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{{Short description|Quantum-mechanical operator associated with rotational symmetry}}

{{Quantum book backlink|Mathematical structure and systems}}
In [[Physics:quantum mechanics|quantum mechanics]], the '''angular momentum operator''' is the operator associated with rotational motion and rotational symmetry, and is the quantum analogue of [[Physics:Angular momentum|angular momentum]] in classical mechanics. Classically, angular momentum is described by a vector <math>\mathbf{L} = (L_x, L_y, L_z)</math>, whose components can all be specified simultaneously. In quantum mechanics, these components become operators <math>\hat{L}_x, \hat{L}_y, \hat{L}_z</math>, representing measurements of angular momentum along each axis. However, unlike in classical physics, these operators do not commute, satisfying relations such as <math>[\hat{L}_x, \hat{L}_y] = i\hbar \hat{L}_z</math> (and cyclic permutations), which implies that the components <math>L_x, L_y, L_z</math> cannot be simultaneously known exactly. Instead, one can simultaneously determine the total angular momentum <math>\hat{L}^2 = \hat{L}_x^2 + \hat{L}_y^2 + \hat{L}_z^2</math> and a single component, usually <math>\hat{L}_z</math>. Quantum states are therefore labeled by the quantum numbers <math>\ell</math> (total angular momentum) and <math>m</math> (its projection along a chosen axis). Geometrically, this corresponds to a situation in which the magnitude of the angular momentum vector is well-defined while only one of its directional components is sharp, so the vector cannot be assigned a definite direction in three-dimensional space. Instead, it is often visualized as lying on a sphere of fixed radius (set by <math>L^2</math>) with uncertainty in its orientation, forming a “cone” of possible directions. As an [[Physics:Observable|observable]], angular momentum is represented by operators whose eigenstates correspond to states with definite angular momentum, and whose eigenvalues give the possible results of measurement; it plays a central role in atomic physics, molecular physics, spectroscopy, and quantum theory more generally.

Angular momentum is one of the fundamental conserved quantities of motion, together with [[Physics:Angular momentum|linear momentum]] and [[Physics:Energy|energy]].<ref name="Liboff">Introductory Quantum Mechanics, [[Richard L. Liboff]], 2nd Edition, {{ISBN|0-201-54715-5}}</ref> In quantum mechanics, several related angular momentum operators appear:

* the '''orbital angular momentum''' operator, usually denoted <math>\mathbf{L}</math>,
* the '''spin angular momentum''' operator, usually denoted <math>\mathbf{S}</math>,
* the '''total angular momentum''' operator, usually denoted <math>\mathbf{J}</math>.

These are related by
<math display="block">\mathbf{J}=\mathbf{L}+\mathbf{S}.</math>

Depending on context, the expression ''angular momentum operator'' may refer either to orbital angular momentum or to total angular momentum. For a closed system, the total angular momentum is conserved, in accordance with rotational symmetry and [[Physics:Noether's theorem|Noether's theorem]].
[[File:Quantum angular momentum operator in physics.jpg|thumb|450px]]
{{quantum mechanics}}
==Overview==
[[File:LS coupling (corrected).png|thumb|250x250px|''Vector cones'' of total angular momentum <math>\mathbf{J}</math> (green), orbital angular momentum <math>\mathbf{L}</math> (blue), and spin angular momentum <math>\mathbf{S}</math> (red). The cone structure reflects the fact that not all components can be known simultaneously; see [[#Visual interpretation|Visual interpretation]].]]

In quantum mechanics, angular momentum appears in three closely related forms: orbital angular momentum, spin angular momentum, and total angular momentum.

===Orbital angular momentum===
The classical angular momentum of a particle is
<math display="block">\mathbf{L}=\mathbf{r}\times\mathbf{p}.</math>
The same formal expression holds in quantum mechanics, except that <math>\mathbf{r}</math> and <math>\mathbf{p}</math> are now the [[Physics:Position operator|position operator]] and [[Physics:Momentum operator|momentum operator]]:
<math display="block">\mathbf{L}=\mathbf{r}\times\mathbf{p}.</math>

Thus <math>\mathbf{L}</math> is a [[vector operator]], with components
<math display="block">\mathbf{L}=(L_x,L_y,L_z).</math>

For a single spinless, uncharged particle in the position representation,
<math display="block">\mathbf{L}=-i\hbar(\mathbf{r}\times\nabla),</math>
where <math>\nabla</math> is the gradient operator.

