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{{quantum mechanics}}
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In [[quantum mechanics]], the '''angular momentum operator''' is one of several related [[operator (mathematics)|operators]] analogous to classical [[angular momentum]]. The angular momentum operator plays a central role in the theory of [[atomic physics]] and other quantum problems involving [[rotational symmetry]].  In both classical and quantum mechanical systems, angular momentum (together with linear momentum and energy) is one of the three fundamental properties of motion.<ref name="Liboff">Introductory Quantum Mechanics, [[Richard L. Liboff]], 2nd Edition, ISBN 0-201-54715-5</ref>
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There are several angular momentum operators: '''total angular momentum''' (usually denoted '''J'''), '''orbital angular momentum''' (usually denoted '''L'''), and '''spin angular momentum''' ('''spin''' for short, usually denoted '''S'''). The term "angular momentum operator" can (confusingly) refer to either the total or the orbital angular momentum. Total angular momentum is always [[conservation of angular momentum|conserved]], see [[Noether's theorem]].
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==Spin, orbital, and total angular momentum==
'''MathML'''
{{main|Spin (physics)}}
:<math forcemathmode="mathml">E=mc^2</math>
[[File:LS coupling.svg|250px|thumb|"Vector cones" of total angular momentum '''J''' (purple), orbital '''L''' (blue), and spin '''S''' (green). The cones arise due to [[quantum uncertainty]] between measuring angular momentum components ([[#Visual interpretation|see below]]).]]


The [[angular momentum|classical definition of angular momentum]] is <math>\mathbf{L}=\mathbf{r}\times\mathbf{p}</math>. This 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 ''orbital angular momentum operator''. Specifically, '''L''' is a ''vector operator'', meaning <math>\mathbf{L}=(L_x,L_y,L_z)</math>, where ''L''<sub>x</sub>, ''L''<sub>y</sub>, ''L''<sub>z</sub> are three different operators.
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:<math forcemathmode="png">E=mc^2</math>


However, there is another type of angular momentum, called [[spin (physics)|''spin angular momentum'']] (more often shortened to ''spin''), represented by the spin operator '''S'''. Almost all [[elementary particle]]s have spin. 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. All [[elementary particles]] have a characteristic spin, for example [[electron]]s always have "spin 1/2" while [[photon]]s always have "spin 1".
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


Finally, there is [[total angular momentum]] '''J''', which combines both the spin and orbital angular momentum of a particle or system:
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:<math>\mathbf{J}=\mathbf{L}+\mathbf{S}.</math>
[[Conservation of angular momentum]] states that '''J''' for a closed system, or '''J''' for the whole universe, is conserved. However, '''L''' and '''S''' are ''not'' generally conserved. For example, the [[spin–orbit interaction]] allows angular momentum to transfer back and forth between '''L''' and '''S''', with the total '''J''' remaining constant.


==Orbital angular momentum operator==
==Demos==
Orbital angular momentum '''L''' is mathematically defined as the [[cross product]] of a wave function's [[position operator]] ('''r''') and [[momentum operator]] ('''p'''):


:<math>\mathbf{L}=\mathbf{r}\times\mathbf{p}</math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


This is analogous to the definition of [[angular momentum]] in classical physics.


In the special case of a single particle with no [[electric charge]] and no [[spin (physics)|spin]], the angular momentum operator can be written in the position basis as a single vector equation:
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:<math>\mathbf{L}=-i\hbar(\mathbf{r}\times\nabla)</math>
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==Commutation relations==
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===Commutation relations between components===
==Bug reporting==
The orbital angular momentum operator is a vector operator, meaning it can be written in terms of its vector components <math>\mathbf{L}=(L_x,L_y,L_z)</math>. The components have the following [[commutation relation]]s with each other:<ref>{{cite book|url=http://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}}</ref>
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
:<math>[L_x,L_y]=i\hbar L_z, \;\; [L_y,L_z]=i\hbar L_x, \;\; [L_z,L_x]=i\hbar L_y</math>
or in symbols,
:<math>[L_l, L_m ] = i \hbar \varepsilon_{lmn} L_n</math>,
where ''ε<sub>lmn</sub>'' denotes the [[Levi-Civita symbol]], and ''l,m,n'' are Cartesian coordinates (each can be ''x'', ''y'' or ''z''), and [, ] is the [[commutator]]
:<math>[X,Y] \equiv XY-YX</math>.
These can be proved as a direct consequence of the [[canonical commutation relation]]s <math>[x_l,p_m]=i \hbar \delta_{lm}</math>, where ''δ<sub>lm</sub>'' is the [[Kronecker delta]].
 
