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:''"Spherical tensor operator" redirects to here. For the closely related concept see [[spherical basis]].''

In [[pure mathematics|pure]] and [[applied mathematics]], particularly [[quantum mechanics]] and [[computer graphics]] and applications therefrom, a '''tensor operator''' generalizes the notion of [[operator (mathematics)|operator]]s which are [[scalar (mathematics)|scalar]]s and [[vector (mathematics and physics)|vector]]s. A special class of these are '''spherical tensor operators''' which apply the notion of the [[spherical basis]] and [[spherical harmonic]]s. The spherical basis closely relates to the description of [[angular momentum]] in quantum mechanics and spherical harmonic functions.

== Rotations of quantum states ==

===Quantum rotation operator===

The [[Rotation operator (quantum mechanics)|rotation operator]] about the [[unit vector]] '''n''' (defining the axis of rotation) through angle ''θ'' is

:<math>U[R(\theta, \hat{\mathbf{n}})] = \exp\left(-\frac{i\theta}{\hbar}\hat{\mathbf{n}}\cdot\widehat{\mathbf{J}}\right)</math>

where '''J''' = (''J<sub>x</sub>'', ''J<sub>y</sub>'', ''J<sub>z</sub>'') are the rotation generators (also the angular momentum matrices):

:<math>J_x = \begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & -i \\
0 & i & 0
\end{pmatrix}\,\quad J_y = \begin{pmatrix}
0 & 0 & i \\
0 & 0 & 0 \\
-i & 0 & 0
\end{pmatrix}\,\quad J_z = \begin{pmatrix}
0 & -i & 0 \\
i & 0 & 0 \\
0 & 0 & 0
\end{pmatrix}</math>

and let <math>\widehat{R} = \widehat{R}(\theta,\hat{\mathbf{n}})</math> is a [[rotation matrix]]. Taylor expanding to first order in ''θ'' gives the infinitesimal rotation operator:

:<math>U[R(\theta, \hat{\mathbf{n}})] = 1 - \frac{i\theta}{\hbar}\hat{\mathbf{n}}\cdot\widehat{\mathbf{J}} </math>

An operator <math>\widehat{\Omega}</math> is invariant under a unitary transformation ''U'' if:

:<math>\widehat{\Omega} = {U}^\dagger \widehat{\Omega} U </math>

in this case for the rotation <math>\widehat{U}(R)</math>: 

:<math>\widehat{\Omega} = {U(R)}^\dagger \widehat{\Omega} U(R) = \exp\left(\frac{i\theta}{\hbar}\hat{\mathbf{n}}\cdot\widehat{\mathbf{J}}\right) \widehat{\Omega} \exp\left(-\frac{i\theta}{\hbar}\hat{\mathbf{n}}\cdot\widehat{\mathbf{J}}\right) </math>

===Angular momentum eigenkets ===

The orthonormal basis set for total angular momentum is {{ket|''j'', ''m''}}, where ''j'' is the total angular momentum quantum number and ''m'' is the magnetic angular momentum quantum number, which takes values −''j'', −''j'' + 1, ..., ''j'' − 1, ''j''. A general state

:<math> | \psi \rangle = \sum_m |j,m\rangle </math>

in the space rotates to a new state {{ket|{{overline|''ψ''}}}} by:

:<math> |\bar{\psi} \rangle = U(R)|\psi \rangle </math>

Using the [[completeness condition]]:

:<math> I = \sum_m |j , m' \rangle \langle j, m' | </math>

we have

:<math> | \bar{\psi} \rangle = I U(R)|\psi \rangle = \sum_{mm'} | j , m' \rangle \langle j, m' | U(R) |j , m \rangle </math>

Introducing the [[Wigner D matrix]] elements:

:<math>{D(R)}^{(j)}_{m'm} = \langle j, m' | U(R) |j,m \rangle</math>

gives the matrix multiplication:

:<math> | \bar{\psi} \rangle = \sum_{mm'} D^{(j)}_{m'm} | j , m \rangle \quad \Rightarrow \quad | \bar{\psi} \rangle = D^{(j)} | \psi \rangle </math>

For one basis ket:

:<math> | \overline{j , m} \rangle = \sum_{mm'} {D(R)}^{(j)}_{m'm} | j , m \rangle </math>

For the case of orbital angular momentum, the eigenstates {{ket|''l'', ''m''}} of the orbital [[angular momentum operator]] '''L''' and solutions of [[Laplace's equation]] on a 3d sphere are [[spherical harmonic]]s:

:<math> Y_\ell^m( \theta , \varphi ) = \langle \theta,\phi | \ell , m \rangle =  \sqrt{{(2\ell+1)\over 4\pi}{(\ell-m)!\over (\ell+m)!}}  \, P_\ell^m ( \cos{\theta} ) \, e^{i m \phi } </math>

where ''P''<sub>{{ell}}</sub><sup>''m''</sup> is an [[associated Legendre polynomial]], {{ell}} is the orbital angular momentum quantum number, and ''m'' is the orbital magnetic [[quantum number]] which takes the values −{{ell}}, −{{ell}} + 1, ... {{ell}} − 1, {{ell}} The formalism of spherical harmonics have wide applications in applied mathematics, and are closely related to the formalism of spherical tensors, as shown below.

