Editing Angular momentum coupling (section)

===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.