===Spin angular momentum===
{{main|Physics:Spin}}

In addition to orbital angular momentum, quantum systems may possess an intrinsic form of angular momentum called '''spin''', represented by
<math display="block">\mathbf{S}=(S_x,S_y,S_z).</math>

Spin has no exact classical counterpart. It is often illustrated as if a particle were spinning about an axis, but this picture is only heuristic; spin is an intrinsic quantum property.<ref name="ohanian">{{Cite journal |last=Ohanian |first=Hans C. |date=1986-06-01 |title=What is spin? |url=https://physics.mcmaster.ca/phys3mm3/notes/whatisspin.pdf |journal=American Journal of Physics |volume=54 |issue=6 |pages=500–505 |doi=10.1119/1.14580 |bibcode=1986AmJPh..54..500O |issn=0002-9505}}</ref> Elementary particles have fixed intrinsic spin: for example, electrons have spin <math>\tfrac12</math>, while photons have spin 1.

===Total angular momentum===
The '''total angular momentum''' operator combines orbital and spin contributions:
<math display="block">\mathbf{J}=\mathbf{L}+\mathbf{S}.</math>

For a closed system, the total angular momentum is conserved. By contrast, <math>\mathbf{L}</math> and <math>\mathbf{S}</math> need not be conserved separately. For example, in [[Physics:Spin–orbit interaction|spin–orbit interaction]], angular momentum may be exchanged between orbital and spin parts while the total <math>\mathbf{J}</math> remains constant.

==Commutation relations==

===Commutation relations between components===
The components of the orbital angular momentum operator satisfy the commutation relations<ref>{{cite book|chapter-url=https://books.google.com/books?id=dRsvmTFpB3wC&pg=PA171|title=Quantum Mechanics|first=G.|last=Aruldhas|page=171|chapter=formula (8.8)|isbn=978-81-203-1962-2|date=2004-02-01|publisher=Prentice Hall India}}</ref>
<math display="block">[L_x,L_y]=i\hbar L_z,\qquad [L_y,L_z]=i\hbar L_x,\qquad [L_z,L_x]=i\hbar L_y,</math>
where the [[commutator]] is defined by
<math display="block">[X,Y]\equiv XY-YX.</math>

In index notation,
<math display="block">[L_l,L_m]=i\hbar\sum_{n=1}^{3}\varepsilon_{lmn}L_n,</math>
or, using Einstein summation convention,
<math display="block">[L_l,L_m]=i\hbar\varepsilon_{lmn}L_n.</math>

These relations can also be written compactly as<ref>{{cite book |last1=Shankar|first1=R.|title=Principles of Quantum Mechanics|url=https://archive.org/details/principlesquantu00shan_139|url-access=limited|date=1994|publisher=Kluwer Academic / Plenum|location=New York|isbn=9780306447907|page=[https://archive.org/details/principlesquantu00shan_139/page/n338 319]|edition=2nd}}</ref>
<math display="block">\mathbf{L}\times\mathbf{L}=i\hbar\mathbf{L}.</math>

They follow from the canonical commutation relations
<math display="block">[x_l,p_m]=i\hbar\delta_{lm}.</math>

The same algebra holds for spin and total angular momentum:
<math display="block">[S_l,S_m]=i\hbar\sum_{n=1}^{3}\varepsilon_{lmn}S_n,\qquad
[J_l,J_m]=i\hbar\sum_{n=1}^{3}\varepsilon_{lmn}J_n.</math>

These commutation relations show that angular momentum operators generate the Lie algebra associated with three-dimensional rotations, usually written in physics as the algebra of [[Physics:SU(2) color superconductivity|SU(2)]] or [[SO(3)]].

