the atomic Hamiltonian is a sum of kinetic energies of the electrons and the spherically symmetric electron–nucleus interactions. The individual electron angular momenta 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. 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 combine 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 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 = 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. Both operators, 1 and 2, are conserved. However, if we switch on the electron–electron interaction that depends on the distance (1,2) between the electrons, then only a simultaneous and equal rotation of the two electrons will leave (1,2) invariant. In such a case neither 1 nor 2 is a constant of motion in general, but the total orbital angular momentum = 1 + 2 is. Given the eigenstates of 1 and 2, the construction of eigenstates of (which still is conserved) is the coupling of the angular momenta of electrons 1 and 2. The total orbital angular momentum quantum number is restricted to integer values