Angular momentum coupling
[1][2] In astronomy, spin–orbit coupling reflects the general law of conservation of angular momentum, which holds for celestial systems as well.
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:[citation needed] In both cases the angular momentum operator commutes with the Hamiltonian of the system.
By Heisenberg's uncertainty relation this means that the angular momentum and the energy (eigenvalue of the Hamiltonian) can be measured at the same time.
In this model the atomic Hamiltonian is a sum of kinetic energies of the electrons and the spherically symmetric electron–nucleus interactions.
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.
The total orbital angular momentum quantum number L is restricted to integer values and must satisfy the triangular condition that
Reiterating slightly differently the above: one expands the quantum states of composed systems (i.e. made of subunits like two hydrogen atoms or two electrons) in basis sets which are made of tensor products of quantum states 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 z axis).
The subsystems are therefore correctly described by a pair of ℓ, m quantum numbers (see angular momentum for details).
The behavior of atoms and smaller particles is well described by the theory of quantum mechanics, in which each particle has an intrinsic angular momentum called spin and specific configurations (of e.g. electrons in an atom) are described by a set of quantum numbers.
Angular momentum coupling is a category including some of the ways that subatomic particles can interact with each other.
In solid-state physics, the spin coupling with the orbital motion can lead to splitting of energy bands due to Dresselhaus or Rashba effects.
In light atoms (generally Z ≤ 30[4]), electron spins si interact among themselves so they combine to form a total spin angular momentum S. The same happens with orbital angular momenta ℓi, forming a total orbital angular momentum L. The interaction between the quantum numbers L and S is called Russell–Saunders coupling (after Henry Norris Russell and Frederick Saunders) or LS coupling.
In larger magnetic fields, these two momenta decouple, giving rise to a different splitting pattern in the energy levels (the Paschen–Back effect), and the size of LS coupling term becomes small.
[8] Term symbols are used to represent the states and spectral transitions of atoms, they are found from coupling of angular momenta mentioned above.
In the Rydberg formula the frequency or wave number of the light emitted by a hydrogen-like atom is proportional to the difference between the two terms of a transition.
The series known to early spectroscopy were designated sharp, principal, diffuse, and fundamental and consequently the letters S, P, D, and F were used to represent the orbital angular momentum states of an atom.
Thus, for example, uranium molecular orbital diagrams must directly incorporate relativistic symbols when considering interactions with other atoms.