There is an analogous relationship to the commutator in classical physics which is central to the theory of canonical transformations of [[Hamilton's equations]] of motion:<ref>H. Goldstein, C. P. Poole and J. Safko, ''Classical Mechanics, 3rd Edition'', Addison-Wesley 2002, pp. 388 &#64256;.</ref>
:<math>[u, v]_{q,p} = \frac{\partial u}{\partial q_i} \frac{\partial v}{\partial p_i} - \frac{\partial u}{\partial p_i} \frac{\partial v}{\partial q_i}\!</math>
(summation over generalized coordinate index i implied) where, in this bilinear expression, <math>[u, v]_{q,p}</math> is the [[Poisson bracket]] of two functions <math>u, v</math> with respect to the canonical (generalized) coordinates <math>(p,q)</math>.  "... identification of the canonical angular momentum as the generator of rigid rotation of [a system of particles] leads to a number of interesting and important Poisson bracket relations."<ref>Idem, pp. 408-411.</ref>  Among these are:
:<math>[L_l, L_m ] = \varepsilon_{lmn} L_n. \!</math>
Here, <math>L_z</math>, for example, is a transformation generated by the generalized momentum conjugate to <math>q_i</math>:
:<math>L_z(q, p) = p_i</math>.
It can be shown<ref>Idem, p. 404.</ref> (using Cartesian coordinates x, y and z for each particle i in the system) that
:<math>L_z = x_ip_{iy} - y_ip_{ix}</math>.
This generating function <math>L_z</math> has the physical significance of being the ''z''-component of the total angular momentum:
:<math>L_z \ \equiv \ (r_i \ \times \ p_i)_z</math>.
It is important to recognize that the Poisson bracket is an analogue, not the commutator in disguise.  Hamilton's equations do not generalize to quantum mechanics because they assume that the position and momentum of a particle can be known simultaneously to infinite precision at any point in time.  See the section "Generalization to quantum mechanics through Poisson bracket" in the article on [[Hamiltonian mechanics]] for details and additional references.
 
The same commutation relations apply for the other angular momentum operators (spin and total angular momentum):<ref name=littlejohn/>
:<math>[S_l, S_m ] = i \hbar \sum_{n=1}^{3} \varepsilon_{lmn} S_n, \quad [J_l, J_m ] = i \hbar \sum_{n=1}^{3} \varepsilon_{lmn} J_n</math>.
These can be ''assumed'' to hold in analogy with '''L'''. Alternatively, they can be ''derived'' as discussed [[#Connection to commutation relations|below]].
 
These commutation relations mean that '''L''' has the mathematical structure of a [[Lie algebra]]. In this case, the Lie algebra is [[SU(2)]] or [[SO(3)]], the rotation group in three dimensions. The same is true of '''J''' and '''S'''. The reason is discussed [[#Total angular momentum as a generator of rotations|below]].
 
These commutation relations are relevant for measurement and uncertainty, as discussed further below.
 
===Commutation relations involving vector magnitude===
Like any vector, a [[Euclidean norm|magnitude]] can be defined for the orbital angular momentum operator,
:<math>L^2 \equiv L_x^2 + L_y^2 + L_z^2</math> .
'''''L'''''<sup>2</sup> is another quantum [[operator (mathematics)|operator]]. It commutes with the components of '''''L''''',
:<math>[L^2,L_x] = [L^2,L_y] = [L^2,L_z] = 0~.\,</math>
 
One way to prove that these operators commute is to start from the [''L''<sub>ℓ</sub>,  ''L''<sub>''m''</sub>] commutation relations in the previous section:
 
:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
!Click [show] on the right to see a proof of [''L''<sup>2</sup>, ''L''<sub>x</sub>] = 0, starting from the [''L''<sub>ℓ</sub>, ''L''<sub>''m''</sub>] commutation relations<ref>{{cite book | author=Griffiths, David J. | title=Introduction to Quantum Mechanics | publisher=[[Prentice Hall]] | year=1995 | page=146}}</ref>
|-
|<math>[L^2, L_x] = (L_x^2 + L_y^2 + L_z^2)L_x - L_x (L_x^2 + L_y^2 + L_z^2) </math>
:::<math> = L_y(L_y L_x - L_x L_y) + (L_y L_x - L_x L_y) L_y + L_z(L_z L_x - L_x L_z) + (L_z L_x - L_x L_z)L_z </math>
:::<math> = L_y [L_y, L_x] + [L_y, L_x] L_y + L_z [L_z, L_x] + [L_z, L_x] L_z </math>
:::<math> = L_y (-i \hbar L_z) + (-i \hbar L_z) L_y + L_z (i \hbar L_y) + (i \hbar L_y) L_z </math>
:::<math> = 0 </math>
|}
 
Mathematically,  '''''L'''''<sup>2</sup> is a [[Casimir invariant]] of the [[Lie algebra]] SO(3) spanned by '''''L'''''.
 