Spherical harmonics are functions of the polar and azimuthal angles, ''θ'' and ''ϕ'' respectively, which can be conveniently by collected into a unit vector '''n'''(''θ'', ''ϕ'') pointing in the direction of those angles, in the Cartesian basis it is:

:<math>\hat{\mathbf{n}}(\theta,\phi) = \cos\theta \sin\phi \mathbf{e}_x + \sin\theta \sin\phi \mathbf{e}_y + \cos\phi \mathbf{e}_z</math>

So a spherical harmonic can also be written ''Y''<sub>{{ell}}</sub><sup>''m''</sup> = {{bra-ket|'''n'''|{{ell}}, ''m''}}. Spherical harmonic states {{ket|''m'', {{ell}}}} rotate according to the inverse rotation matrix ''U''(''R''<sup>−1</sup>), while {{ket|'''n'''}} rotates by the initial rotation matrix <math>\widehat{U}(R)</math>.

:<math>| \overline{\ell , m} \rangle = \sum_{m'} D_{m'm}^{(\ell)} U(R^{-1}) | \ell , m' \rangle\,,\quad | \overline{\hat{\mathbf{n}}} \rangle = U(R) | \hat{\mathbf{n}}\rangle </math>

==Rotations of tensorial operators==

===Scalar operators===

A scalar operator is invariant under rotations:<ref name="Abers ch 5">{{cite book|title=Quantum Mechanics|author=E. Abers|publisher=Addison Wesley|year=2004|chapter=5|isbn=978-0-13-146100-0}}</ref>

:<math> U(R)^\dagger \widehat{S} U(R) = \widehat{S}</math>

and we have a simple result, that the scalar operator commutes with the rotation generators:

:<math> \left[ \widehat{S}_a , \widehat{J}_b \right] = 0 </math>

Examples of a scalar operators include 
*the [[energy operator]]:
::<math> \widehat{E} \psi = i\hbar\frac{\partial}{\partial t} \psi </math>
*[[potential energy]] ''V'' 
::<math>\widehat{V}(\mathbf{r},t)\psi = V(\mathbf{r},t)\psi </math>
*[[kinetic energy]] ''T'':
::<math>\widehat{T}(\mathbf{r},t)\psi = -\frac{\hbar^2}{2m} \nabla^2 (\mathbf{r},t)</math>
*as well as the operator coupling term in [[spin-orbit coupling]]:
:<math>\widehat{\mathbf{L}}\cdot\widehat{\mathbf{S}} = \widehat{L}_x \widehat{S}_x + \widehat{L}_y \widehat{S}_y + \widehat{L}_z \widehat{S}_z \,. </math>

===Vector operators===

Vector operators (as well as [[pseudovector]] operators) can be rotated according to:<ref name="Abers ch 5"/>

:<math> {U(R)}^\dagger \widehat{V}_i U(R) = \sum_j R_{ij} \widehat{V}_j </math>

from this and the infinitesimal rotation operator and its Hermitian conjugate, one can derive the commutation relation with the rotation generator:

:<math> \left[ \widehat{V}_a , \widehat{J}_b \right] = i \hbar \varepsilon_{abc} \widehat{V}_c </math>

where ''ε<sub>ijk</sub>'' is the [[Levi-Civita symbol]], which all vector operators must satisfy, by construction. As the symbol ''ε<sub>ijk</sub>'' is a [[pseudotensor]], pseudovector operators are invariant [[up to]] a sign: +1 for [[proper rotation]]s and −1 for [[improper rotation]]s.

Vector operators include 
*the [[position operator]]:
::<math>\widehat{\mathbf{r}} \psi = \mathbf{r} \psi </math>
*the [[momentum operator]]:
::<math>\widehat{\mathbf{p}} \psi = -i\hbar \nabla \psi </math>
and peusodovector operators include 
*the orbital [[angular momentum operator]]: 
::<math>\widehat{\mathbf{L}} \psi = -i\hbar \mathbf{r} \times \nabla \psi </math>
*as well the [[spin (physics)|spin operator]] '''S''', and hence the total angular momentum
::<math>\widehat{\mathbf{J}} = \widehat{\mathbf{L}}+\widehat{\mathbf{S}}\,.</math>

In Dirac notation:

:<math>\langle\bar{\psi}|\widehat{V}_a|\bar{\psi}\rangle = \langle \psi | {U(R)}^\dagger \widehat{V}_a U(R) | \psi \rangle = \sum_b R_{ab} \langle \psi | \widehat{V}_b | \psi \rangle </math>

and since {{ket|''ψ''}} is any quantum state, the same result follows:

:<math>\widehat{V}_a = {U(R)}^\dagger \widehat{V}_a U(R) = \sum_b R_{ab} \widehat{V}_b </math>

Note that here, the term "vector" is used two different ways: kets such as {{ket|''ψ''}} are elements of abstract Hilbert spaces, while the vector operator is defined as a quantity whose components transform in a certain way under rotations.

