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Product Operator Formalism
In NMR spectroscopy, the product operator formalism is a method used to determine the outcome of pulse sequences in a rigorous but straightforward way. With this method it is possible to predict how the bulk magnetization evolves with time under the action of pulses applied in different directions. It is a net improvement from the semi-classical vector model which is not able to predict many of the results in NMR spectroscopy and is a simplification of the complete density matrix formalism. In this model, for a single spin, four base operators exist: I_x, I_y, I_z and E/2 which represent respectively polarization (population difference between the two spin states), single quantum coherence (magnetization on the xy plane) and the unit operator. Many other, non-classical operators exist for coupled systems. Using this approach, the evolution of the magnetization under free precession is represented by I_z and corresponds to a rotation about the z-axis with a phase angle proportion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NMR Spectroscopy
Nuclear magnetic resonance spectroscopy, most commonly known as NMR spectroscopy or magnetic resonance spectroscopy (MRS), is a spectroscopic technique to observe local magnetic fields around atomic nuclei. The sample is placed in a magnetic field and the NMR signal is produced by excitation of the nuclei sample with radio waves into nuclear magnetic resonance, which is detected with sensitive radio receivers. The intramolecular magnetic field around an atom in a molecule changes the resonance frequency, thus giving access to details of the electronic structure of a molecule and its individual functional groups. As the fields are unique or highly characteristic to individual compounds, in modern organic chemistry practice, NMR spectroscopy is the definitive method to identify monomolecular organic compounds. The principle of NMR usually involves three sequential steps: # The alignment (polarization) of the magnetic nuclear spins in an applied, constant magnetic field B0. # The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-adjoint 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 positi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boltzmann Distribution
In statistical mechanics and mathematics, a Boltzmann distribution (also called Gibbs distribution Translated by J.B. Sykes and M.J. Kearsley. See section 28) is a probability distribution or probability measure that gives the probability that a system will be in a certain state as a function of that state's energy and the temperature of the system. The distribution is expressed in the form: :p_i \propto e^ where is the probability of the system being in state , is the energy of that state, and a constant of the distribution is the product of the Boltzmann constant and thermodynamic temperature . The symbol \propto denotes proportionality (see for the proportionality constant). The term ''system'' here has a very wide meaning; it can range from a collection of 'sufficient number' of atoms or a single atom to a macroscopic system such as a natural gas storage tank. Therefore the Boltzmann distribution can be used to solve a very wide variety of problems. The distribu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
J-coupling
In nuclear chemistry and nuclear physics, ''J''-couplings (also called spin-spin coupling or indirect dipole–dipole coupling) are mediated through chemical bonds connecting two spins. It is an indirect interaction between two nuclear spins that arises from hyperfine interactions between the nuclei and local electrons. In NMR spectroscopy, ''J''-coupling contains information about relative bond distances and angles. Most importantly, ''J''-coupling provides information on the connectivity of chemical bonds. It is responsible for the often complex splitting of resonance lines in the NMR spectra of fairly simple molecules. ''J''-coupling is a frequency ''difference'' that is not affected by the strength of the magnetic field, so is always stated in Hz. Vector model and manifestations for chemical structure assignments The origin of ''J''-coupling can be visualized by a vector model for a simple molecule such as hydrogen fluoride (HF). In HF, the two nuclei have spin . Four states ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distortionless Enhancement By Polarization Transfer
Carbon-13 (C13) nuclear magnetic resonance (most commonly known as carbon-13 NMR spectroscopy or 13C NMR spectroscopy or sometimes simply referred to as carbon NMR) is the application of nuclear magnetic resonance (NMR) spectroscopy to carbon. It is analogous to proton NMR ( NMR) and allows the identification of carbon atoms in an organic molecule just as proton NMR identifies hydrogen atoms. 13C NMR detects only the isotope. The main carbon isotope, is not detected. Although much less sensitive than 1H NMR spectroscopy, 13C NMR spectroscopy is widely used for characterizing organic and organometallic compounds. Chemical shifts 13C NMR chemical shifts follow the same principles as those of 1H, although the typical range of chemical shifts is much larger than for 1H (by a factor of about 20). The chemical shift reference standard for 13C is the carbons in tetramethylsilane (TMS), whose chemical shift is considered to be 0.0 ppm. ImageSize = width:540 height:440 AlignBars ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generator (mathematics)
In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to it, that result in the creation of a larger collection of objects, called the generated set. The larger set is then said to be generated by the smaller set. It is commonly the case that the generating set has a simpler set of properties than the generated set, thus making it easier to discuss and examine. It is usually the case that properties of the generating set are in some way preserved by the act of generation; likewise, the properties of the generated set are often reflected in the generating set. List of generators A list of examples of generating sets follow. * Generating set or spanning set of a vector space: a set that spans the vector space * Generating set of a group: A subset of a group that is not contained in any subgro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group ''G'' is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if ''G'' is GL(n, \mathbb), the Lie group of real ''n''-by-''n'' invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible ''n''-by-''n'' matrix g to an endomorphism of the vector space of all linear transformations of \mathbb^n defined by: x \mapsto g x g^ . For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of ''G'' on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields. Definition Let ''G'' be a Lie group, and let :\Psi: G \to \operatorname(G) be the mapping , with Aut(''G'') the automorphism group of ''G'' and given by the inner automorphism (conjugation) :\Psi_g(h)= ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superoperator
In physics, a superoperator is a linear operator acting on a vector space of linear operators.John Preskill, Lecture notes for Quantum Computation course at CaltechCh. 3 Sometimes the term refers more specially to a completely positive map which also preserves or does not increase the trace of its argument. This specialized meaning is used extensively in the field of quantum computing, especially quantum programming, as they characterise mappings between density matrices. The use of the super- prefix here is in no way related to its other use in mathematical physics. That is to say superoperators have no connection to supersymmetry and superalgebra which are extensions of the usual mathematical concepts defined by extending the ring of numbers to include Grassmann numbers. Since superoperators are themselves operators the use of the super- prefix is used to distinguish them from the operators upon which they act. Left/Right Multiplication Defining the left and right multipl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