In the classical case, '''''L''''' is the orbital angular momentum of the entire system of particles, '''''n''''' is the unit vector along one of the Cartesian axes and we also have Poisson pseudo-commutation of '''''L''''' with each of its Cartesian components:<ref>Goldstein et al, p. 410</ref>
:<math>[\mathbf L \cdot \mathbf L, \mathbf L \cdot \mathbf n] = [L^2, \mathbf L \cdot \mathbf n] = 0</math>
 
with <math> \mathbf n </math> selecting one of the Cartesian components of <math> \mathbf L </math>.
 
The same commutation relations apply for the other angular momentum operators (spin and total angular momentum):
:<math>[S^2,S_i]=0, \quad [J^2,J_i]=0</math> .
 
===Uncertainty principle===
{{main|Uncertainty principle|Uncertainty principle derivations}}
In general, in quantum mechanics, when two [[observable|observable operators]] do not commute, they are called ''incompatible observables''. Two incompatible observables cannot be measured simultaneously; instead they satisfy an [[uncertainty principle]]. The more accurately one observable is known, the less accurately the other one can be known. Just as there is an uncertainty principle relating position and momentum, there are uncertainty principles for angular momentum.
 
The [[uncertainty principle|Robertson–Schrödinger relation]] gives the following uncertainty principle:
:<math>\sigma_{L_x} \sigma_{L_y} \geq \frac{\hbar}{2} \left| \langle L_z \rangle \right|.</math>
where <math>\sigma_X</math> is the [[standard deviation]] in the measured values of ''X'' and <math>\langle X \rangle</math> denotes the [[Expectation value (quantum mechanics)|expectation value]] of ''X''. This inequality is also true if ''x,y,z'' are rearranged, or if ''L'' is replaced by ''J'' or ''S''.
 
Therefore, two orthogonal components of angular momentum cannot be simultaneously known or measured, except in special cases such as <math>L_x=L_y=L_z=0</math>.
 
It is, however, possible to simultaneously measure or specify ''L''<sup>2</sup> and any one component of ''L''; for example, ''L''<sup>2</sup> and ''L''<sub>z</sub>. This is often useful, and the values are characterized by [[azimuthal quantum number]] and [[magnetic quantum number]], as discussed further below.
 