===Spherical vector operators===

A vector operator in the [[spherical basis]] is {{nowrap|''V'' {{=}} (''V''<sub>+1</sub>, ''V''<sub>0</sub>, ''V''<sub>−1</sub>)}} where the components are:<ref name="Abers ch 5"/>

:<math>V_{+1}=-\frac{1}{\sqrt{2}}(V_x + i V_y)\,\quad V_{-1}=\frac{1}{\sqrt{2}}(+ V_x - i V_y)\,,\quad V_0 = V_z \,, </math>

and the commutators with the rotation generators are:

:<math>\left[J_z, V_q\right] = q V_q </math>
:<math>\left[J_\pm, V_0\right] = \sqrt{2} V_\pm </math>
:<math>\left[J_\pm, V_\mp\right] = \sqrt{2} V_0 </math>
:<math>\left[J_\pm, V_\pm\right] = 0 </math>

where ''q'' is a placeholder for the spherical basis labels (+1, 0, −1), and:

:<math>J_\pm = J_x \pm i J_y \,,</math>

(some authors may place a factor of 1/2 on the left hand side of the equation) and raise (''J''<sub>+</sub>) or lower (''J''<sub>−</sub>) the total magnetic [[quantum number]] ''m'' by one unit. In the spherical basis the generators are:

:<math>J_{\pm 1} = \mp \frac{1}{\sqrt{2}} J_\pm \,,\quad J_0 = J_z</math>

The rotation transformation in the spherical basis (originally written in the Cartesian basis) is then:

:<math> {U(R)}^\dagger \widehat{V}_q U(R) = \sum_{q'} {D(R)}^{(1)}_{qq'} \widehat{V}_{q'} </math>

One can generalize the ''vector'' operator concept easily to ''tensorial operators'', shown next.

===Tensor operators and their reducible and irreducible representations===

A tensor operator can be rotated according to:<ref name="Abers ch 5"/>

:<math> U(R)^\dagger \widehat{T}_{pqr\cdots} U(R) = R_{pi}R_{qj}R_{rk}\cdots \widehat{T}_{ijk\cdots}</math>

Consider a dyadic tensor with components ''T'' = ''a<sub>i</sub>b<sub>j</sub>'', this rotates infinitesimally according to:

:<math> U(R)^\dagger \widehat{T}_{pq} U(R) = R_{pi} R_{qj} \widehat{T}_{ij} = R_{pi} \widehat{a}_i R_{qj} \widehat{b}_j </math>

Cartesian dyadic tensors of the form 

:<math>\hat{\mathbf{T}} = \mathbf{e}_i \widehat{a}_i \otimes \mathbf{e}_j \widehat{b}_j = \mathbf{e}_i \otimes \mathbf{e}_j \widehat{a}_i \widehat{b}_j </math> 

where '''a''' and '''b''' are two vector operators:

:<math>\hat{\mathbf{a}} = \mathbf{e}_i \widehat{a}_i \,,\quad \hat{\mathbf{b}} = \mathbf{e}_j \widehat{b}_j </math> 

are reducible, which means they can be re-expressed in terms of '''a''' and '''b''' as a rank 0 tensor (scalar), plus a rank 1 tensor (an antisymmetric tensor), plus a rank 2 tensor (a symmetric tensor with zero [[trace (mathematics)|trace]]):

:<math>\mathbf{T} = \mathbf{T}^{(1)} + \mathbf{T}^{(2)} + \mathbf{T}^{(3)} </math>

where the first term 

:<math>\widehat{T}^{(1)}_{ij} = \frac{\widehat{a}_k \widehat{b}_k}{3}\delta_{ij} </math>

includes just one component, a scalar equivalently written ('''a'''·'''b''')/3,  the second 

:<math>\widehat{T}^{(2)}_{ij} = \frac{1}{2}[\widehat{a}_i \widehat{b}_j - \widehat{a}_j \widehat{b}_i] = \widehat{a}_{[i} \widehat{b}_{j]} </math>

includes three independent components, equivalently the components of ('''a'''×'''b''')/2, and the third 

:<math>\widehat{T}^{(3)}_{ij} = \frac{1}{2}(\widehat{a}_i \widehat{b}_j + \widehat{a}_j \widehat{b}_i) - \frac{\widehat{a}_k \widehat{b}_k}{3}\delta_{ij} = \widehat{a}_{(i} \widehat{b}_{j)} - T^{(1)}_{ij}</math>

includes five independent components. Throughout, ''δ<sub>ij</sub>'' is the [[Kronecker delta]], the components of the [[identity matrix]]. The number in the superscripted brackets denotes the tensor rank. These three terms are irreducible, which means they cannot be decomposed further and still be tensors satisfying the defining transformation laws under which they must be invariant. These also correspond to the number of spherical harmonic functions 2{{ell}} + 1 for {{ell}} = 0, 1, 2, the same as the ranks for each tensor. Each of the irreducible representations '''T'''<sup>(1)</sup>, '''T'''<sup>(2)</sup> ...  transform like angular momentum eigenstates according to the number of independent components.

