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{{Short description|Coupling in quantum physics}}
{{Coupling in molecules}}
In [[quantum mechanics]], '''angular momentum coupling''' is the procedure of constructing [[eigenstates]] of total
In both cases the separate angular momenta are no longer [[constants of motion]], but the sum of the two angular momenta usually still is. Angular momentum coupling in atoms is of importance in atomic [[spectroscopy]]. Angular momentum coupling of [[electron spin]]s is
In [[astronomy]], '''
==General theory and detailed origin==
[[File:Vector model of orbital angular momentum.svg|250px
===Angular momentum conservation===
[[Conservation of angular momentum]] is the principle that the total angular momentum of a system has a constant magnitude and direction if the system is subjected to no external [[torque]]. [[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:{{cn|date=February 2024}}
#The system experiences a spherically symmetric potential field.
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In both cases the angular momentum operator [[commutator|commutes]] with the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of the system. By Heisenberg's [[Heisenberg Uncertainty Principle|uncertainty relation]] this means that the angular momentum and the energy (eigenvalue of the Hamiltonian) can be measured at the same time.
An example of the first situation is an atom whose [[electron]]s only
An example of the second situation is a [[rigid rotor]] moving in field-free space. A rigid rotor has a well-defined, time-independent,
These two situations originate in classical mechanics. The third kind of conserved angular momentum, associated with [[
In general the conservation of angular momentum implies full rotational symmetry
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===Examples===
As an example we consider two electrons
However, if we switch on the
and equal rotation of the two electrons will leave
'''{{mvar|l}}'''<sub>1</sub> nor '''{{mvar|l}}'''<sub>2</sub> is a constant of
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
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
▲As an example we consider two electrons, 1 and 2, in an atom (say the [[helium]] atom). If there is no electron-electron interaction, but only electron-nucleus interaction, the two electrons can be rotated around the nucleus independently of each other; nothing happens to their energy. Both operators, '''l'''<sub>1</sub> and '''l'''<sub>2</sub>, are conserved.
▲However, if we switch on the electron-electron interaction that depends on the distance ''d''(1,2) between the electrons, then only a simultaneous
▲and equal rotation of the two electrons will leave ''d''(1,2) invariant. In such a case neither
▲'''l'''<sub>1</sub> nor '''l'''<sub>2</sub> is a constant of motion in general, but the total orbital angular momentum '''L''' = '''l'''<sub>1</sub> + '''l'''<sub>2</sub>
▲is. Given the eigenstates of '''l'''<sub>1</sub> and '''l'''<sub>2</sub>, the construction of eigenstates of '''L''' (which still is conserved) is the ''coupling of the angular momenta of electrons 1 and 2''.
In [[quantum mechanics]], coupling also exists between angular momenta belonging to different [[Hilbert space]]s of a single object, e.g. its [[
▲such that the three nonnegative integer values could correspond to the three sides of a triangle.<ref>Merzbacher, Eugen (1998). Quantum Mechanics (3rd ed.). John Wiley. pp. 428–9. ISBN 0-471-88702-1.</ref>
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.
▲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 1/2 for an electron, then the total (orbital plus spin) angular momentum will also be restricted to half-integer values.
The subsystems are therefore correctly described by a
▲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 ''z'' axis).
▲The subsystems are therefore correctly described by a set of ''{{ell}}'', ''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.
The behavior of [[atoms]] and smaller [[Subatomic particle|particles]] is well described by the theory of [[quantum mechanics]], in which each particle has an intrinsic angular momentum called [[
▲==Spin-orbit coupling==
▲{{Main article|Spin-orbit coupling}}
In [[atomic physics]], [[
▲The behavior of [[atoms]] and smaller [[Subatomic particle|particles]] is well described by the theory of [[quantum mechanics]], in which each particle has an intrinsic angular momentum called [[spin (physics)|spin]] and specific configurations (of e.g. electrons in an atom) are described by a set of [[quantum numbers]]. Collections of particles also have angular momenta and corresponding quantum numbers, and under different circumstances the angular momenta of the parts couple in different ways to form the angular momentum of the whole. 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 [[Electronic band structure|energy bands]] due to [[Dresselhaus effect|Dresselhaus]] or [[Rashba effect|Rashba]] effects.
▲In [[atomic physics]], [[spin-orbit coupling]], also known as '''spin-pairing''', describes a weak magnetic interaction, or [[coupling (physics)|coupling]], of the particle [[spin (physics)|spin]] and the [[orbital motion (quantum)|orbital motion]] of this particle, e.g. the [[electron]] spin and its motion around an [[atom]]ic [[atomic nucleus|nucleus]]. One of its effects is to separate the energy of internal states of the atom, e.g. spin-aligned and spin-antialigned that would otherwise be identical in energy. This interaction is responsible for many of the details of atomic structure.
In the [[macroscopic]] world of [[astrodynamics|orbital mechanics]], the term ''
===LS coupling===
[[File:LS coupling (corrected).
