Editing Angular momentum (section)

== Angular momentum in quantum mechanics ==
{{Main|Angular momentum operator}}
In [[quantum mechanics]], angular momentum (like other quantities) is expressed as an [[operator (physics)|operator]]<!-- note that [[angular momentum operator]] is linked from [[#In quantum mechanics]]-->, and its one-dimensional projections have [[point spectrum|quantized eigenvalues]]. Angular momentum is subject to the [[Heisenberg uncertainty principle]], implying that at any time, only one [[vector projection|projection]] (also called "component") can be measured with definite precision; the other two then remain uncertain. Because of this, the axis of rotation of a quantum particle is undefined. Quantum particles ''do'' possess a type of non-orbital angular momentum called "spin", but this angular momentum does not correspond to a spinning motion.<ref>{{cite book |title=Understanding the Properties of Matter |edition=2nd, illustrated, revised |first1=Michael |last1=de Podesta |publisher=CRC Press |year=2002 |isbn=978-0-415-25788-6 |page= [https://books.google.com/books?id=ya-4enCDoWQC&pg=PA29 29] |url=https://books.google.com/books?id=ya-4enCDoWQC}}</ref> In [[relativistic quantum mechanics#Relativistic quantum angular momentum|relativistic quantum mechanics]] the above relativistic definition becomes a tensorial operator.

=== 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.

=== Quantization ===
{{Further|Angular momentum operator}}
In [[quantum mechanics]], angular momentum is [[angular momentum quantization|quantized]] – that is, it cannot vary continuously, but only in "[[Quantum number|quantum leaps]]" between certain allowed values. For any system, the following restrictions on measurement results apply, where <math>\hbar</math> is the [[reduced Planck constant]] and <math>\hat n</math> is any [[Euclidean vector]] such as x, y, or z:
{| class="wikitable"
|-
| '''If you [[measurement in quantum mechanics|measure]]...'''
| '''The result can be...'''
|-
| <math>L_\hat{n}</math>
| <math>\ldots, -2\hbar, -\hbar, 0, \hbar, 2\hbar, \ldots</math>
|-
| <math>S_\hat{n}</math> or <math>J_\hat{n}</math>
| <math>\ldots, -\frac{3}{2}\hbar, -\hbar, -\frac{1}{2}\hbar, 0, \frac{1}{2}\hbar, \hbar, \frac{3}{2}\hbar, \ldots</math>
|-
| <math>\begin{align}
      &L^2 \\
  ={} &L_x^2 + L_y^2 + L_z^2
\end{align}</math>
| <math>\left[\hbar^2 n(n + 1)\right]</math>, where <math>n = 0, 1, 2, \ldots</math>
|-
| <math>S^2</math> or <math>J^2</math>
| <math>\left[\hbar^2 n(n + 1)\right]</math>, where <math>n = 0, \tfrac{1}{2}, 1, \tfrac{3}{2}, \ldots</math>
|}
[[File:Circular Standing Wave.gif|thumb|right|In this [[standing wave]] on a circular string, the circle is broken into exactly 8 [[wavelength]]s. A standing wave like this can have 0, 1, 2, or any integer number of wavelengths around the circle, but it ''cannot'' have a non-integer number of wavelengths like 8.3. In quantum mechanics, angular momentum is quantized for a similar reason.]]
The [[reduced Planck constant]] <math>\hbar</math> is tiny by everyday standards, about 10<sup>−34</sup> [[Joule-second|J s]], and therefore this quantization does not noticeably affect the angular momentum of macroscopic objects. However, it is very important in the microscopic world. For example, the structure of [[electron shell]]s and subshells in chemistry is significantly affected by the quantization of angular momentum.

Quantization of angular momentum was first postulated by [[Niels Bohr]] in [[Bohr model|his model]] of the atom and was later predicted by [[Erwin Schrödinger]] in his [[Schrödinger equation#Quantization|Schrödinger equation]].

=== Uncertainty ===
In the definition <math>\mathbf{L}=\mathbf{r}\times\mathbf{p}</math>, six operators are involved: The [[position operator]]s <math>r_x</math>, <math>r_y</math>, <math>r_z</math>, and the [[momentum operator]]s <math>p_x</math>, <math>p_y</math>, <math>p_z</math>. However, the [[uncertainty principle|Heisenberg uncertainty principle]] tells us that it is not possible for all six of these quantities to be known simultaneously with arbitrary precision. Therefore, there are limits to what can be known or measured about a particle's angular momentum. It turns out that the best that one can do is to simultaneously measure both the angular momentum vector's [[magnitude (vector)|magnitude]] and its component along one axis.

The uncertainty is closely related to the fact that different components of an angular momentum operator do not [[commutator|commute]], for example <math>L_xL_y \neq L_yL_x</math>. (For the precise [[commutation relation]]s, see [[angular momentum operator]].)

=== Total angular momentum as generator of rotations ===
As mentioned above, orbital angular momentum '''L''' is defined as in classical mechanics: <math>\mathbf{L}=\mathbf{r}\times\mathbf{p}</math>, but ''total'' angular momentum '''J''' is defined in a different, more basic way: '''J''' is defined as the "generator of rotations".<ref name=littlejohn>{{cite web|url=http://bohr.physics.berkeley.edu/classes/221/1011/notes/spinrot.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://bohr.physics.berkeley.edu/classes/221/1011/notes/spinrot.pdf |archive-date=2022-10-09 |url-status=live|author-link1=Robert Grayson Littlejohn|title= Lecture notes on rotations in quantum mechanics|first= Robert |last=Littlejohn|access-date= 13 Jan 2012|work=Physics 221B Spring 2011|year=2011}}</ref> More specifically, '''J''' is defined so that the operator
<math display="block">R(\hat{n},\phi) \equiv \exp\left(-\frac{i}{\hbar}\phi\, \mathbf{J}\cdot \hat{\mathbf{n} }\right)</math>
is the [[Rotation operator (quantum mechanics)|rotation operator]] that takes any system and rotates it by angle <math>\phi</math> about the axis <math>\hat{\mathbf{n}}</math>. (The "exp" in the formula refers to [[matrix exponential|operator exponential]].) To put this the other way around, whatever our quantum Hilbert space is, we expect that the [[rotation group SO(3)]] will act on it. There is then an associated action of the Lie algebra so(3) of SO(3); the operators describing the action of so(3) on our Hilbert space are the (total) angular momentum operators.

The relationship between the angular momentum operator and the rotation operators is the same as the relationship between [[Lie algebra]]s and [[Lie group]]s in mathematics. The close relationship between angular momentum and rotations is reflected in [[Noether's theorem]] that proves that angular momentum is conserved whenever the laws of physics are rotationally invariant.