Angular momentum coupling

In quantum mechanics, the procedure of constructing total angular momentum eigenstates (states of a system with well-defined values of angular momentum) from the individual angular momentum eigenstates is called angular momentum coupling . It is used when, due to a physical interaction between two angular momenta, they are no longer independent constants of motion (their individual values no longer follow conservation laws), but the sum of the two angular momenta normally is. For example, the spin and motion of an electron can interact by spin-orbit coupling, in which case it is useful to couple its orbital and spin angular momenta. Or two charged particles, each with a well-defined angular momentum, can interact by Coulomb forces, and then it is useful to couple the angular momenta of each particle resulting in a total angular momentum, as a step towards solving the Schrödinger equation of two particles.

The coupling of angular momenta in atoms is important to explain atomic spectroscopy experiments. The coupling of angular momenta of electronic spins is of importance in the part of quantum chemistry that studies magnetochemistry, and in the part of quantum physics that studies condensed matter physics.

In astronomy, coupling of angular momenta reflects the general law of conservation of angular momentum that is also valid for celestial objects. In simple cases, the direction of the angular momentum vector is neglected, and the spin-orbit coupling is the ratio between the frequency with which a planet or other celestial body rotates on its own axis and that with which it orbits another body. This is commonly known as orbital resonance. Frequently, the underlying physical effects are tidal forces.

General theory and details of the origin

Angular momentum is a property of physical systems, and it is a constant of motion (conserved property, independent of time and well defined) in two situations: (i) The system is subject to a potential field of spherical symmetry. (ii) The system moves -in a quantum mechanical sense- in isotropic space. In both cases the angular momentum operator commutes with the Hamiltonian of the system. By Heisenberg's uncertainty principle this means that the value of the angular momentum and the energy of the system can have arbitrarily precise values simultaneously.

An example of the first situation is an atom whose electrons are only exposed to the Coulomb field of its nucleus. In this model, the atomic Hamiltonian is the sum of the kinetic energies of the electrons and of the electron-nucleus interactions, of spherical symmetry. Thus, neglecting interelectronic interaction (and other minor perturbations such as spin-orbit coupling), the orbital angular momentum l of each electron commutes with that of the total Hamiltonian.

An example of the second situation is a rigid rotor moving in field-free space. A rigid rotor has a well-defined angular momentum that is independent of time.

These two situations originate from classical mechanics. A third type of conserved angular momentum, associated with the quantum magnitude of spin, has no classical analogue. However, all the rules for angular momentum coupling apply to spin as well.

In general, conservation of angular momentum implies complete rotational symmetry (described by the symmetry groups SO(3) and SU(2)), and vice versa. If two or more physical systems have separate angular momenta conservation, it may be useful to sum these moments into a total angular momentum, which will be a conserved property of the combined system. The construction of eigenstates of the total angular momentum from the eigenstates of the angular momenta of the individual subsystems is called angular momentum coupling.

The application of angular momentum coupling is useful when there is an interaction between subsystems that, without it, would have conserved angular momenta. The interaction breaks the spherical symmetry of the subsystems, but the total angular momentum remains a constant of motion, which is useful for solving the Schrödinger equation.

In quantum mechanics, this type of coupling also occurs between angular momenta belonging to different Hilbert spaces of the same object, for example its spin and its orbital magnetic moment.

Spin-orbit coupling

The behavior of subatomic particles is well described by the theory of quantum mechanics, in which each particle has an intrinsic angular momentum called spin, and where specific configurations - for example of electrons in an atom - are described by a series of quantum numbers. Collectives of particles also have angular momenta and corresponding quantum numbers, and under different conditions the angular momenta of the parts add up in different ways resulting in different overall angular momenta. The spin-orbit interaction is the interaction between a magnetic moment associated with the spin and its spatial movement under a potential (usually of electrostatic origin).

In atomic physics, spin-orbit coupling describes an interaction of the magnetic moment associated with the spin of the electrons and their orbital motion around the nucleus. The effect occurs in the spectrum of the atom or molecule, where the spectral lines that coincided separate. Spectral lines are associated with energy levels of the system. In the case where energy levels seem to coincide, for example they have the same energy if the electron was spin aligned or anti-aligned with the orbital angular momentum, now they are separated a bit due to the spin-orbit interaction that prefers one alignment over the other..

This interaction is responsible for many details of atomic and molecular structure. Commonly, we find it when in an ion, an atom or a molecule, in addition to unpaired electrons (which provide spin magnetic moment) we have an electronic configuration with orbital degeneracy. In these cases, electrostatic interaction (Coulomb repulsion) coexists with relativistic electromagnetic effects (spin-orbit interaction).