===Commutation relations involving the magnitude===
The square of the orbital angular momentum operator is defined by
<math display="block">L^2\equiv L_x^2+L_y^2+L_z^2.</math>

This operator commutes with each component of <math>\mathbf{L}</math>:
<math display="block">[L^2,L_x]=[L^2,L_y]=[L^2,L_z]=0.</math>

Thus one may simultaneously specify the values of <math>L^2</math> and one chosen component, usually <math>L_z</math>. The same property holds for spin and total angular momentum:
<math display="block">[S^2,S_i]=0,\qquad [J^2,J_i]=0.</math>

Mathematically, <math>L^2</math> is a [[Casimir element|Casimir invariant]] of the rotation algebra.

===Uncertainty principle===
{{main|Uncertainty principle}}

Because different components of angular momentum do not commute, they cannot in general be measured simultaneously with arbitrary precision. For example, the Robertson–Schrödinger relation gives
<math display="block">\sigma_{L_x}\sigma_{L_y}\ge \frac{\hbar}{2}\left|\langle L_z\rangle\right|,</math>
where <math>\sigma_X</math> is the standard deviation of measurements of <math>X</math>, and <math>\langle X\rangle</math> is the expectation value.

Thus two orthogonal components, such as <math>L_x</math> and <math>L_y</math>, are complementary observables. By contrast, <math>L^2</math> and one component such as <math>L_z</math> can be measured simultaneously.

==Quantization==
{{see also|Physics:Azimuthal quantum number|Physics:Magnetic quantum number}}

In quantum mechanics, angular momentum is quantized: only certain discrete measurement results are allowed. For orbital angular momentum,
{| class="wikitable"
|-
! Quantity
! Allowed values
! Notes
|-
| <math>L^2</math>
| <math>\hbar^2\ell(\ell+1)</math>, where <math>\ell=0,1,2,\ldots</math>
| <math>\ell</math> is the orbital or azimuthal quantum number.
|-
| <math>L_z</math>
| <math>\hbar m_\ell</math>, where <math>m_\ell=-\ell,-\ell+1,\ldots,\ell</math>
| <math>m_\ell</math> is the magnetic quantum number.
|-
| <math>S^2</math>
| <math>\hbar^2 s(s+1)</math>, where <math>s=0,\tfrac12,1,\tfrac32,\ldots</math>
| <math>s</math> is the spin quantum number.
|-
| <math>S_z</math>
| <math>\hbar m_s</math>, where <math>m_s=-s,-s+1,\ldots,s</math>
| <math>m_s</math> is the spin projection quantum number.
|-
| <math>J^2</math>
| <math>\hbar^2 j(j+1)</math>, where <math>j=0,\tfrac12,1,\tfrac32,\ldots</math>
| <math>j</math> is the total angular momentum quantum number.
|-
| <math>J_z</math>
| <math>\hbar m_j</math>, where <math>m_j=-j,-j+1,\ldots,j</math>
| <math>m_j</math> is the total angular momentum projection quantum number.
|}

For orbital angular momentum, both <math>\ell</math> and <math>m_\ell</math> are always integers. For spin and total angular momentum, half-integer values are also possible.

===Derivation using ladder operators===
{{main|Physics:Ladder_operator#Angular_momentum}}

A standard derivation of the allowed values uses the ladder operators
<math display="block">J_+\equiv J_x+iJ_y,\qquad J_-\equiv J_x-iJ_y.</math>

If <math>|\psi\rangle</math> is a simultaneous eigenstate of <math>J^2</math> and <math>J_z</math>, then <math>J_+|\psi\rangle</math> and <math>J_-|\psi\rangle</math> are either zero or new simultaneous eigenstates with the same value of <math>J^2</math> but with the <math>J_z</math> eigenvalue shifted by <math>\pm\hbar</math>. Repeated application of these operators leads to the quantization rules above.

Since <math>\mathbf{L}</math> obeys the same algebra as <math>\mathbf{J}</math>, the same ladder-operator argument applies to orbital angular momentum. In the case of <math>\mathbf{L}</math>, single-valuedness of the wavefunction in the azimuthal angle <math>\phi</math> imposes the further restriction that <math>\ell</math> and <math>m_\ell</math> must be integers.