In light atoms (generally ''Z'' ≤ 30<ref>[http://chemwiki.ucdavis.edu/Physical_Chemistry/Spectroscopy/Electronic_Spectroscopy/The_atomic_spectrum/Atomic_Term_Symbols/The_Russell_Saunders_Coupling_Scheme The Russell Saunders Coupling Scheme] R. J. Lancashire, UCDavis ChemWiki (accessed 26 Dec.2015)</ref>), electron spins '''s'''<sub>''i''</sub> interact among themselves so they combine to form a total spin angular momentum '''S'''. The same happens with orbital angular momenta '''ℓ'''<sub>''i''</sub>, forming a total orbital angular momentum '''L'''. The interaction between the quantum numbers '''L''' and '''S''' is called '''Russell–Saunders coupling''' (after [[Henry Norris
▲In light atoms (generally ''Z'' ≤ 30<ref>[http://chemwiki.ucdavis.edu/Physical_Chemistry/Spectroscopy/Electronic_Spectroscopy/The_atomic_spectrum/Atomic_Term_Symbols/The_Russell_Saunders_Coupling_Scheme The Russell Saunders Coupling Scheme] R.J.Lancashire, UCDavis ChemWiki (accessed 26 Dec.2015)</ref>), electron spins '''s'''<sub>''i''</sub> interact among themselves so they combine to form a total spin angular momentum '''S'''. The same happens with orbital angular momenta '''ℓ'''<sub>''i''</sub>, forming a total orbital angular momentum '''L'''. The interaction between the quantum numbers '''L''' and '''S''' is called ''[[Henry Norris Russell|Russell]]–Saunders coupling'' or ''LS coupling''. Then '''S''' and '''L''' couple together and form a total angular momentum '''J''':<ref>{{cite book|title = Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles|edition=2nd|author=R. Resnick, R. Eisberg|publisher=John Wiley & Sons|year=1985|page=281|isbn=978-0-471-87373-0}}</ref><ref>{{cite book|title = Physics of Atoms and Molecules|author=B.H. Bransden, C.J.Joachain|publisher=Longman|year=1983|pages=339–341|isbn=0-582-44401-2}}</ref>
:<math>\mathbf J = \mathbf L + \mathbf S, \, </math>
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: <math>\mathbf L = \sum_i \boldsymbol{\ell}_i, \ \mathbf S = \sum_i \mathbf{s}_i. \, </math>
This is an approximation which is good as long as any external magnetic fields are weak. In larger magnetic fields, these two momenta decouple, giving rise to a different splitting pattern in the energy levels (the
For an extensive example on how LS-coupling is practically applied, see the article on [[term symbol]]s.
===jj coupling===
In heavier atoms the situation is different. In atoms with bigger nuclear charges,
:<math>\mathbf J = \sum_i \mathbf j_i = \sum_i (\boldsymbol{\ell}_i + \mathbf{s}_i).</math>
This description, facilitating calculation of this kind of interaction, is known as ''jj coupling''.
==
{{See also|J-coupling|Dipolar coupling|NMR spectroscopy}}
'''
provide detailed information about the structure and conformation of molecules.
== Term symbols ==
{{Main article|Term symbol}}
Term symbols are used to represent the states and spectral transitions of atoms, they are found from coupling of angular momenta mentioned above. When the state of an atom has been specified with a term symbol, the allowed transitions can be found through [[selection rule]]s by considering which transitions would conserve [[angular momentum]]. A [[photon]] has spin 1, and when there is a transition with emission or absorption of a photon the atom will need to change state to conserve angular momentum. The term symbol selection rules are
The expression "term symbol" is derived from the "term series" associated with the [[Rydberg state]]s of an atom and their [[Atomic energy level#Orbital state energy level|energy levels]]. 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 [[Atomic spectral line|spectroscopy]] were designated ''sharp'', ''principal'', ''diffuse'', and ''fundamental'' and consequently the letters {{math|S, P, D,}} and {{math|F}} were used to represent the orbital angular momentum states of an atom.<ref>{{cite book |last=Herzberg |first=Gerhard |title=Atomic Spectra and Atomic Structure |url=https://archive.org/details/atomicspectraato00herz_877 |url-access=limited |publisher=Dover |location=New York |year=1945 |isbn=0-486-60115-3 |pages=
== Relativistic effects ==
In very heavy atoms, relativistic shifting of the energies of the electron energy levels accentuates
== Nuclear coupling ==
In atomic nuclei, the
== See also ==
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<references/>
==
* [http://hyperphysics.phy-astr.gsu.edu/hbase/atomic/lcoup.html#c1 LS and jj coupling]
* [http://hyperphysics.phy-astr.gsu.edu/hbase/atomic/term.html#c1 Term symbol]
* [http://nucracker.volya.net/index.php?p=spins Web calculator of spin couplings: shell model, atomic term symbol]
[[Category:Angular momentum]]
[[Category:Atomic physics]]
[[Category:Rotational symmetry]]
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