Editing Angular momentum (section)

=== Spin, orbital, and total angular momentum ===
{{Main|Spin (physics)}}

[[File:Classical angular momentum.svg|upright=1.25|thumb|Angular momenta of a ''classical'' object.{{ubl
| '''Left:''' "spin" angular momentum '''S''' is really orbital angular momentum of the object at every point.
| '''Right:''' extrinsic orbital angular momentum '''L''' about an axis.
| '''Top:''' the [[moment of inertia tensor]] '''I''' and [[angular velocity]] '''ω''' ('''L''' is not always parallel to '''ω''').<ref>{{cite book|title=Feynman's Lectures on Physics (volume 2)|author1=R.P. Feynman |author2=R.B. Leighton |author3=M. Sands |publisher=Addison–Wesley|year=1964|pages=31–7|isbn=978-0-201-02117-2}}</ref>
| '''Bottom:''' momentum '''p''' and its radial position '''r''' from the axis. The total angular momentum (spin plus orbital) is '''J'''. For a ''quantum'' particle the interpretations are different; [[Spin (physics)|particle spin]] does ''not'' have the above interpretation.}}
]]

The classical definition of angular momentum as <math>\mathbf{L} = \mathbf{r}\times\mathbf{p}</math> can be carried over to quantum mechanics, by reinterpreting '''r''' as the quantum [[position operator]] and '''p''' as the quantum [[momentum operator]]. '''L''' is then an [[Operator (physics)|operator]], specifically called the ''[[angular momentum operator|orbital angular momentum operator]]''. The components of the angular momentum operator satisfy the commutation relations of the Lie algebra so(3). Indeed, these operators are precisely the infinitesimal action of the rotation group on the quantum [[Hilbert space]].<ref>{{harvnb|Hall|2013}} Section 17.3</ref> (See also the discussion below of the angular momentum operators as the generators of rotations.)

However, in quantum physics, there is another type of angular momentum, called ''spin angular momentum'', represented by the spin operator '''S'''. Spin is often depicted as a particle literally spinning around an axis, but this is a misleading and inaccurate picture: spin is an intrinsic property of a particle, unrelated to any sort of motion in space and fundamentally different from orbital angular momentum. All [[elementary particle]]s have a characteristic spin (possibly zero),<ref>{{cite book |title=Facts And Mysteries In Elementary Particle Physics |edition=revised |first1=Martinus J G |last1=Veltman |publisher=World Scientific |year=2018 |isbn=978-981-323-707-0 |url=https://books.google.com/books?id=xWdhDwAAQBAJ&pg=PT351}}</ref> and almost all [[elementary particle]]s have nonzero spin.<ref>{{cite book |title=Advanced Visual Quantum Mechanics |edition=illustrated |first1=Bernd |last1=Thaller |publisher=Springer Science & Business Media |year=2005 |isbn=978-0-387-27127-9 |page=[https://books.google.com/books?id=iq1Gi6hmTRAC&pg=PA114 114] |url=https://books.google.com/books?id=iq1Gi6hmTRAC}}</ref> For example [[electron]]s have "spin 1/2" (this actually means "spin [[reduced Planck constant|ħ]]/2"), [[photon]]s have "spin 1" (this actually means "spin ħ"), and [[Pion|pi-mesons]] have spin 0.<ref>{{cite book |title=Relativistic Quantum Mechanics: With Applications in Condensed Matter and Atomic Physics |edition=illustrated |first1=Paul |last1=Strange |publisher=Cambridge University Press |year=1998 |isbn=978-0-521-56583-7 |page=[https://books.google.com/books?id=sdVrBM2w0OwC&pg=PA64 64] |url=https://books.google.com/books?id=sdVrBM2w0Ow}}</ref>

Finally, there is [[total angular momentum]] '''J''', which combines both the spin and orbital angular momentum of all particles and fields. (For one particle, {{nowrap|'''J''' {{=}} '''L''' + '''S'''}}.) [[Conservation of angular momentum]] applies to '''J''', but not to '''L''' or '''S'''; for example, the [[spin–orbit interaction]] allows angular momentum to transfer back and forth between '''L''' and '''S''', with the total remaining constant. Electrons and photons need not have integer-based values for total angular momentum, but can also have half-integer values.<ref>{{Cite journal|doi=10.1126/sciadv.1501748|pmid=28861467|title=There are many ways to spin a photon: Half-quantization of a total optical angular momentum|journal=Science Advances|volume=2|issue=4|pages=e1501748|year=2016|last1=Ballantine|first1=K. E.|last2=Donegan|first2=J. F.|last3=Eastham|first3=P. R.|bibcode = 2016SciA....2E1748B|pmc=5565928}}</ref>

In molecules the total angular momentum '''F''' is the sum of the rovibronic (orbital) angular momentum '''N''', the electron spin angular momentum '''S''', and the nuclear spin angular momentum '''I'''. For electronic singlet states the rovibronic angular momentum is denoted '''J''' rather than '''N'''. As explained by Van Vleck,<ref>
{{cite journal
| author1 = J. H. Van Vleck
| title = The Coupling of Angular Momentum Vectors in Molecules
| journal = Rev. Mod. Phys.
| volume = 23 | page= 213 | year = 1951
| issue = 3
| doi = 10.1103/RevModPhys.23.213
| bibcode = 1951RvMP...23..213V
}}</ref> the components of the molecular rovibronic angular momentum referred to molecule-fixed axes have different commutation relations from those for the components about space-fixed axes.