===Visual interpretation===
[[File:Vector model of orbital angular momentum.svg|250px|right|thumb|Heuristic vector model of orbital angular momentum.]]

Although angular momentum in quantum mechanics is represented by operators rather than classical vectors, it is often illustrated heuristically by vectors of fixed length whose tip can lie only on a cone. For a state with given <math>\ell</math> and <math>m_\ell</math>, the magnitude is
<math display="block">|L|=\sqrt{L^2}=\hbar\sqrt{\ell(\ell+1)},</math>
while the component <math>L_z</math> has the definite value
<math display="block">L_z=\hbar m_\ell.</math>

The uncertainty in the transverse components <math>L_x</math> and <math>L_y</math> is represented by the circular spread around the cone.

===Quantization in macroscopic systems===
The same quantum rules apply in principle to macroscopic bodies. In practice, however, the allowed steps in angular momentum are so small compared with the total angular momentum of ordinary objects that the spectrum appears continuous for all observable purposes.

==Angular momentum as the generator of rotations==
{{see also|Physics:Total angular momentum quantum number}}

The most fundamental characterization of angular momentum is that it generates rotations.<ref name=littlejohn>{{cite web|url=http://bohr.physics.berkeley.edu/classes/221/1011/notes/spinrot.pdf|title=Lecture notes on rotations in quantum mechanics|first=Robert|last=Littlejohn|access-date=13 Jan 2012|work=Physics 221B Spring 2011|year=2011|archive-date=26 August 2014|archive-url=https://web.archive.org/web/20140826003155/http://bohr.physics.berkeley.edu/classes/221/1011/notes/spinrot.pdf|url-status=dead}}</ref> If <math>R(\hat{n},\phi)</math> denotes the operator that rotates a system by an angle <math>\phi</math> about the axis <math>\hat{n}</math>, then the angular momentum component along that axis is defined by
<math display="block">J_{\hat n}\equiv i\hbar \lim_{\phi\to 0}\frac{R(\hat n,\phi)-1}{\phi}
=\left.i\hbar\frac{\partial R(\hat n,\phi)}{\partial\phi}\right|_{\phi=0}.</math>

Equivalently,
<math display="block">R(\hat n,\phi)=\exp\left(-\frac{i\phi J_{\hat n}}{\hbar}\right).</math>

Thus angular momentum governs how quantum states transform under rotations.

[[File:RotationOperators.svg|thumb|300px|Different kinds of [[Physics:Rotation operator (quantum mechanics)|rotation operators]]]].
{{ordered list
| list-style-type = upper-alpha
| <math>R</math>, associated with <math>\mathbf{J}</math>, rotates the entire system.
| <math>R_{\text{spatial}}</math>, associated with <math>\mathbf{L}</math>, rotates positions in space.
| <math>R_{\text{internal}}</math>, associated with <math>\mathbf{S}</math>, rotates internal spin degrees of freedom.
}}]]

The orbital and spin operators similarly generate spatial and internal rotations:
<math display="block">R_{\text{spatial}}(\hat n,\phi)=\exp\left(-\frac{i\phi L_{\hat n}}{\hbar}\right),</math>
<math display="block">R_{\text{internal}}(\hat n,\phi)=\exp\left(-\frac{i\phi S_{\hat n}}{\hbar}\right).</math>

The relation
<math display="block">\mathbf{J}=\mathbf{L}+\mathbf{S}</math>
reflects the corresponding decomposition of a full rotation into spatial and internal parts.

===SU(2), SO(3), and 360° rotations===
{{main|Physics:Spin}}

In classical mechanics, a rotation by <math>360^\circ</math> is identical to doing nothing. In quantum mechanics, however, a state with half-integer total angular momentum may change sign under a full <math>360^\circ</math> rotation:
<math display="block">R(\hat n,360^\circ)=-1</math>
for half-integer <math>j</math>, whereas
<math display="block">R(\hat n,360^\circ)=+1</math>
for integer <math>j</math>.<ref name=littlejohn/>

This reflects the fact that quantum rotations are described by [[SU(2)]], which is the double cover of [[SO(3)]]. Orbital angular momentum, by contrast, corresponds to ordinary spatial rotations and therefore only allows integer quantum numbers.

