overcounting in Feynman path integrals The necessity for Faddeev–Popov ghosts follows from the requirement that quantum field theories yield unambiguous, non-singular solutions. This is not possible in the path integral formulation when a gauge symmetry is present since there is no procedure for selecting among physically equivalent solutions related by gauge transformation. The path integrals overcount field configurations corresponding to the same physical state; the measure of the path integrals contains a factor which does not allow obtaining various results directly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauge Symmetry
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, the Lagrangian is invariant under these transformations. The term "gauge" refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian of a physical system. The transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the ''symmetry group'' or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called gauge invariance). When such a theory is quantized, the quanta of the g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Feynman Diagram
In theoretical physics, a Feynman diagram is a pictorial representation of the mathematical expressions describing the behavior and interaction of subatomic particles. The scheme is named after American physicist Richard Feynman, who introduced the diagrams in 1948. The calculation of probability amplitudes in theoretical particle physics requires the use of large, complicated integrals over a large number of variables. Feynman diagrams instead represent these integrals graphically. Feynman diagrams give a simple visualization of what would otherwise be an arcane and abstract formula. According to David Kaiser, "Since the middle of the 20th century, theoretical physicists have increasingly turned to this tool to help them undertake critical calculations. Feynman diagrams have revolutionized nearly every aspect of theoretical physics." While the diagrams apply primarily to quantum field theory, they can be used in other areas of physics, such as solid-state theory. Frank Wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Virtual Particle
A virtual particle is a theoretical transient particle that exhibits some of the characteristics of an ordinary particle, while having its existence limited by the uncertainty principle, which allows the virtual particles to spontaneously emerge from vacuum at short time and space ranges. The concept of virtual particles arises in the perturbation theory (quantum mechanics), perturbation theory of quantum field theory (QFT) where interactions between ordinary particles are described in terms of exchanges of virtual particles. A process involving virtual particles can be described by a schematic representation known as a Feynman diagram, in which virtual particles are represented by internal lines. Virtual particles do not necessarily carry the same mass as the corresponding ordinary particle, although they always conserve energy and momentum. The closer its characteristics come to those of ordinary particles, the longer the virtual particle exists. They are important in the ph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Optical Theorem
In physics, the optical theorem is a general law of wave scattering theory, which relates the zero-angle scattering amplitude to the total cross section of the scatterer. It is usually written in the form :\sigma=\frac~\mathrm\,f(0), where (0) is the scattering amplitude with an angle of zero, that is the amplitude of the wave scattered to the center of a distant screen and is the wave vector in the incident direction. Because the optical theorem is derived using only conservation of energy, or in quantum mechanics from conservation of probability, the optical theorem is widely applicable and, in quantum mechanics, \sigma_\mathrm includes both elastic and inelastic scattering. The generalized optical theorem, first derived by Werner Heisenberg, follows from the unitary condition and is given by :f(\mathbf,\mathbf') -f^*(\mathbf',\mathbf)= \frac \int f(\mathbf,\mathbf'')f^*(\mathbf',\mathbf'')\,d\Omega'' where f(\mathbf,\mathbf') is the scattering amplitude that depends on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bound State
A bound state is a composite of two or more fundamental building blocks, such as particles, atoms, or bodies, that behaves as a single object and in which energy is required to split them. In quantum physics, a bound state is a quantum state of a particle subject to a potential energy, potential such that the particle has a tendency to remain localized in one or more regions of space. The potential may be external or it may be the result of the presence of another particle; in the latter case, one can equivalently define a bound state as a state representing two or more particles whose interaction energy exceeds the total energy of each separate particle. One consequence is that, given a potential vanish at infinity, vanishing at infinity, negative-energy states must be bound. The energy spectrum of the set of bound states are most commonly discrete, unlike scattering states of Free particle, free particles, which have a continuous spectrum. Although not bound states in the stric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical System
A physical system is a collection of physical objects under study. The collection differs from a set: all the objects must coexist and have some physical relationship. In other words, it is a portion of the physical universe chosen for analysis. Everything outside the system is known as the '' environment'', which is ignored except for its effects on the system. The split between system and environment is the analyst's choice, generally made to simplify the analysis. For example, the water in a lake, the water in half of a lake, or an individual molecule of water in the lake can each be considered a physical system. An '' isolated system'' is one that has negligible interaction with its environment. Often a system in this sense is chosen to correspond to the more usual meaning of system, such as a particular machine. In the study of quantum coherence, the "system" may refer to the microscopic properties of an object (e.g. the mean of a pendulum bob), while the relevant "env ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
