User:Maschen/Spherical basis

"Spherical tensor" redirects to here.

In pure and applied mathematics, a spherical basis is the basis used to express spherical tensors. While spherical polar coordinates are one orthogonal coordinate system for expressing vectors and tensors using polar and azimuthal angles and radial distance, the spherical basis are constructed from the standard basis and complex numbers. The form the spherical basis takes closely relates to the description of angular momentum in quantum mechanics, and spherical harmonic functions.

A spherical tensor is a special case of a Cartesian tensor.

Basis vectors in three dimensions

Definition

The spherical basis vectors can be defined in terms of the Cartesian basis (e_x, e_y, e_z) = (e₁, e₂, e₃) using complex-valued coefficients in the xy plane:

${\begin{aligned}\mathbf {e} _{+}&=-{\frac {1}{\sqrt {2}}}\mathbf {e} _{x}-{\frac {i}{\sqrt {2}}}\mathbf {e} _{y}\\\mathbf {e} _{-}&=+{\frac {1}{\sqrt {2}}}\mathbf {e} _{x}-{\frac {i}{\sqrt {2}}}\mathbf {e} _{y}\\\end{aligned}}\quad \rightleftharpoons \quad \mathbf {e} _{\pm }=\mp {\frac {1}{\sqrt {2}}}\left(\mathbf {e} _{x}\pm i\mathbf {e} _{y}\right)\,$

where i denotes the imaginary unit, and one normal to the plane in the z direction:

$\mathbf {e} _{0}=\mathbf {e} _{z}$

The relations can be summarized by a change of basis:

${\begin{pmatrix}\mathbf {e} _{+}\\\mathbf {e} _{-}\\\mathbf {e} _{0}\end{pmatrix}}={\begin{pmatrix}-{\frac {1}{\sqrt {2}}}&-{\frac {i}{\sqrt {2}}}&0\\+{\frac {1}{\sqrt {2}}}&-{\frac {i}{\sqrt {2}}}&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}\mathbf {e} _{x}\\\mathbf {e} _{y}\\\mathbf {e} _{z}\end{pmatrix}}$

It is convenient to provide a name for this unitary matrix:

$\mathbf {U} ={\begin{pmatrix}-{\frac {1}{\sqrt {2}}}&-{\frac {i}{\sqrt {2}}}&0\\+{\frac {1}{\sqrt {2}}}&-{\frac {i}{\sqrt {2}}}&0\\0&0&1\end{pmatrix}}$

in other words, its Hermitian conjugate (complex conjugate and matrix transposed) is also the inverse matrix:

$\mathbf {U} ^{\dagger }=\mathbf {U} ^{\star \mathrm {T} }=\mathbf {U} ^{-1}={\begin{pmatrix}-{\frac {1}{\sqrt {2}}}&+{\frac {1}{\sqrt {2}}}&0\\+{\frac {i}{\sqrt {2}}}&+{\frac {i}{\sqrt {2}}}&0\\0&0&1\end{pmatrix}}$

Inverting the basis equations in matrix form:

${\begin{pmatrix}\mathbf {e} _{x}\\\mathbf {e} _{y}\\\mathbf {e} _{z}\end{pmatrix}}=\mathbf {U} ^{-1}{\begin{pmatrix}\mathbf {e} _{+}\\\mathbf {e} _{-}\\\mathbf {e} _{0}\end{pmatrix}}$

or explictly:

${\begin{aligned}\mathbf {e} _{x}&=-{\frac {1}{\sqrt {2}}}\mathbf {e} _{+}+{\frac {1}{\sqrt {2}}}\mathbf {e} _{-}\\\mathbf {e} _{y}&=+{\frac {i}{\sqrt {2}}}\mathbf {e} _{+}+{\frac {i}{\sqrt {2}}}\mathbf {e} _{-}\\\end{aligned}}$

$\mathbf {e} _{z}=\mathbf {e} _{0}$

Properties

Orthonormal basis

The spherical basis is an orthonormal basis, since the inner product ⟨ , ⟩ of every pair vanishes meaning the basis vectors are all mutually orthogonal:

$\left\langle \mathbf {e} _{+},\mathbf {e} _{-}\right\rangle =\left\langle \mathbf {e} _{-},\mathbf {e} _{0}\right\rangle =\left\langle \mathbf {e} _{0},\mathbf {e} _{+}\right\rangle =0$

and each basis vector is a unit vector:

$\left\langle \mathbf {e} _{+},\mathbf {e} _{+}\right\rangle =\left\langle \mathbf {e} _{-},\mathbf {e} _{-}\right\rangle =\left\langle \mathbf {e} _{0},\mathbf {e} _{0}\right\rangle =1$

hence the need for the normalizing factor of 1/√2.

