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Sum Rule In Quantum Mechanics
In quantum mechanics, a sum rule is a formula for transitions between energy levels, in which the sum of the transition strengths is expressed in a simple form. Sum rules are used to describe the properties of many physical systems, including solids, atoms, atomic nuclei, and nuclear constituents such as protons and neutrons. The sum rules are derived from general principles, and are useful in situations where the behavior of individual energy levels is too complex to be described by a precise quantum-mechanical theory. In general, sum rules are derived by using Heisenberg's quantum-mechanical algebra to construct operator equalities, which are then applied to the particles or energy levels of a system. Derivation of sum rules Assume that the Hamiltonian \hat has a complete set of eigenfunctions , n\rangle with eigenvalues E_n: : \hat , n\rangle = E_n , n\rangle. For the Hermitian operator In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science. Classical physics, the collection of theories that existed before the advent of quantum mechanics, describes many aspects of nature at an ordinary (macroscopic) scale, but is not sufficient for describing them at small (atomic and subatomic) scales. Most theories in classical physics can be derived from quantum mechanics as an approximation valid at large (macroscopic) scale. Quantum mechanics differs from classical physics in that energy, momentum, angular momentum, and other quantities of a bound system are restricted to discrete values ( quantization); objects have characteristics of both particles and waves ( wave–particle duality); and there ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Werner Heisenberg
Werner Karl Heisenberg () (5 December 1901 – 1 February 1976) was a German theoretical physicist and one of the main pioneers of the theory of quantum mechanics. He published his work in 1925 in a breakthrough paper. In the subsequent series of papers with Max Born and Pascual Jordan, during the same year, his matrix formulation of quantum mechanics was substantially elaborated. He is known for the uncertainty principle, which he published in 1927. Heisenberg was awarded the 1932 Nobel Prize in Physics "for the creation of quantum mechanics". Heisenberg also made contributions to the theories of the hydrodynamics of turbulent flows, the atomic nucleus, ferromagnetism, cosmic rays, and subatomic particles. He was a principal scientist in the German nuclear weapons program during World War II. He was also instrumental in planning the first West German nuclear reactor at Karlsruhe, together with a research reactor in Munich, in 1957. Following World War II, he was app ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian (quantum Mechanics)
Hamiltonian may refer to: * Hamiltonian mechanics, a function that represents the total energy of a system * Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system ** Dyall Hamiltonian, a modified Hamiltonian with two-electron nature ** Molecular Hamiltonian, the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule * Hamiltonian (control theory), a function used to solve a problem of optimal control for a dynamical system * Hamiltonian path, a path in a graph that visits each vertex exactly once * Hamiltonian group, a non-abelian group the subgroups of which are all normal * Hamiltonian economic program The Hamiltonian economic program was the set of measures that were proposed by American Founding Father and first Secretary of the Treasury Alexander Hamilton in four notable reports and implemented by the US Congress during George Washington's ..., the economic policies advocated by Alexander Hamilton ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. If ''V'' is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of ''A'' is a Hermitian matrix, i.e., equal to its conjugate transpose ''A''. By the finite-dimensional spectral theorem, ''V'' has an orthonormal basis such that the matrix of ''A'' relative to this basis is a diagonal matrix with entries in the real numbers. In this article, we consider generalizations of this concept to operators on Hilbert spaces of arbitrary dimension. Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oscillator Strength
In spectroscopy, oscillator strength is a dimensionless quantity that expresses the probability of absorption or emission of electromagnetic radiation in transitions between energy levels of an atom or molecule. For example, if an emissive state has a small oscillator strength, nonradiative decay will outpace radiative decay. Conversely, "bright" transitions will have large oscillator strengths. The oscillator strength can be thought of as the ratio between the quantum mechanical transition rate and the classical absorption/emission rate of a single electron oscillator with the same frequency as the transition. Theory An atom or a molecule can absorb light and undergo a transition from one quantum state to another. The oscillator strength f_ of a transition from a lower state , 1\rangle to an upper state , 2\rangle may be defined by : f_ = \frac\frac(E_2 - E_1) \sum_ , \langle 1 m_1 , R_\alpha , 2 m_2 \rangle , ^2, where m_e is the mass of an electron and \hbar is the re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sum Rules (quantum Field Theory)
In quantum field theory, a ''sum rule'' is a relation between a static quantity and an integral over a dynamical quantity. Therefore, they have a form such as: \int A(x) dx = B where A(x) is the dynamical quantity, for example a structure function characterizing a particle, and B is the static quantity, for example the mass or the charge of that particle. Quantum field theory sum rules should not be confused with sum rules in quantum chromodynamics or quantum mechanics. Properties Many sum rules exist. The validity of a particular sum rule can be sound if its derivation is based on solid assumptions, or on the contrary, some sum rules have been shown experimentally to be incorrect, due to unwarranted assumptions made in their derivation. The list of sum rules below illustrate this. Sum rules are usually obtained by combining a dispersion relation with the optical theorem, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
QCD Sum Rules
In quantum chromodynamics, the confining and strong coupling nature of the theory means that conventional perturbative techniques often fail to apply. The QCD sum rules (or Shifman– Vainshtein–Zakharov sum rules) are a way of dealing with this. The idea is to work with gauge invariant operators and operator product expansions of them. The vacuum to vacuum correlation function for the product of two such operators can be reexpressed as :\left\langle 0 , T\left\ , 0 \right\rangle where we have inserted hadronic particle states on the right hand side. Overview Instead of a model-dependent treatment in terms of constituent quarks, hadrons are represented by their interpolating quark currents taken at large virtualities. The correlation function of these currents is introduced and treated in the framework of the operator product expansion (OPE), where the short and long-distance quark-gluon interactions are separated. The former are calculated using QCD perturbation theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