==Quantization==
{{see also|Azimuthal quantum number|Magnetic quantum number}}
In [[quantum mechanics]], angular momentum is ''quantized'' – that is, it cannot vary continuously, but only in "quantum leaps" between certain allowed values. For any system, the following restrictions on measurement results apply, where <math>\hbar</math> is [[reduced Planck constant]]:
{| class="wikitable"
|-
!If you [[measurement in quantum mechanics|measure]]...
!...the result can be...
!Notes
|-
|''L''<sub>z</sub>
|<math>(\hbar m)</math>, where <math>m\in\{\ldots, -2, -1, 0, 1, 2, \ldots\}</math>
|''m'' is sometimes called "[[magnetic quantum number]]".<br />This same quantization rule holds for any component of '''L''', e.g. ''L''<sub>x</sub> or ''L''<sub>y</sub>.<br /> This rule is sometimes called '''spatial quantization'''.<ref>I''ntroduction to quantum mechanics: with applications to chemistry'', by Linus Pauling, Edgar Bright Wilson, page 45, [http://books.google.com/books?id=D48aGQTkfLgC&pg=PA45&dq=spatial+quantization google books link]</ref>
|-
|''S''<sub>z</sub> or ''J''<sub>z</sub>
|<math>(\hbar m)</math>, where <math>m\in\{\ldots, -1, -0.5, 0, 0.5, 1, 1.5, \ldots\}</math>
|For ''S''<sub>z</sub>, ''m'' is sometimes called "[[spin quantum number|spin projection quantum number]]".<br /> For ''J''<sub>z</sub>, ''m'' is sometimes called "[[Azimuthal quantum number#Total angular momentum of an electron in the atom|total angular momentum projection quantum number]]".<br />This same quantization rule holds for any component of '''S''' or '''J''', e.g. ''S''<sub>x</sub> or ''J''<sub>y</sub>.
|-
|<math>L^2</math>
|<math>(\hbar^2 \ell (\ell+1))</math>, where <math>\ell \in \{0,1,2,\ldots\}</math>
|''L''<sup>2</sup> is defined by <math>L^2 \equiv L_x^2 +L_y^2 + L_z^2</math>.<br /><math>\ell</math> is sometimes called "[[azimuthal quantum number]]" or "orbital quantum number".
|-
|<math>S^2</math>
|<math>(\hbar^2 s(s+1))</math>, where <math>s \in \{ 0,0.5,1,1.5, \ldots \}</math>
|''s'' is called [[spin quantum number]] or just "spin". For example, a [[spin-½|spin-½ particle]] is a particle where ''s''=½.
|-
|<math>J^2</math>
|<math>(\hbar^2 j(j+1))</math>, where <math>j \in \{ 0,0.5,1,1.5, \ldots \} </math>
|''j'' is sometimes called "[[Azimuthal quantum number#Total angular momentum of an electron in the atom|total angular momentum quantum number]]".
|-
|<math>L^2</math> and <math>L_z</math><br />simultaneously
|<math>(\hbar^2 \ell(\ell+1))</math> for <math>L^2</math>, and <math>(\hbar m_\ell)</math> for <math>L_z</math><br />where <math>\ell \in \{ 0,1,2,\ldots \}</math> and <br /><math>m_\ell \in \{ -\ell, (-\ell+1), \ldots, (\ell-1),\ell \}</math>
|(See above for terminology.)
|-
|<math>S^2</math> and <math>S_z</math><br />simultaneously
|<math>(\hbar^2 s(s+1))</math> for <math>S^2</math>, and <math>(\hbar m_s)</math> for <math>S_z</math><br />where <math>s \in \{0,0.5,1,1.5,\ldots\}</math> and<br /><math>m_s \in \{ -s, (-s+1), \ldots, (s-1),s\}</math>
|(See above for terminology.)
|-
|<math>J^2</math> and <math>J_z</math><br />simultaneously
|<math>(\hbar^2 j(j+1))</math> for <math>J^2</math>, and <math>(\hbar m_j)</math> for <math>J_z</math><br />where <math>j \in \{ 0,0.5,1,1.5,\ldots \}</math> and <br /><math>m_j \in \{ -j, (-j+1), \ldots, (j-1),j \}</math>
|(See above for terminology.)
|}
[[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.]]
 
===Derivation using ladder operators===
{{main|Ladder operator#Angular momentum}}
A common way to derive the quantization rules above is the method of ''[[ladder operator]]s''.<ref name=Griffithsladder>{{cite book | author=Griffiths, David J. | title=Introduction to Quantum Mechanics | publisher=[[Prentice Hall]] | year=1995 | pages=147–149}}</ref> The ladder operators are defined:
:<math>J_+ \equiv J_x + i J_y, \quad J_- \equiv J_x - i J_y</math>
Suppose a state <math>| \psi \rangle</math> is a state in the simultaneous eigenbasis of <math>J^2</math> and <math>J_z</math> (i.e., a state with a single, definite value of <math>J^2</math> and a single, definite value of <math>J_z</math>). Then using the commutation relations, one can prove that <math>J_+|\psi\rangle</math> and <math>J_-|\psi\rangle</math> are ''also'' in the simultaneous eigenbasis, with the same value of <math>J^2</math>, but where <math>J_z |\psi\rangle</math> is increased or decreased by <math>\hbar</math>, respectively. (It is also possible that one or both of these vectors is the zero vector.) (For a proof, see [[ladder operator#angular momentum]].)
 
By manipulating these ladder operators and using the commutation rules, it is possible to prove almost all of the quantization rules above.
 
{| class="toccolours collapsible collapsed" width="75%" style="text-align:left"
!Click [show] on the right to see more details in the ladder-operator proof of the quantization rules<ref name=Griffithsladder/>
|-
|Before starting the main proof, we will note a useful fact: That <math>J_x^2,J_y^2,J_z^2</math> are [[positive-semidefinite matrix|positive-semidefinite operator]]s, meaning that all their eigenvalues are nonnegative. That also implies that the same is true for their sums, including <math>J^2 = J_x^2 + J_y^2 + J_z^2</math> and <math>(J^2 - J_z^2) = (J_x^2 + J_y^2)</math>. The reason is because the square of ''any'' [[Hermitian operator]] is always positive semidefinite. (A Hermitian operator has real eigenvalues, so the squares of those eigenvalues are nonnegative.)
 