===Spherical tensor operators===

Continuing the previous example of the second order dyadic tensor '''T''' = '''a''' &otimes; '''b''', casting each of '''a''' and '''b''' into the spherical basis and substituting into '''T''' gives the spherical tensor operators of the second order, which are:

:<math>\widehat{T}^{(2)}_{\pm 2} = \widehat{a}_{\pm 1} \widehat{b}_{\pm 1}</math>
:<math>\widehat{T}^{(2)}_{\pm 1} = \frac{1}{\sqrt{2}}\left( \widehat{a}_{\pm 1} \widehat{b}_0 + \widehat{a}_0 \widehat{b}_{\pm 1} \right)</math>
:<math>\widehat{T}^{(2)}_{0} = \frac{1}{\sqrt{6}}\left( \widehat{a}_{+1} \widehat{b}_{-1} + \widehat{a}_{-1} \widehat{b}_{+1} + 2 \widehat{a}_0 \widehat{b}_0 \right)</math>

Using the infinitesimal rotation operator and its Hermitian conjugate, one can derive the commutation relation in the spherical basis:

:<math>\left[J_a, \widehat{T}^{(2)}_{q} \right] = \sum_{q'} {D(J)}^{(2)}_{qq'} \widehat{T}_{q'}^{(2)} </math>

and the finite rotation transformation in the spherical basis is:

:<math>{U(R)}^\dagger \widehat{T}^{(2)}_q U(R) = \sum_{q'} {D(R)}^{(2)}_{qq'} \widehat{T}_{q'}^{(2)}</math>

In general, tensor operators can be constructed from two perspectives.<ref name="Nielsen Robinson"/>

One way is to specify how spherical tensors transform under a physical rotation - a [[group theory|group theoretical]] definition. A rotated angular momentum eigenstate can be decomposed into a linear combination of the initial eigenstates: the coefficients in the linear combination consist of Wigner rotation matrix entries. Spherical tensor operators are sometimes defined as the set of operators that transform just like the eigenkets under a rotation.

A spherical tensor ''T''<sub>''q''</sub><sup>(''k'')</sup> of rank ''k'' is defined to rotate into ''T''<sub>''q''&prime;</sub><sup>(''k'')</sup> according to:

:<math>{U(R)}^\dagger \widehat{T}_{q'}^{(k)} U(R) = \sum_{q} D^{(k)}_{qq'} \widehat{T}_{q}^{(k)} </math>

where ''q'' = ''k'', ''k'' − 1, ..., −''k'' + 1, −''k''. For spherical tensors, ''k'' and ''q'' are analogous labels to {{ell}} and ''m'' respectively, for spherical harmonics. Some authors write ''T''<sub>''k''</sub><sup>''q''</sup> instead of ''T''<sub>''q''</sub><sup>(''k'')</sup>, with or without the brackets enclosing the rank number ''k''.

Another related procedure requires that the spherical tensors satisfy certain commutation relations with respect to the rotation generators ''J<sub>x</sub>'', ''J<sub>y</sub>'', ''J<sub>z</sub>'' - an algebraic definition.

The commutation relations of the angular momentum components with the tensor operators are:

:<math>\left[ J_\pm , \widehat{T}_{k}^{q} \right] = \hbar \sqrt{( k \mp q )(k \pm q + 1)} \widehat{T}_{k}^{q \pm 1} </math>
:<math>\left[ J_z , \widehat{T}_{k}^{q} \right] = \hbar q \widehat{T}_{k}^{q} </math>

Consider any 3d vector, not just a unit vector, and not just the [[position vector]]:

:<math>\mathbf{a} = a_x \mathbf{e}_x + a_y \mathbf{e}_y + a_z \mathbf{e}_z</math>

then a spherical tensor is a spherical harmonic as a function of this vector '''a''', and in Dirac notation:

:<math> T_q^{(k)} = Y_{\ell = k }^{m = q} (\mathbf{a}) = \langle \mathbf{a}|k,q\rangle</math>

(the super and subscripts switch places for the corresponding labels ''{{ell}}'' ↔ ''k'' and ''m'' ↔ ''q'' which spherical tensors and spherical harmonics use).

Spherical harmonic states and spherical tensors can also be constructed out of the [[Clebsch–Gordan coefficients]]. Irreducible spherical tensors can build higher rank spherical tensors; if ''A''<sub>''q''1</sub><sup>(''k''1)</sup> and ''B''<sub>''q''2</sub><sup>(''k''2)</sup> are two spherical tensors of ranks ''k''<sub>1</sub> and ''k''<sub>2</sub> respectively, then:

:<math>T_{q}^{(k)} = \sum_q \langle k_1 , k_2 , q_1 , q_2 | k , q , q_1 , q_2 \rangle A_{q_1}^{(k_1)} B_{q_2}^{(k_2)} </math>

is a spherical tensor of rank ''k''.