===Connection to representation theory===
{{main|Physics:Particle physics and representation theory|Representation theory of SU(2)|Rotation group SO(3)#A note on Lie algebras}}

When rotation operators act on quantum states, they define a representation of the rotation group. Correspondingly, angular momentum operators define a representation of the associated Lie algebra. The classification of possible angular momentum quantum numbers is therefore a representation-theoretic problem for [[Physics:SU(2) color superconductivity|SU(2)]] and [[SO(3)]].

===Connection to commutation relations===
Rotations about different axes do not commute. This noncommutativity is reflected at the operator level in the angular momentum commutation relations. Thus the algebra of angular momentum is a direct expression of the geometry of rotations in three-dimensional space.

==Conservation of angular momentum==
If the [[Hamiltonian (quantum mechanics)|Hamiltonian]] <math>H</math> is rotationally invariant, then angular momentum is conserved. Rotational invariance means
<math display="block">RHR^{-1}=H,</math>
where <math>R</math> is a rotation operator. This implies
<math display="block">[H,R]=0,</math>
and therefore
<math display="block">[H,\mathbf{J}]=\mathbf{0}.</math>

By the [[Ehrenfest theorem]], the total angular momentum is then conserved. For a spinless particle in a [[central potential]], this reduces to conservation of orbital angular momentum. When spin is present, spin–orbit coupling may transfer angular momentum between <math>\mathbf{L}</math> and <math>\mathbf{S}</math>, while the total <math>\mathbf{J}</math> remains conserved.

==Angular momentum coupling==
{{main|Physics:Angular momentum coupling|Physics:Clebsch–Gordan coefficients}}

When a system contains multiple sources of angular momentum, the individual contributions may combine to form a conserved total angular momentum. For example, for two angular momenta <math>\mathbf{J}_1</math> and <math>\mathbf{J}_2</math>,
<math display="block">\mathbf{J}=\mathbf{J}_1+\mathbf{J}_2.</math>

The corresponding total quantum number satisfies
<math display="block">j\in\{|j_1-j_2|,\ |j_1-j_2|+1,\ldots,j_1+j_2\}.</math>

Transformations between uncoupled and coupled angular momentum bases are described by [[Physics:Clebsch–Gordan coefficients|Clebsch–Gordan coefficients]]. In atomic and molecular physics, this structure underlies term symbols and the classification of energy levels.

==Orbital angular momentum in spherical coordinates==
Angular momentum operators naturally arise in problems with spherical symmetry. In the position representation and [[spherical coordinates]], the orbital angular momentum operator is<ref>{{Cite book
| publisher = Springer Berlin Heidelberg
| last = Bes
| first = Daniel R.
| isbn = 978-3-540-46215-6
| title = Quantum Mechanics
| series = Advanced Texts in Physics
| location = Berlin, Heidelberg
| year = 2007
| page=70
| doi = 10.1007/978-3-540-46216-3
| bibcode = 2007qume.book.....B
}}</ref>
<math display="block">\begin{align}
\mathbf L &= i \hbar \left(\frac{\hat{\boldsymbol{\theta}}}{\sin(\theta)} \frac{\partial}{\partial\phi} - \hat{\boldsymbol{\phi}} \frac{\partial}{\partial\theta}\right) \\
&= i\hbar\left(
\hat{\mathbf{x}} \left(\sin(\phi) \frac{\partial}{\partial\theta} + \cot(\theta)\cos(\phi) \frac{\partial}{\partial\phi}\right)
+\hat{\mathbf{y}} \left(-\cos(\phi)\frac{\partial}{\partial\theta} + \cot(\theta)\sin(\phi)\frac{\partial}{\partial\phi}\right)
-\hat{\mathbf z}\frac{\partial}{\partial\phi}
\right), \\
L_+ &= \hbar e^{i\phi} \left( \frac{\partial}{\partial\theta} + i\cot(\theta) \frac{\partial}{\partial\phi} \right), \\
L_- &= \hbar e^{-i\phi} \left( -\frac{\partial}{\partial\theta} + i\cot(\theta) \frac{\partial}{\partial\phi} \right), \\
L^2 &= -\hbar^2 \left(\frac{1}{\sin(\theta)} \frac{\partial}{\partial\theta} \left(\sin(\theta) \frac{\partial}{\partial\theta}\right) + \frac{1}{\sin^2(\theta)}\frac{\partial^2}{\partial\phi^2}\right), \\
L_z &= -i \hbar \frac{\partial}{\partial\phi}.
\end{align}</math>