As above, assume that a state <math>| \psi \rangle</math> is a state in the simultaneous eigenbasis of <math>J^2</math> and <math>J_z</math>. Its eigenvalue with respect to <math>J^2</math> can be written in the form <math>\hbar^2 j (j+1)</math> for some real number ''j'' > 0 (because as mentioned in the previous paragraph, <math>J^2</math> has nonnegative eigenvalues), and its eigenvalue with respect to <math>J_z</math> can be written <math>\hbar m</math> for some real number ''m''. Instead of <math>| \psi \rangle</math> we will use the more descriptive notation <math>|\psi \rangle = | j,m \rangle</math>.
 
Next, consider the sequence ("ladder") of states
:<math>\{\ldots \; , \; J_- J_- | j,m \rangle \; , \; J_- | j,m \rangle \; , \; | j,m \rangle \; , \; J_+ | j,m \rangle \; , \; J_+ J_+ | j,m \rangle \; , \; \ldots \} </math>
Some entries in this infinite sequence may be the [[zero vector]] (as we will see). However, as described above, all the nonzero entries have the same value of <math>J^2</math>, and among the nonzero entries, each entry has a value of <math>J_z</math> which is exactly <math>\hbar</math> more than the previous entry.
 
In this ladder, there can only be a finite number of nonzero entries, with infinite copies of the zero vector on the left and right. The reason is, as mentioned above, <math>(J^2 - J_z^2)</math> is positive-semidefinite, so if any quantum state is an eigenvector of both <math>J^2</math> and <math>J_z^2</math>, the former eigenvalue is larger. The states in the ladder all have the same <math>J^2</math> eigenvalue, but going very far to the left or the right, the <math>J_z^2</math> eigenvalue gets larger and larger. The only possible resolution is, as mentioned, that there are only finitely many nonzero entries in the ladder.
 
Now, consider the last nonzero entry to the right of the ladder, <math>|j,m_{max} \rangle</math>. This state has the property that <math>J_+ |j,m_{max}\rangle = 0</math>. As proven in the [[ladder operator]] article,
:<math>J_+ |j,m \rangle = \hbar \sqrt{j(j+1) - m(m+1)}|j,m+1\rangle</math>
If this is zero, then <math>j(j+1) = m_{max}(m_{max}+1)</math>, so <math>j=m</math> or <math>j = -m - 1</math>. However, because <math>J^2 - J_z^2</math> is positive-semidefinite, <math>\hbar^2 j(j+1) \geq (\hbar m)^2</math>, which means that the only possibility is <math>m_{max} = j</math>.
 
Similarly, consider the first nonzero entry on the left of the ladder, <math>|j, m_{min}\rangle</math>. This state has the property that
<math>J_- |j,m_{min}\rangle = 0</math>. As proven in the [[ladder operator]] article,
:<math>J_- |j,m\rangle = \hbar \sqrt{j(j+1) - m(m-1)} |j,m-1\rangle</math>
As above, the only possibility is that <math>m_{min} = -j</math>
 
Since ''m'' changes by 1 on each step of the ladder, <math>(j - (-j))</math> is an integer, so ''j'' is an integer or half-integer (0 or 0.5 or 1 or 1.5...).
|}
 
Since '''S''' and '''L''' have the same commutation relations as '''J''', the same ladder analysis works for them.
 
The ladder-operator analysis does '''''not''''' explain one aspect of the quantization rules above: the fact that '''L''' (unlike '''J''' and '''S''') cannot have half-integer quantum numbers. This fact can be proven (at least in the special case of one particle) by writing down every possible eigenfunction of ''L''<sup>2</sup> and ''L''<sub>z</sub>, (they are the [[spherical harmonic]]s), and seeing explicitly that none of them have half-integer quantum numbers.<ref>{{cite book | author=Griffiths, David J. | title=Introduction to Quantum Mechanics | publisher=[[Prentice Hall]] | year=1995 | pages=148–153}}</ref> An alternative derivation is [[#SU(2), SO(3), and 360° rotations|below]].
 
===Visual interpretation===
[[File:Vector model of orbital angular momentum.svg|250px|"250px"|right|thumb|Illustration of the vector model of orbital angular momentum.]]
{{main|Vector model of the atom}}
Since the angular momenta are quantum operators, they cannot be drawn as vectors like in classical mechanics. Nevertheless, it is common to depict them heuristically in this way. Depicted on the right is a set of states with quantum numbers <math>\ell=2</math>, and <math>m_\ell=-2,-1,0,1,2</math> for the five cones from bottom to top. Since <math>|L|=\sqrt{L^2}=\hbar \sqrt{6}</math>, the vectors are all shown with length <math>\hbar \sqrt{6}</math>. The rings represent the fact that <math>L_z</math> is known with certainty, but <math>L_x</math> and  <math>L_y</math> are unknown; therefore every classical vector with the appropriate length and ''z''-component is drawn, forming a cone. The expected value of the angular momentum for a given ensemble of systems in the quantum state characterized by <math> \ell</math> and <math>m_\ell</math> could be somewhere on this cone while it cannot be defined for a single system (since the components of <math>L</math> do not commute with each other).
 