In solid state physics, the difference between energy bands given by spin-orbit coupling is observed when the magnetic moment associated with charge carriers in a solid interacts with the electrostatic field of the crystal lattice due to the relative motion between carriers and ions. This phenomenon is remarkable when the lattice lacks certain symmetries, for example the Dresselhaus spin-orbit effect appears in solid systems without inversion symmetry. Another example is the so-called quantum dots (also called artificial atoms) where the interaction appears due to the restrictions of movement over a small space.

In the macroscopic world of orbital mechanics, the term spin-orbit coupling is sometimes used in the same sense as spin-orbit resonance.

Usually, it is calculated using perturbation theory: the other interactions are assumed to be much stronger, and spin-orbit coupling is treated as a minor perturbation. In this sense, different approaches are common, depending on the case, as detailed below. It must be taken into account that at high magnetic fields, these two moments become uncoupled, giving rise to a different pattern of energy levels described by the Zeeman effect, and the relative importance of the spin-orbit coupling term decreases.

LS or Russell-Saunders scheme

The so-called spinal-orbit interaction is directly related to the null field and the Landé g factor: if there is axial elongation in the coordination of the metal ion, a ${displaystyle lambda }$ positive implies a g. 2 and a D. Axial and ${displaystyle lambda }$ negatives have the same result, while the other two combinations have the opposite result (g/2003/2 and D/2003/0).

This scheme assumes that the electrostatic interaction is much stronger than the magnetic one, and also that the external magnetic fields are weak. In light atoms (usually Z<30, or metals of the first transition series), the electron spins s_i interact with each other and result in in an angular momentum of spin S. Similarly, the orbital angular momenta l_i form a total orbital angular momentum L. The interaction between the quantum numbers L and S is called Russell-Saunders coupling or LS coupling. S and L add up to form a total angular momentum J:

${displaystyle {vec {J}}={vec {L}}}+{vec {S}}}qquad therefore qquad {vec {L}}}=sum _{i}{vec {l}}}}{qquad {vec {vec {s}}{s}{vec}}{$.

The Hamiltonian model that describes the effect on energy is as follows:

${displaystyle {hat {mathcal {H}}}_{LS}=lambda {vec {L}}cdot {vec {s}}}}}$,

where ${displaystyle lambda }$ is positive for layers of orbital d less than semi-filled (d1-d4), negative for layers more than semi-filled (d6-d9) and null for empty and semi-filled layers.

To measure average energy values, the wave function cannot be used in the standard base, ${displaystyle Δlsm_{1}m_{2}rangle }$since you cannot express the hamiltonian with that base. For this purpose the normal base is used ${displaystyle Δlsjm_{j}rangle }$ that does have that property and the Coefficients Clebsch—Gordan that link one base and another:

${displaystyle Δlsjm_{j}rangle =sum _{m_{1},m_{2}}}}{C_{m_{1}m_{2}}{2}}{jm_{j}}{sm_{1}m_{2}rangle }}}}$

J-j scheme

The situation is different in heavier atoms, where spin-orbit interactions are often of comparable size to spin-spin and/or orbit-orbit interactions, so S and J are no longer good quantum numbers for those systems. The j-j scheme is the opposite of the Russell-Saunders scheme, and is based on a magnetic interaction that is much stronger than the electrostatic one. It is valid especially for the actinides, and, to a lesser extent, for the lanthanides.

In cases where spin-orbit interactions are comparatively small, it is best to combine each individual orbital angular momentum l_i with its corresponding spin angular momentum s_i, giving rise to individual angular momenta j _i, and add these to get the total angular momentum J

${displaystyle mathbf {J} =sum _{i}mathbf {j} _{i}=sum _{i}(mathbf {l} _{i}+mathbf {s} _{i}). !$

Intermediate Couplings

When a simultaneous application of the two interactions is necessary, because they are of comparable magnitude, the resolution of the problem presents a much greater complexity, and it is not possible to arrive at general analytical solutions. In these cases, it is useful and illustrative to represent correlation diagrams between the two previous approaches. In general, these will not present a one-to-one correspondence between states, but will show mixtures, due to the non-crossing rule.

Spin-spin coupling

The spin-spin coupling is what occurs between the spin angular momenta, intrinsic to different particles. Between nuclear spins, it is of great importance in nuclear magnetic resonance, since it provides information on the structure of the molecules under study. Between nuclear and electronic spins, it defines the hyperfine structure, for example in atomic spectroscopy or in electron spin resonance spectroscopy. Between electronic spins, magnetic exchange is the basis of magnetochemistry, giving rise, for example, to ferromagnetic or antiferromagnetic couplings.