The angular part of the [[Laplace operator]] can be written in terms of <math>L^2</math>:
<math display="block">\Delta=\frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial}{\partial r}\right)-\frac{L^2}{\hbar^2 r^2}.</math>

The simultaneous eigenstates of <math>L^2</math> and <math>L_z</math> satisfy
<math display="block">\begin{align}
L^2|\ell,m\rangle &= \hbar^2\ell(\ell+1)|\ell,m\rangle,\\
L_z|\ell,m\rangle &= \hbar m|\ell,m\rangle,
\end{align}</math>
with wavefunctions
<math display="block">\langle \theta,\phi|\ell,m\rangle=Y_{\ell,m}(\theta,\phi),</math>
where <math>Y_{\ell,m}</math> are the [[spherical harmonics]].<ref>Sakurai, JJ & Napolitano, J (2010), ''[[Modern Quantum Mechanics]]'' (2nd edition), Pearson, {{ISBN|978-0805382914}}</ref>

==See also==
{{#invoke:PhysicsQC|tocHeadingAndList|Physics:Quantum basics/See also}}
{{colbegin}}
* [[Physics:Runge–Lenz vector|Runge–Lenz vector]]
* [[Holstein–Primakoff transformation]]
* [[Jordan map]]
* [[Pauli–Lubanski pseudovector]]
* [[Angular momentum diagrams (quantum mechanics)]]
* [[Spherical basis]]
* [[Tensor operator]]
* [[Orbital magnetization]]
* [[Orbital angular momentum of free electrons]]
* [[Orbital angular momentum of light]]
{{colend}}

==Notes==
{{NoteFoot}}

==References==
{{reflist|3}}

==Further reading==
* {{Cite book|title=Quantum Mechanics|first=E.|last=Abers|publisher=Addison Wesley, Prentice Hall Inc|year=2004|isbn=978-0-13-146100-0}}
* {{Cite book |last1=Biedenharn |first1=L. C. |url=https://www.cambridge.org/core/books/angular-momentum-in-quantum-physics/53AFDEE1D64D0256AD874534F084C402 |title=Angular Momentum in Quantum Physics: Theory and Application |last2=Louck |first2=James D. |date=1984 |publisher=Cambridge University Press |isbn=978-0-521-30228-9 |series=Encyclopedia of Mathematics and its Applications |location=Cambridge |doi=10.1017/cbo9780511759888|bibcode=1984amqp.book.....B |author-link=Lawrence Biedenharn}}
* {{Cite book|title=Physics of Atoms and Molecules|first1=B.H.|last1=Bransden|first2=C.J.|last2=Joachain|publisher=Longman|year=1983|isbn=0-582-44401-2}}
* {{Cite book|chapter-url=https://www.feynmanlectures.caltech.edu/III_18.html|title=The Feynman Lectures on Physics Vol. III|edition=The New Millennium|chapter=Ch. 18: Angular Momentum|first1=Richard P.|last1=Feynman|first2=Robert B.|last2=Leighton|first3=Matthew|last3=Sands}}
* {{Cite book|title=Quantum Mechanics Demystified|first=D.|last=McMahon|publisher=McGraw-Hill|year=2006|isbn=0-07-145546-9}}
* {{Cite book |last=Zare|first=R.N.|title=Angular Momentum: Understanding Spatial Aspects in Chemistry and Physics|publisher=Wiley-Interscience|year=1991|isbn=978-0-471-85892-8}}
{{Author|Harold Foppele}}
{{Physics operator}}

[[Category:Rotation]]
[[Category:Conservation laws]]
[[Category:Rotational symmetry]]

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