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Table Of Clebsch–Gordan Coefficients
This is a table of Clebsch–Gordan coefficients used for adding angular momentum values in quantum mechanics. The overall sign of the coefficients for each set of constant j_1, j_2, j is arbitrary to some degree and has been fixed according to the Condon–Shortley and Wigner sign convention as discussed by Baird and Lawrence Biedenharn, Biedenharn. Tables with the same sign convention may be found in the Particle Data Group's ''Review of Particle Properties'' and in online tables. Formulation The Clebsch–Gordan coefficients are the solutions to : , j_1,j_2;J,M\rangle = \sum_^ \sum_^ , j_1,m_1;j_2,m_2\rangle \langle j_1,j_2;m_1,m_2\mid j_1,j_2;J,M\rangle Explicitly: : \begin & \langle j_1,j_2;m_1,m_2\mid j_1,j_2;J,M\rangle \\[6pt] = & \delta_ \sqrt\ \times \\[6pt] &\sqrt\ \times \\[6pt] &\sum_k \frac. \end The summation is extended over all integer for which the argument of every factorial is nonnegative. For brevity, solutions with and are omitted. They may b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clebsch–Gordan Coefficients
In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum decomposition of the tensor product of two irreducible representations (i.e., a reducible representation into irreducible representations, in cases where the numbers and types of irreducible components are already known abstractly). The name derives from the German mathematicians Alfred Clebsch and Paul Gordan, who encountered an equivalent problem in invariant theory. From a vector calculus perspective, the CG coefficients associated with the SO(3) group can be defined simply in terms of integrals of products of spherical harmonics and their complex conjugates. The addition of spins in qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angular Momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed system remains constant. Angular momentum has both a direction and a magnitude, and both are conserved. Bicycles and motorcycles, frisbees, rifled bullets, and gyroscopes owe their useful properties to conservation of angular momentum. Conservation of angular momentum is also why hurricanes form spirals and neutron stars have high rotational rates. In general, conservation limits the possible motion of a system, but it does not uniquely determine it. The three-dimensional angular momentum for a point particle is classically represented as a pseudovector , the cross product of the particle's position vector (relative to some origin) and its momentum vector; the latter is in Newtonian mechanics. Unlike linear momentum, angular m ... [...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 are limits to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lawrence Biedenharn
Lawrence Christian Biedenharn, Jr. (18 November 1922, Vicksburg, Mississippi – 12 February 1996, Austin, Texas) was an American theoretical nuclear physicist and mathematical physicist, a leading expert on applications of Lie group theory to physics. Biedenharn studied at MIT with an interruption in World War II from 1942 to 1946 as a lieutenant in the Signal Corps in the Pacific theater, where in 1946 he was stationed in Tokyo for a year as a radio officer. He received his bachelor's degree in absentia from MIT. After World War II, he returned to MIT where he received his PhD in physics under Victor Weisskopf in 1950. As an MIT graduate student he shared an office with J. David Jackson, then a graduate student, and John Blatt, then a post-doc. Biedenharn was employed at Oak Ridge National Laboratory, became an assistant professor at Yale University, and then an associate professor at Rice University. From 1961 he was a professor at Duke University, where he became in 1987 "Ja ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle Data Group
The Particle Data Group (or PDG) is an international collaboration of particle physicists that compiles and reanalyzes published results related to the properties of particles and fundamental interactions. It also publishes reviews of theoretical results that are phenomenologically relevant, including those in related fields such as cosmology. The PDG currently publishes the ''Review of Particle Physics'' and its pocket version, the ''Particle Physics Booklet'', which are printed biennially as books, and updated annually via the World Wide Web. In previous years, the PDG has published the ''Pocket Diary for Physicists'', a calendar with the dates of key international conferences and contact information of major high energy physics institutions, which is now discontinued. PDG also further maintains the standard numbering scheme for particles in event generators, in association with the event generator authors. ''Review of Particle Physics'' The ''Review of Particle Physics'' (former ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Java (programming Language)
Java is a high-level, class-based, object-oriented programming language that is designed to have as few implementation dependencies as possible. It is a general-purpose programming language intended to let programmers ''write once, run anywhere'' ( WORA), meaning that compiled Java code can run on all platforms that support Java without the need to recompile. Java applications are typically compiled to bytecode that can run on any Java virtual machine (JVM) regardless of the underlying computer architecture. The syntax of Java is similar to C and C++, but has fewer low-level facilities than either of them. The Java runtime provides dynamic capabilities (such as reflection and runtime code modification) that are typically not available in traditional compiled languages. , Java was one of the most popular programming languages in use according to GitHub, particularly for client–server web applications, with a reported 9 million developers. Java was originally developed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory Of Lie Groups
In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras. Finite-dimensional representations Representations A complex representation of a group is an action by a group on a finite-dimensional vector space over the field \mathbb C. A representation of the Lie group ''G'', acting on an ''n''-dimensional vector space ''V'' over \mathbb C is then a smooth group homomorphism :\Pi:G\rightarrow\operatorname(V), where \operatorname(V) is the general linear group of all invertible linear transformations of V under their composition. Since all ''n''-dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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