==Angular momentum and spherical harmonics==

===Orbital angular momentum and spherical harmonics===

Orbital angular momentum operators have the ladder operators:

:<math>L_\pm = L_x \pm i L_y </math>

which raise or lower the orbital magnetic quantum number ''m<sub>{{ell}}</sub>'' by one unit. This has almost exactly the same form as the spherical basis, aside from constant multiplicative factors.

===Spherical tensor operators and quantum spin===

{{See also|Spin-weighted spherical harmonics|spin spherical harmonics}}

Spherical tensors can also be formed from algebraic combinations of the spin operators ''S<sub>x</sub>'', ''S<sub>y</sub>'', ''S<sub>z</sub>'', as matrices, for a spin system with total quantum number ''j'' = {{ell}} + ''s'' (and {{ell}} = 0). Spin operators have the ladder operators:

:<math>S_\pm = S_x \pm i S_y </math>

which raise or lower the spin magnetic quantum number ''m<sub>s</sub>'' by one unit. 

==Applications==

Spherical bases have broad applications in pure and applied mathematics and physical sciences where spherical geometries occur.

===Magnetic resonance===

The spherical tensor formalism provides a common platform for treating coherence and relaxation in [[nuclear magnetic resonance]]. In [[nuclear magnetic resonance|NMR]] and [[Electron paramagnetic resonance|EPR]], spherical tensor operators are employed to express the quantum dynamics of [[spin (physics)|particle spin]], by means of an equation of motion for the [[density matrix]] entries, or to formulate dynamics in terms of an equation of motion in [[Liouville space]]. The Liouville space equation of motion governs the observable averages of spin variables. When relaxation is formulated using a spherical tensor basis in Liouville space, insight is gained because the relaxation matrix exhibits the cross-relaxation of spin observables directly.<ref name="Nielsen Robinson">{{cite article|title=The Spherical Tensor Formalism Applied to Relaxation in Magnetic Resonance|author=R.D. Nielsen, B.H. Robinson|year=2006|page=270-271|url=http://www.fast-mariecurie-rtn-project.eu/espacePrive/dvormondt/nielsen06.pdf}}</ref>

===Image processing and computer graphics===

==See also==

*[[Wigner–Eckart theorem]]
*[[Structure tensor]]

==References==

===Notes===

{{reflist}}

*{{cite book|title=Translational Dynamics and Magnetic Resonance:Principles of Pulsed Gradient Spin Echo NMR
 |author=P. T. Callaghan|year=2011|publisher=Oxford University Press|isbn=0-191-621-048|url=http://books.google.co.uk/books?id=PocotAhdrtQC&pg=PA89&dq=spherical+tensor+operators&hl=en&sa=X&ei=eyq3UZL2JML3O7HxgLgM&ved=0CD0Q6AEwAg#v=onepage&q=spherical%20tensor%20operators&f=false}}

*{{cite book|title=Polarization and Correlation Phenomena in Atomic Collisions:
A Practical Theory Course|year=2000 |publisher=Springer|author=V. V. Balashov, A. N. Grum-Grzhimailo, N.M. Kabachnik|url=http://books.google.co.uk/books?id=Z_K9HOndNssC&pg=PA209&dq=tensor+operators&hl=en&sa=X&ei=tiu3UcDxIsfMOa_BgeAF&ved=0CF8Q6AEwCA#v=onepage&q=tensor%20operators&f=false}}

*{{cite book|title=Spherical Tensor Operators: Tables of Matrix Elements and Symmetries|year=1990|publisher=World Scientific|isbn=981-0202-830|author=J. A. Tuszynski|url=http://books.google.co.uk/books?id=0IRIaJ4yLZEC&printsec=frontcover&dq=tensor+operators&hl=en&sa=X&ei=bSu3UfOrBIXuOu7NgHg&ved=0CDEQ6AEwAA}}

*{{cite book|title=Quantum Groups and Their Applications in Physics:
Varenna on Lake Como, Villa Monastero, 28 June-8 July 1994|year=|author=L. Castellani, J. Wess|publisher=Società Italiana di Fisica, IOS|year=1996|isbn=905-199-24-75|url=http://books.google.co.uk/books?id=_b0bZzw426AC&pg=PA19&dq=tensor+operators&hl=en&sa=X&ei=tiu3UcDxIsfMOa_BgeAF&ved=0CGYQ6AEwCQ#v=onepage&q=tensor%20operators&f=false}}

