Editing Angular momentum coupling (section)

==General theory and detailed origin==
[[File:Vector model of orbital angular momentum.svg|250px|right|thumb|Orbital angular momentum (denoted '''l''' or '''L''').]]

===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.

===Examples===
As an example we consider two electrons, in an atom (say the [[helium]] atom) labeled with {{mvar|i}} = 1 and 2. If there is no electron–electron interaction, but only electron–nucleus interaction, then the two electrons can be rotated around the nucleus independently of each other; nothing happens to their energy. The expectation values of both operators, '''{{mvar|l}}'''<sub>1</sub> and '''{{mvar|l}}'''<sub>2</sub>, are conserved.
However, if we switch on the electron–electron interaction that depends on the distance {{mvar|d}}(1,2) between the electrons, then only a simultaneous
and equal rotation of the two electrons will leave {{mvar|d}}(1,2) invariant. In such a case the expectation value of neither
'''{{mvar|l}}'''<sub>1</sub> nor '''{{mvar|l}}'''<sub>2</sub> is a constant of motion in general, but the expectation value of the total orbital angular momentum operator '''{{mvar|L}}''' = '''{{mvar|l}}'''<sub>1</sub> + '''{{mvar|l}}'''<sub>2</sub>
is. Given the eigenstates of '''{{mvar|l}}'''<sub>1</sub> and '''{{mvar|l}}'''<sub>2</sub>, the construction of eigenstates of '''{{mvar|L}}''' (which still is conserved)  is the ''coupling of the angular momenta of electrons'' 1 ''and'' 2.

The total orbital angular momentum quantum number {{mvar|L}} is restricted to integer values and must satisfy the triangular condition that <math>|l_1 - l_2| \leq L \leq l_1 + l_2</math>, such that the three nonnegative integer values could correspond to the three sides of a triangle.<ref>{{cite book |last=Merzbacher |first=Eugen |year=1998 |title=Quantum Mechanics |edition=3rd |publisher=John Wiley |pages=428–429 |ISBN=0-471-88702-1}}</ref>

In [[quantum mechanics]], coupling also exists between angular momenta belonging to different [[Hilbert space]]s of a single object, e.g. its [[Spin (physics)|spin]] and its orbital [[angular momentum]]. If the spin has half-integer values, such as {{sfrac|1|2}} for an electron, then the total (orbital plus spin) angular momentum will also be restricted to half-integer values.

Reiterating slightly differently the above:  one expands the [[quantum state]]s of composed systems (i.e. made of subunits like two [[hydrogen atom]]s or two [[electron]]s) in [[basis (linear algebra)|basis sets]] which are made of [[tensor product]]s of [[quantum state]]s which in turn describe the subsystems individually. We assume that the states of the subsystems can be chosen as eigenstates of their angular momentum operators (and of their component along any arbitrary {{mvar|z}} axis).

The subsystems are therefore correctly described by a pair of {{mvar|ℓ}}, {{mvar|m}} [[quantum number]]s (see [[angular momentum]] for details). When there is interaction among the subsystems, the total Hamiltonian contains terms that do not commute with the angular operators acting on the subsystems only. However, these terms ''do'' commute with the ''total'' angular momentum operator. Sometimes one refers to the non-commuting interaction terms in the Hamiltonian as ''angular momentum coupling terms'', because they necessitate the angular momentum coupling.