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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 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 \\ pt= & \delta_ \sqrt\ \times \\ pt&\sqrt\ \times \\ pt&\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 be calculated using the simple ... [...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 quant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Angular Momentum
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total angular momentum of a closed system remains constant. Angular momentum has both a direction (geometry), direction and a magnitude, and both are conserved. Bicycle and motorcycle dynamics, Bicycles and motorcycles, flying discs, Rifling, 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 mo ... [...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] |
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 "Jame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle Data Group
The Particle Data Group (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'' (fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Java (programming Language)
Java is a High-level programming language, high-level, General-purpose programming language, general-purpose, Memory safety, memory-safe, object-oriented programming, object-oriented programming language. It is intended to let programmers ''write once, run anywhere'' (Write once, run anywhere, WORA), meaning that compiler, compiled Java code can run on all platforms that support Java without the need to recompile. Java applications are typically compiled to Java bytecode, bytecode that can run on any Java virtual machine (JVM) regardless of the underlying computer architecture. The syntax (programming languages), syntax of Java is similar to C (programming language), C and C++, but has fewer low-level programming language, low-level facilities than either of them. The Java runtime provides dynamic capabilities (such as Reflective programming, reflection and runtime code modification) that are typically not available in traditional compiled languages. Java gained popularity sh ... [...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' representation of Lie algebras, representations of Lie algebras. Finite-dimensional representations Representations A complex group representation, 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 o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