*{{cite book|title=Introduction to the Graphical Theory of Angular Momentum|publisher=Springer|author=|year=2009|isbn=364-203-11-96|url=http://books.google.co.uk/books?id=3xLabi3WThgC&pg=PA6&dq=tensor+operators&hl=en&sa=X&ei=lCy3Uai8D8LEOeDWgZgF&ved=0CEoQ6AEwBTgK}}

*{{cite book|title=Angular Momentum in Quantum Mechanics|author=A. R. Edmonds|publisher=Princeton University Press|isbn=0-691-025-894|edition=2nd|year=1996|url=http://books.google.co.uk/books?id=0BSOg0oHhZ0C&printsec=frontcover&dq=spherical+tensor+operators&hl=en&sa=X&ei=eyq3UZL2JML3O7HxgLgM&ved=0CEMQ6AEwAw#v=onepage&q=spherical%20tensor%20operators&f=false}}

*{{cite article|title=Tensors and rotations in NMR|author=L.J. Mueller|year=2011|publisher=Wiley Periodicals|doi=10.1002/cmr.a.20224|page=|url=http://onlinelibrary.wiley.com/doi/10.1002/cmr.a.20224/abstract}}

*{{cite article|title=Spherical Tensor Operators in NMR|author=M.S. Anwar|year=2004|page=|url=http://physlab.lums.edu.pk/images/3/39/Spherical_tensor.pdf}}

*{{cite book|title=Principles of nuclear magnetic resonance microscopy|author=P. Callaghan|page=56-57|year=1993|isbn=0-198-539-975|publisher=Oxford University Press|url=http://books.google.co.uk/books?id=yjrjT_W5hygC&pg=PA56&dq=spherical+tensor+basis&hl=en&sa=X&ei=W9upUY3tB-XI0AWojYHoDg&ved=0CDAQ6AEwADgK#v=onepage&q=spherical%20tensor%20basis&f=false}}

===Further reading===

;Spherical harmonics

*{{cite book|title=Springer Handbook of Atomic, Molecular, and Optical Physics|page=57|author=G.W.F. Drake|publisher=Springer|year=2006|edition=2nd|isbn=0-3872-6308-X|url=http://books.google.co.uk/books?id=Jj-ad_2aNOAC&pg=PA52&dq=spherical+tensor+basis&hl=en&sa=X&ei=utupUYv8Fqn30gX1zIHgDA&ved=0CFAQ6AEwBTge#v=onepage&q=spherical%20tensor%20basis&f=false}}

*{{cite book|title=Theoretical global seismology|page=appendix C|author=F.A. Dahlen, J. Tromp|publisher=Princeton University Press|year=1998|edition=2nd|isbn=0-69100-1243|url=http://books.google.co.uk/books?id=GWnuBws5gBEC&pg=PA877&dq=spherical+tensor+basis&hl=en&sa=X&ei=BtypUanVC--N0wWftIHwDg&ved=0CDAQ6AEwADgo#v=onepage&q=spherical%20tensor%20basis&f=false}}

*{{cite book|title=Review of Progress in Quantitative Nondestructive Evaluation|volume=16|series=Review Of Progress In Quantitative Nondestructive Evaluation|page=1708|author=D.O. Thompson, D.E. Chimenti|publisher=Springer|year=1997|edition=|isbn=0-3064-55978|url=http://books.google.co.uk/books?id=3etLzmHu6bQC&pg=PA1709&dq=spherical+tensor+basis&hl=en&sa=X&ei=BtypUanVC--N0wWftIHwDg&ved=0CGAQ6AEwCTgo#v=onepage&q=spherical%20tensor%20basis&f=false}}

*{{cite book|title=Nuclear Physics with Polarized Particles|author=H. Paetz, G. Schieck|page=31|publisher=Springer|year=2011|volume=842 |series=Lecture Notes in Physics|isbn=364-224-225-1|url=http://books.google.co.uk/books?id=jsJg8VjFe6YC&pg=PA31&dq=spherical+basis+tensor&hl=en&sa=X&ei=yNqpUb_RCqWb0QWVuoGgDg&ved=0CDoQ6AEwAQ#v=onepage&q=spherical%20basis%20tensor&f=false}}

*{{cite book|title=Angular Momentum Techniques in Quantum Mechanics|year=1999|author=V. Devanathan|page=34, 61|volume=108|series=Fundamental Theories of Physics|publisher=Springer|isbn=0-7923-5866-X|url=http://books.google.co.uk/books?id=VeiursWZeoUC&pg=PT1&dq=spherical+basis+tensor&hl=en&sa=X&ei=D9upUYj0FIOn0QXt9IHwDQ&ved=0CDAQ6AEwADgK}}

*{{cite book|title=A Companion to Angular Momentum|author=V.D. Kleiman, R.N. Zare|publisher=John Wiley & Sons|year=1998|chapter=5|page=112|isbn=0-4711-9249-X|url=http://books.google.co.uk/books?id=j-Wi3ZAK2nEC&pg=PA112&dq=spherical+basis+tensor&hl=en&sa=X&ei=yNqpUb_RCqWb0QWVuoGgDg&ved=0CFMQ6AEwBQ#v=onepage&q=spherical%20basis%20tensor&f=false}}

