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## Angular momentum coupling
**spin-orbit coupling**in astronomy reflects the general law of conservation of angular momentum, which holds for celestial systems as well. In simple cases, the direction of the angular momentum vector is neglected, and the spin-orbit coupling is the ratio between the frequency with which a planet or other celestial body spins about its own axis to that with which it orbits another body. This is more commonly known as orbital resonance. Often, the underlying physical effects are tidal forces.
## General theory and detailed originAngular momentum is a property of a physical system that is a constant of motion An example of the first situation is an atom whose electrons only feel the Coulomb field of its nucleus. If we ignore the electron-electron interaction (and other small interactions such as spin-orbit coupling), the An example of the second situation is a rigid rotor moving in field-free space. A rigid rotor has a well-defined, time-independent, angular momentum. These two situations originate in classical mechanics. The third kind of conserved angular momentum, associated with spin, does not have a classical counterpart. However, all rules of angular momentum coupling apply to spin as well. In general the conservation of angular momentum implies full rotational symmetry
(described by the groups SO(3) and SU(2)) and, conversely, spherical symmetry implies conservation of angular momentum. If two or more physical systems have conserved angular momenta, it can be useful to add these momenta to a total angular momentum of the combined system—a conserved property of the total system.
The building of eigenstates of the total conserved angular momentum from the angular momentum eigenstates of the individual subsystems is referred to as Application of angular momentum coupling is useful when there is an interaction between subsystems that, without interaction, would have conserved angular momentum. By the very interaction the spherical symmetry of the subsystems is broken, but the angular momentum of the total system remains a constant of motion. Use of the latter fact is helpful in the solution of the Schrödinger equation. As an example we consider two electrons, 1 and 2, in an atom (say the helium atom). If there is no electron-electron interaction, but only electron nucleus interaction, the two electrons can be rotated around the nucleus independently of each other; nothing happens to their energy. Both operators, In quantum mechanics, coupling also exists between angular momenta belonging to different Hilbert spaces of a single object, Reiterating slightly differently the above: one expands the quantum states of composed systems ( ## Footnote**^**Also referred to as a*conserved*property
## Spin-orbit couplingThe behavior of atoms and smaller particles is well described by the theory of quantum mechanics, in which each particle has an intrinsic angular momentum called spin and specific configurations (of e.g. electrons in an atom) are described by a set of quantum numbers. Collections of particles also have angular momenta and corresponding quantum numbers, and under different circumstances the angular momenta of the parts add in different ways to form the angular momentum of the whole. Angular momentum coupling is a category including some of the ways that subatomic particles can interact with each other. In atomic physics, In the macroscopic world of orbital mechanics, the term ## LS couplingIn light atoms (generally Z<30), electron spins - where and
This is an approximation which is good as long as any external magnetic fields are weak. In larger magnetic fields, these two momenta decouple, giving rise to a different splitting pattern in the energy levels (the For an extensive example on how LS-coupling is practically applied, see the article on Term symbols. ## jj couplingIn heavier atoms the situation is different. In atoms with bigger nuclear charges, spin-orbit interactions are frequently as large or larger than spin-spin interactions or orbit-orbit interactions. In this situation, each orbital angular momentum This description, facilitating calculation of this kind of interaction, is known as ## Spin-spin coupling
## Term symbolsTerm symbols are used to represent the states and spectral transitions of atoms, they are found from coupling of angular momenta mentioned above. When the state of an atom has been specified with a term symbol, the allowed transitions can be found through selection rules by considering which transitions would conserve angular momentum. A photon has spin 1, and when there is a transition with emission or absorption of a photon the atom will need to change state to conserve angular momentum. The term symbol selection rules are. ΔS=0, ΔL=0,±1, Δl=±1, ΔJ=0,±1 ## Relativistic effectsIn very heavy atoms, relativistic shifting of the energies of the electron energy levels accentuates spin-orbit coupling effect. Thus, for example, uranium molecular orbital diagrams must directly incorporate relativistic symbols when considering interactions with other atoms. ## Nuclear couplingIn atomic nuclei, the spin-orbit interaction is much stronger than for atomic electrons, and is incorporated directly into the nuclear shell model. In addition, unlike atomic-electron term symbols, the lowest energy state is not L - S, but rather, l + s. All nuclear levels whose l value (orbital angular momentum) is greater than zero are thus split in the shell model to create states designated by l + s and l - s. Due to the nature of the shell model, which assumes an average potential rather than a central Coulombic potential, the nucleons that go into the l + s and l - s nuclear states are considered degenerate within each orbital (e.g. The 2p3/2 contains four nucleons, all of the same energy. Higher in energy is the 2p1/2 which contains two equal-energy nucleons). ## See alsoClebsch-Gordan coefficients |
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Angular_momentum_coupling". A list of authors is available in Wikipedia. |