===Quantization in macroscopic systems===
The quantization rules are technically true even for macroscopic systems, like the angular momentum '''L''' of a spinning tire. However they have no observable effect. For example, if <math>L_z/\hbar</math> is roughly 100000000, it makes essentially no difference whether the precise value is an integer like 100000000 or 100000001, or a non-integer like 100000000.2—the discrete steps are too small to notice.
 
==Angular momentum as the generator of rotations==
{{see also|Total angular momentum quantum number}}
The most general and fundamental definition of angular momentum is as the ''generator'' of 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|accessdate= 13 Jan 2012|work=[http://bohr.physics.berkeley.edu/classes/221/1011/221.html Physics 221B Spring 2011]|year=2011}}</ref> More specifically, let <math>R(\hat{n},\phi)</math> be a [[Rotation operator (quantum mechanics)|rotation operator]], which rotates any quantum state about axis <math>\hat{n}</math> by angle <math>\phi</math>. As <math>\phi\rightarrow 0</math>, the operator <math>R(\hat{n},\phi)</math> approaches the [[identity operator]], because a rotation of 0° maps all states to themselves. Then the angular momentum operator <math>J_{\hat{n}}</math> about axis <math>\hat{n}</math> is defined as:<ref name=littlejohn/>
:<math>J_{\hat{n}} \equiv i \hbar \lim_{\phi\rightarrow 0} \frac{R(\hat{n},\phi) - 1}{\phi}</math>
where 1 is the [[identity operator]]. Remark also that ''R'' is an additive morphism : <math>R(\hat{n},\phi_1+\phi_2)=R(\hat{n},\phi_1)R(\hat{n},\phi_2)</math> ; as a consequence<ref name=littlejohn/>
:<math>R(\hat{n},\phi) = \exp(-i \phi J_{\hat{n}}/\hbar)</math>
where exp is [[matrix exponential]].
 
In simpler terms, the total angular momentum operator characterizes how a quantum system is changed when it is rotated. The relationship between angular momentum operators and rotation operators is the same as the relationship between [[Lie algebra]]s and [[Lie group]]s in mathematics, as discussed further below.
 
[[File:RotationOperators.svg|thumb|300px|The different types of [[rotation operator (quantum mechanics)|rotation operators]]. <u>Top</u>: Two particles, with spin states indicated schematically by the arrows. <u>(A)</u> The operator ''R'', related to '''J''', rotates the entire system. <u>(B)</u> The operator ''R''<sub>spatial</sub>, related to '''L''', rotates the particle positions without altering their internal spin states. <u>(C)</u> The operator ''R''<sub>internal</sub>, related to '''S''', rotates the particles' internal spin states without changing their positions.]]
Just as '''J''' is the generator for [[rotation operator (quantum mechanics)|rotation operators]], '''L''' and '''S''' are generators for modified partial rotation operators. The operator
:<math>R_\mathrm{spatial}(\hat{n},\phi) = \exp(-i \phi L_{\hat{n}}/\hbar),</math>
rotates the position (in space) of all particles and fields, without rotating the internal (spin) state of any particle. Likewise, the operator
:<math>R_\mathrm{internal}(\hat{n},\phi) = \exp(-i \phi S_{\hat{n}}/\hbar),</math>
rotates the internal (spin) state of all particles, without moving any particles or fields in space. The relation '''J'''='''L'''+'''S''' comes from:
:<math>R(\hat{n},\phi) = R_\mathrm{internal}(\hat{n},\phi) R_\mathrm{spatial}(\hat{n},\phi)</math>
i.e. if the positions are rotated, and then the internal states are rotated, then altogether the complete system has been rotated.
 