;Angular momentum and spin 

*{{cite book|title=Angular Momentum Techniques in Quantum Mechanics|chapter=Vectors and Tensors in Spherical Basis|series=Fundamental Theories of Physics|volume=108|year=2002|page=24-33|doi=10.1007/0-306-47123-X_3|isbn=978-0-306-47123-0 (online)|url=http://link.springer.com/chapter/10.1007%2F0-306-47123-X_3#page-1}}

*{{cite book|title=Quantum mechanics|year=2000|author=K.T. Hecht|series=Graduate texts in contemporary physics|isbn=0-387-989-196|publisher=Springer|url=http://books.google.co.uk/books?id=jcDb7iRvI0AC&pg=PA305&dq=spherical+tensor+basis&hl=en&sa=X&ei=bNupUbCfNKnW0QXeu4DIDg&ved=0CDAQ6AEwADgU#v=onepage&q=spherical%20tensor%20basis&f=false}}

;Condensed matter physics

*{{cite article|title=Molecular Crystal Global Phase Diagrams:I Method of Construction|title=J.A. Mettes, J.B. Keith, R.B. McClurg|year=2002|url=http://journals.iucr.org/a/issues/2004/06/00/pz5009/pz5009sup1.pdf}}

*{{cite book|title=Crystal-Field Engineering of Solid-State Laser Materials|page=49|author=B.Henderson, R.H. Bartram|publisher=Cambridge University Press|year=2005|isbn=0-52101-8013|volume=25|series=Cambridge Studies in Modern Optics|url=http://books.google.co.uk/books?id=Dfpy6jnxh68C&pg=PA49&dq=spherical+tensor+basis&hl=en&sa=X&ei=bNupUbCfNKnW0QXeu4DIDg&ved=0CFwQ6AEwCDgU#v=onepage&q=spherical%20tensor%20basis&f=false}}

*{{cite book|title=Atomic Structure|author=E. U. Condon, Halis Odabasi|publisher=CUP Archive|year=1980|isbn=0-5212-98938|url=http://books.google.co.uk/books?id=hzE8AAAAIAAJ&pg=PA656&dq=spherical+tensor+basis&hl=en&sa=X&ei=utupUYv8Fqn30gX1zIHgDA&ved=0CDAQ6AEwADge#v=onepage&q=spherical%20tensor%20basis&f=false}}

*{{cite book|title=Solid State NMR Spectroscopy: Principles and Applications|author=Melinda J. Duer|author=K.D. Bonin, V.V. Kresin|publisher=John Wiley & Sons|year=2008|chapter=2|page=113|isbn=0-4709-9938-1|url=http://books.google.co.uk/books?id=m3BUm65np-UC&pg=PA113&dq=spherical+tensor+basis&hl=en&sa=X&ei=ONupUdzbL-eV0AWvmoCQDg&ved=0CGUQ6AEwCA#v=onepage&q=spherical%20tensor%20basis&f=false}}

*{{cite book|title=Electric - Dipole Polarizabilities of Atoms, Molecules and Clusters|author=K.D. Bonin, V.V. Kresin|publisher=World Scientific|year=1997|chapter=2|page=14-15|isbn=981-022-493-1|url=http://books.google.co.uk/books?id=1E8M117QOFEC&pg=PA14&dq=spherical+tensor+basis&hl=en&sa=X&ei=ONupUdzbL-eV0AWvmoCQDg&ved=0CEgQ6AEwAw#v=onepage&q=spherical%20tensor%20basis&f=false}}

*{{cite book|title=Solid State NMR Studies of Biopolymers|author=A.E. McDermott, T.Polenova|year=2012|page=42|publisher=John Wiley & Sons|series=EMR handbooks|isbn=111-858-889-4|url=http://books.google.co.uk/books?id=1nW4xEG1XAEC&pg=PA42&dq=spherical+tensor+basis&hl=en&sa=X&ei=utupUYv8Fqn30gX1zIHgDA&ved=0CFoQ6AEwBzge#v=onepage&q=spherical%20tensor%20basis&f=false}}

;Magnetic resonance

*{{cite article|title=Tensors and rotations in NMR|author=L.J. Mueller|year=2011|publisher=Wiley Periodicals|doi=10.1002/cmr.a.20224|page=|url=http://onlinelibrary.wiley.com/doi/10.1002/cmr.a.20224/abstract}}

*{{cite article|title=Spherical Tensor Operators in NMR|author=M.S. Anwar|year=2004|page=|url=http://physlab.lums.edu.pk/images/3/39/Spherical_tensor.pdf}}

*{{cite book|title=Principles of nuclear magnetic resonance microscopy|author=P. Callaghan|page=56-57|year=1993|isbn=0-198-539-975|publisher=Oxford University Press|url=http://books.google.co.uk/books?id=yjrjT_W5hygC&pg=PA56&dq=spherical+tensor+basis&hl=en&sa=X&ei=W9upUY3tB-XI0AWojYHoDg&ved=0CDAQ6AEwADgK#v=onepage&q=spherical%20tensor%20basis&f=false}}

