Angular momentum coupling
In quantum mechanics, angular momentum coupling is the procedure of constructing eigenstates of a system's angular momentum out of angular momentum eigenstates of its subsystems. The historic example of a system is an atom with N > 1 electrons, with the electrons being its subsystems. Each electron has its own orbital angular momentum, i.e., is in an eigenstate of its own angular momentum operator. For an atom, angular momentum coupling is the construction of an N-electron eigenstate of the total angular momentum operator out of the individual electronic eigenstates.
Other examples are the coupling of spin- and orbital-angular momentum of an electron (where we see the spin and the orbital motion as subsystems of a single electron) and the coupling of nucleonic spins in the shell model of the nucleus.
Usefulness
Angular momentum coupling is useful under two conditions. (i) The angular momenta of the subsystems are constants of the motion when the interactions between the subsystems are switched off (or neglected). (ii) Upon switching on the interactions, the angular momenta of the subsystems are no longer constants of the motion, but the total angular momentum remains a constant of the motion.
These two conditions are surprisingly often fulfilled due to the fact that they almost always follow from rotational symmetry—the symmetry of spherical systems and isotropic interactions. First we note that angular momentum is a time-independent and well-defined property of a physical system [1] in either of two situations: (i) The system is spherical symmetric, or (ii) the system moves (in quantum mechanical sense) in isotropic space. It can be shown that in both cases the angular momentum operator of the system commutes with its Hamiltonian. By Heisenberg's uncertainty relation this means that the angular momentum of the system can assume a sharp value simultaneously with the energy (eigenvalue of the Hamiltonian) of the system. The standard example of a spherical symmetric system is an atom, while a molecule moving in a field-free space is an example of the second kind of system.
Footnote
- ↑ A constant of the motion, also referred to as a conserved property
See also
(To be continued)