Editing Angular momentum coupling (section)

===Angular momentum conservation===

[[Conservation of angular momentum]] is the principle that the total angular momentum of a system has a constant magnitude and direction if the system is subjected to no external [[torque]]. [[Angular momentum]] is a property of a physical system that is a [[constant of motion]] (also referred to as a ''conserved'' property, time-independent and well-defined) in two situations:{{cn|date=February 2024}}

#The system experiences a spherically symmetric potential field. 
#The system moves (in quantum mechanical sense) in isotropic space.

In both cases the angular momentum operator [[commutator|commutes]] with the  [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the system. By Heisenberg's [[Heisenberg Uncertainty Principle|uncertainty relation]] this means that the angular momentum and the energy (eigenvalue of the Hamiltonian) can be measured at the same time.

An example of the first situation is an atom whose [[electron]]s only experience the [[Coulomb force]] of its [[atomic nucleus]]. If we ignore the electron–electron interaction (and other small interactions such as [[spin–orbit coupling]]), the ''orbital angular momentum'' {{mvar|l}} of each electron commutes with the total Hamiltonian. In this model the atomic Hamiltonian is a sum of kinetic energies of the electrons and the spherically symmetric electron–nucleus interactions. The individual electron angular momenta {{mvar|l<sub>i</sub>}} commute with this Hamiltonian. That is, they are conserved properties of this approximate model of the atom.

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.{{cn|date=February 2024}}

These two situations originate in classical mechanics. The third kind of conserved angular momentum, associated with [[Spin (physics)|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 combine these momenta to a total angular momentum of the combined system&mdash;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 ''angular momentum coupling''.

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.