;Image processing

*{{cite book|title=Tensors in Image Processing and Computer Vision|editor=S. Aja-Fernández|author=M. Reisert, H. Burkhardt|publisher=Springer|year=2009|isbn=184-8822-995|url=http://books.google.co.uk/books?id=yS_YbH-SAxgC&pg=PA153&lpg=PA153&dq=spherical+basis+tensor&source=bl&ots=hsc5YdfqJx&sig=sUa_xG7yg_jokN4IHvpkMEdTYUE&hl=en&sa=X&ei=rq2eUaSxGsz20gWh_ICACw&ved=0CFcQ6AEwCTgK#v=onepage&q=spherical%20basis%20tensor&f=false}}

*{{cite book|title=Visualization and Processing of Tensor Fields: Advances and Perspectives|author=D.H. Laidlaw, J. Weickert|publisher=Springer|year=2009|series=Mathematics and Visualization|isbn=354-088-378-9|url=http://books.google.co.uk/books?id=Kh5YLrjSOeAC&pg=PA44&dq=spherical+tensor+basis&hl=en&sa=X&ei=bNupUbCfNKnW0QXeu4DIDg&ved=0CFIQ6AEwBjgU#v=onepage&q=spherical%20tensor%20basis&f=false}}

*{{cite book|title=Energy Tensors: Quadratic, Phase Invariant Image Operators|author=M. Felsberg,
E. Jonsson |publisher=Springer|year=2005|volume=3663|series=Lecture Notes in Computer Science|page=493-500|url=http://link.springer.com/chapter/10.1007/11550518_61}}

*{{cite book|title=Tensor Operator Algebra for Point Groups|author= E. König, S. Kremer |publisher=Springer|year=1979|volume=3663|series=Lecture Notes in Computer Science|page=13-20|url=http://link.springer.com/chapter/10.1007%2F978-1-4613-3003-5_3}}

==External links==

*[http://scipp.ucsc.edu/~haber/ph216/clebsch.pdf (2012) ''Clebsch-Gordon coefficients and the tensor spherical harmonics'']
*[http://scipp.ucsc.edu/~haber/ph214/tensor_harmonics.pdf ''The tensor spherical harmonics'']
*[http://bohr.physics.berkeley.edu/classes/221/1011/notes/wigeck.pdf (2010) ''Irreducible Tensor Operators and the Wigner-Eckart Theorem'']
*[http://cc.oulu.fi/~tf/tiedostot/pub/kvmIII/english/2004/09_tensop.pdf ''Tensor operators'']
*[http://galileo.phys.virginia.edu/classes/752.mf1i.spring03/TensorOperators.htm M. Fowler (2008), ''Tensor Operators'']
*[http://www.physics.rutgers.edu/~steves/502/Lectures_Preliminary/Lec08_Tensor_Operators.pdf ''Tensor_Operators'']
*[http://www.lepp.cornell.edu/~dlr/teaching/p6572/TensorOperatorsWE.pdf (2009) ''Tensor Operators and the Wigner Eckart Theorem'']
*[http://www.core.org.cn/NR/rdonlyres/Chemistry/5-73Fall-2005/BEE27FD3-DE39-468F-B7B7-2EB89C58B89D/0/sec9.pdf ''The Wigner-Eckart theorem'']
*[http://ocw.mit.edu/courses/chemistry/5-74-introductory-quantum-mechanics-ii-spring-2004/lecture-notes/03_lecnotes_rwf.pdf (2004) ''Rotational Transformations and Spherical Tensor Operators'']
*[http://www.glue.umd.edu/afs/glue.umd.edu/department/phys/courses/Phys622/public_html/ji/lecture27.pdf ''Tensor operators'']
*[http://www.princeton.edu/~romalis/PHYS551/TensorOperators.pdf ''Evaluation of the matrix elements for radiative transitions'']
*[http://www.phys.ufl.edu/~cmueller/Homework3.pdf C. Mueller (2009) ''Tensor Operators and the Wigner-Eckart Theorem'']
*[http://www.phy.iitb.ac.in/~dkg/wigner.pdf D.K. Ghosh, (2013) ''Angular Momentum - III : Wigner- Eckart Theorem'']
*[http://info.phys.unm.edu/~ideutsch/classes/phys531f11/sphericaltensors.pdf B. Baragiola (2002) ''Tensor Operators'']
*[http://www.chemie.uni-hamburg.de/nmr/insensitive/tutorial/en.lproj/spherical_tensors.html ''Spherical Tensors'']

{{tensor}}

[[Category:Image processing]]
[[Category:Quantum mechanics]]
[[Category:Condensed matter physics]]
[[Category:Linear algebra]]
[[Category:Tensors]]
[[Category:Spherical geometry]]