===SU(2), SO(3), and 360° rotations===
{{main|Spin (physics)}}
Although one might expect <math>R(\hat{n},360^\circ) = 1</math> (a rotation of 360° is the identity operator), this is ''not'' assumed in quantum mechanics, and it turns out it is often not true: When the total angular momentum quantum number is a half-integer (1/2, 3/2, etc.), <math>R(\hat{n},360^\circ) = -1</math>, and when it is an integer, <math>R(\hat{n},360^\circ) = +1</math>.<ref name=littlejohn/> Mathematically, the structure of rotations in the universe is ''not'' [[SO(3)]], the [[Lie group|group]] of three-dimensional rotations in classical mechanics. Instead, it is [[SU(2)]], which is identical to SO(3) for small rotations, but where a 360° rotation is mathematically distinguished from a rotation of 0°. (A rotation of 720° is, however, the same as a rotation of 0°.)<ref name=littlejohn/>
 
On the other hand, <math>R_\mathrm{spatial}(\hat{n},360^\circ) = +1</math> in all circumstances, because a 360° rotation of a ''spatial'' configuration is the same as no rotation at all. (This is different from a 360° rotation of the ''internal'' (spin) state of the particle, which might or might not be the same as no rotation at all.) In other words, the <math>R_\mathrm{spatial}</math> operators carry the structure of [[SO(3)]], while <math>R</math> and <math>R_\mathrm{internal}</math> carry the structure of [[SU(2)]].
 
From the equation <math>+1=R_\mathrm{spatial}(\hat{z},360^\circ) = \exp(-2\pi i L_z /\hbar)</math>, one picks an eigenstate <math> L_z |\psi\rangle = m\hbar |\psi\rangle</math> and draws
:<math> e^{-2\pi i m} = 1 </math>
which is to say that the orbital angular momentum quantum numbers can only be integers, not half-integers.
 
===Connection to representation theory===
{{main|Particle physics and representation theory|Representation theory of SU(2)}}
Starting with a certain quantum state <math>|\psi_0\rangle</math>, consider the set of states <math>R(\hat{n},\phi)|\psi_0\rangle</math> for all possible <math>\hat{n}</math> and <math>\phi</math>, i.e. the set of states that come about from rotating the starting state in every possible way. This is a [[vector space]], and therefore the manner in which the rotation operators map one state onto another is a [[group representation|''representation'']] of the group of rotation operators.
:''When rotation operators act on quantum states, it forms a [[group representation|representation]] of the [[Lie group]] [[SU(2)]] (for R and R<sub>internal</sub>), or [[SO(3)]] (for R<sub>spatial</sub>).''
From the relation between '''J''' and rotation operators,
:''When angular momentum operators act on quantum states, it forms a [[group representation|representation]] of the [[Lie algebra]] [[SU(2)]]  or [[SO(3)]].''
(The Lie algebras of SU(2) and SO(3) are identical.)
 
The ladder operator derivation above is a method for classifying the representations of the Lie algebra SU(2).
 
===Connection to commutation relations===
Classical rotations do not commute with each other: For example, rotating 1° about the ''x''-axis then 1° about the ''y''-axis gives a slightly different overall rotation than rotating 1° about the ''y''-axis then 1° about the ''x''-axis. By carefully analyzing this noncommutativity, the commutation relations of the angular momentum operators can be derived.<ref name=littlejohn/>
 
(This same calculational procedure is one way to answer the mathematical question "What is the [[Lie algebra]] of the [[Lie group]]s [[SO(3)]] or [[SU(2)]]?")
 
==Conservation of angular momentum==
The [[Hamiltonian (quantum mechanics)|Hamiltonian]] ''H'' represents the energy and dynamics of the system. In a spherically-symmetric situation, the Hamiltonian is invariant under rotations:
:<math>RHR^{-1}=H</math>
where ''R'' is a [[rotation operator (quantum mechanics)|rotation operator]]. As a consequence, <math>[H,R]=0</math>, and then <math>[H,\mathbf{J}]=0</math> due to the relationship between '''J''' and ''R''. By the [[Ehrenfest theorem]], it follows that '''J''' is conserved.
 
To summarize, if ''H'' is rotationally-invariant (spherically symmetric), then total angular momentum '''J''' is conserved. This is an example of [[Noether's theorem]].
 
If ''H'' is just the Hamiltonian for one particle, the total angular momentum of that one particle is conserved when the particle is in a [[central potential]] (i.e., when the potential energy function depends only on <math>|\mathbf{r}|</math>). Alternatively, ''H'' may be the Hamiltonian of all particles and fields in the universe, and then ''H'' is ''always'' rotationally-invariant, as the fundamental laws of physics of the universe are the same regardless of orientation. This is the basis for saying [[conservation of angular momentum]] is a general principle of physics.
 
For a particle without spin, '''J'''='''L''', so orbital angular momentum is conserved in the same circumstances. When the spin is nonzero, the [[spin-orbit interaction]] allows angular momentum to transfer from '''L''' to '''S''' or back. Therefore, '''L''' is not, on its own, conserved.
 
==Angular momentum coupling==
{{main|Angular momentum coupling|Clebsch–Gordan coefficients}}
 
Often, two or more sorts of angular momentum interact with each other, so that angular momentum can transfer from one to the other. For example, in [[spin-orbit coupling]], angular momentum can transfer between '''L''' and '''S''', but only the total '''J'''='''L'''+'''S''' is conserved. In another example, in an atom with two electrons, each has its own angular momentum '''J'''<sub>1</sub> and '''J'''<sub>2</sub>, but only the total '''J'''='''J'''<sub>1</sub>+'''J'''<sub>2</sub> is conserved.
 
In these situations, it is often useful to know the relationship between, on the one hand, states where <math>(J_1)_z, (J_1)^2, (J_2)_z, (J_2)^2</math> all have definite values, and on the other hand, states where <math>(J_1)^2, (J_2)^2, J^2, J_z</math> all have definite values, as the latter four are usually conserved (constants of motion). The procedure to go back and forth between these [[basis (linear algebra)|bases]] is to use [[Clebsch–Gordan coefficients]].
 
One important result in this field is that a relationship between the quantum numbers for <math>(J_1)^2, (J_2)^2, J^2</math>:
:<math> j \in \{ |j_1-j_2|, (|j_1-j_2|+1), \ldots, (j_1 + j_2) \} </math>.
 
For an atom or molecule with '''J''' = '''L''' + '''S''', the [[term symbol]] gives the quantum numbers associated with the operators <math>L^2, S^2, J^2</math>.
 
==Orbital angular momentum in spherical coordinates==
 
Angular momentum operators usually occur when solving a problem with [[spherical symmetry]] in [[spherical coordinates]]. The angular momentum in space representation is
<ref>{{Cite book
| publisher = Springer Berlin Heidelberg
| isbn = 978-3-540-46215-6
| title = Quantum Mechanics
| location = Berlin, Heidelberg
| accessdate = 2011-03-29
| year = 2007
| url = http://www.springerlink.com/index/10.1007/978-3-540-46216-3
| page= 70
}}</ref>
 
: <math>L_{x}=i\hbar\left(\sin\phi\frac{\partial}{\partial\theta}+\cot\theta\cos\phi\frac{\partial}{\partial\phi}\right), </math>
 
: <math>L_{y}=i\hbar\left(-\cos\phi\frac{\partial}{\partial\theta}+\cot\theta\sin\phi\frac{\partial}{\partial\phi}\right), </math>
 
: <math>L_{z}=-i\hbar\frac{\partial}{\partial\phi,}</math>
 
and
 
: <math>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). </math>
 
When solving to find [[eigenstate]]s of this operator, we obtain the following
: <math> L^2 \mid l, m \rang = {\hbar}^2 l(l+1) | l, m \rang </math>
: <math> L_z \mid l, m \rang = \hbar m | l, m \rang </math>
where
:<math> \lang \theta , \phi | l, m \rang = Y_{l,m}(\theta,\phi)</math>
are the [[spherical harmonic]]s.
 
==See also==
*[[Runge–Lenz vector]] (used to describe the shape and orientation of bodies in orbit)
*[[Holstein–Primakoff transformation]]
*[[Vector model of the atom]]
*[[Pauli–Lubanski pseudovector]]
*[[Angular momentum diagrams (quantum mechanics)]]
*[[Spherical basis]]
*[[Tensor operator]]
*[[Orbital magnetization]]
 
==References==
<references/>
 
==Further reading==
 
* ''Quantum Mechanics Demystified'', D. McMahon, Mc Graw Hill (USA), 2006, ISBN(10-) 0-07-145546 9
* ''Quantum mechanics'', E. Zaarur, Y. Peleg, R. Pnini, Schaum’s Easy Oulines Crash Course, Mc Graw Hill (USA), 2006, ISBN (10-)007-145533-7 ISBN (13-)978-007-145533-6
* ''Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles (2nd Edition)'', R. Eisberg, R. Resnick, John Wiley & Sons, 1985, ISBN 978-0-471-87373-0
* ''Quantum Mechanics'', E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004, ISBN 978-0-13-146100-0
* ''Physics of Atoms and Molecules'', B.H. Bransden, C.J.Joachain, Longman, 1983, ISBN 0-582-44401-2
 
{{Physics operator}}
 
[[Category:Rotational symmetry]]
[[Category:Quantum mechanics]]

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