In general, for two vectors with complex coefficients in the same real-valued orthonormal basis e_i, with the property e_i·e_j = δ_ij , we have:

$\left\langle \mathbf {a} ,\mathbf {b} \right\rangle =\mathbf {a} \cdot \mathbf {b} ^{\star }=\sum _{j}a_{j}b_{j}^{\star }$

where · is the usual dot product and the complex conjugate * must be used to keep the magnitude (or "norm") of the vector positive definite.

Coordinates in the Cartesian and spherical basis

A vector A can be written:

$\mathbf {A} =A_{x}\mathbf {e} _{x}+A_{y}\mathbf {e} _{y}+A_{z}\mathbf {e} _{z}=A_{+}\mathbf {e} _{+}+A_{-}\mathbf {e} _{-}+A_{0}\mathbf {e} _{0}$

where the coordinates of A can be expressed easily in the standard basis, in Cartesian notation:

$A_{x}=\left\langle \mathbf {A} ,\mathbf {e} _{x}\right\rangle \,,\quad A_{y}=\left\langle \mathbf {A} ,\mathbf {e} _{y}\right\rangle \,,\quad A_{z}=\left\langle \mathbf {A} ,\mathbf {e} _{z}\right\rangle \quad \rightleftharpoons \quad A_{j}=\left\langle \mathbf {A} ,\mathbf {e} _{j}\right\rangle$

where the index j takes the values x, y, z, or 1, 2, 3, and this translates to the corresponding components for the spherical basis like so:

${\begin{aligned}A_{+}&=\left\langle \mathbf {A} ,\mathbf {e} _{+}\right\rangle =-{\frac {A_{x}}{\sqrt {2}}}+{\frac {iA_{y}}{\sqrt {2}}}\\A_{-}&=\left\langle \mathbf {A} ,\mathbf {e} _{-}\right\rangle =+{\frac {A_{x}}{\sqrt {2}}}+{\frac {iA_{y}}{\sqrt {2}}}\\\end{aligned}}\quad \rightleftharpoons \quad A_{\pm }=\left\langle \mathbf {A} ,\mathbf {e} _{\pm }\right\rangle ={\frac {1}{\sqrt {2}}}\left(\mp A_{x}+iA_{y}\right)$

$A_{0}=\left\langle \mathbf {A} ,\mathbf {e} _{0}\right\rangle =\left\langle \mathbf {A} ,\mathbf {e} _{z}\right\rangle =A_{z}$

or in matrix form:

${\begin{pmatrix}A_{+}\\A_{-}\\A_{0}\end{pmatrix}}={\begin{pmatrix}-{\frac {1}{\sqrt {2}}}&+{\frac {i}{\sqrt {2}}}&0\\+{\frac {1}{\sqrt {2}}}&+{\frac {i}{\sqrt {2}}}&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}A_{x}\\A_{y}\\A_{z}\end{pmatrix}}=\mathbf {U} ^{\mathrm {T} }{\begin{pmatrix}A_{x}\\A_{y}\\A_{z}\end{pmatrix}}$

Inverting these gives in matrix form:

${\begin{pmatrix}A_{x}\\A_{y}\\A_{z}\end{pmatrix}}={\begin{pmatrix}-{\frac {1}{\sqrt {2}}}&+{\frac {1}{\sqrt {2}}}&0\\-{\frac {i}{\sqrt {2}}}&-{\frac {i}{\sqrt {2}}}&0\\0&0&1\end{pmatrix}}{\begin{pmatrix}A_{+}\\A_{-}\\A_{0}\end{pmatrix}}=(\mathbf {U} ^{\mathrm {T} })^{-1}{\begin{pmatrix}A_{+}\\A_{-}\\A_{0}\end{pmatrix}}$

or explicitly:

$A_{x}={\frac {1}{\sqrt {2}}}(-A_{+}+A_{-})$

$A_{y}={\frac {i}{\sqrt {2}}}(A_{+}+A_{-})$

$A_{z}=A_{0}$

Inner product

The inner product between two vectors A and B in the spherical basis follows from the above definition of the inner product:

$\left\langle \mathbf {A} ,\mathbf {B} \right\rangle =A_{+}B_{+}^{\star }+A_{-}B_{-}^{\star }+A_{0}B_{0}^{\star }$

Cartesian and spherical tensor operators in 3d

The state space of angular momentum eigenkets

The rotation operator about the unit vector n (defining the axis of rotation) through angle θ is

$U[R(\theta ,\mathbf {n} )]=\exp \left(-{\frac {i\theta }{\hbar }}\mathbf {n} \cdot {\hat {\mathbf {J} }}\right)$

where J = (J_x, J_y, J_z) are the total angular momentum matrices:

$J_{x}={\begin{pmatrix}0&0&0\\0&0&-i\\0&i&0\end{pmatrix}}\,\quad J_{y}={\begin{pmatrix}0&0&i\\0&0&0\\-i&0&0\end{pmatrix}}\,\quad J_{z}={\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}}$

and let R = R(θ, n) is a rotation matrix.

The orthonormal basis set for total angular momentum is |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

$|\alpha \rangle =\sum _{m}|j,m\rangle$

in the space rotates to a new state |α⟩ by:

$|{\bar {\alpha }}\rangle =U(R)|\alpha \rangle$

using the completeness condition:

$I=\sum _{m}|j,m'\rangle \langle j,m'|$

we have

$|{\bar {\alpha }}\rangle =IU(R)|\alpha \rangle =\sum _{mm'}|j,m'\rangle \langle j,m'|U(R)|j,m\rangle$

introducing the Wigner matrix elements:

$D_{m'm}^{(j)}=\langle j,m'|U(R)|j,m\rangle$

gives the matrix multiplication:

$|{\bar {\alpha }}\rangle =\sum _{mm'}D_{m'm}^{(j)}|j,m\rangle$

For one basis ket:

$|{\overline {j,m}}\rangle =\sum _{mm'}D_{m'm}^{(j)}|j,m\rangle$

An operator Ω is invariant under a unitary transformation U if:

${\hat {\Omega }}={U}^{\dagger }{\hat {\Omega }}U$

in this case for the rotation U(R):

${\hat {\Omega }}={U(R)}^{\dagger }{\hat {\Omega }}U(R)=\exp \left({\frac {i\theta }{\hbar }}\mathbf {n} \cdot {\hat {\mathbf {J} }}\right){\hat {\Omega }}\exp \left(-{\frac {i\theta }{\hbar }}\mathbf {n} \cdot {\hat {\mathbf {J} }}\right)$

The infinitesimal rotation operator is:

$U[R(\theta ,\mathbf {n} )]=1-{\frac {i\theta }{\hbar }}\mathbf {n} \cdot {\hat {\mathbf {J} }}$

Scalar operators

A scalar operator is invariant under rotations:

$U(R)^{\dagger }{\hat {S}}U(R)={\hat {S}}$

Vector operators

A vector operator can be rotated according to:

$U(R)^{\dagger }{\hat {V}}_{i}{U(R)}=R_{ij}{\hat {V}}_{j}$

from this one can derive the commutation relation:

$\left[{\hat {V}}_{a},{\hat {J}}_{b}\right]=i\hbar \varepsilon _{abc}{\hat {V}}_{c}$

where ε_ijk is the Levi-Civita symbol, which all vector operators must satisfy, by construction.

Tensor operators

A tensor operator can be rotated according to:

$U(R)^{\dagger }{\hat {T}}_{pqr\cdots }U(R)=R_{pi}R_{qj}R_{rk}\cdots {\hat {T}}_{ijk\cdots }$

Consider a dyadic tensor with components T = a_ib_j, this rotates infinitesimally according to:

$U(R)^{\dagger }{\hat {T}}_{pq}U(R)=R_{pi}R_{qj}{\hat {T}}_{ij}=R_{pi}{\hat {a}}_{i}R_{qj}{\hat {b}}_{j}$

Cartesian dyadic tensors are reducible, which means they can be re-expressed in terms of the two vector operators

${\hat {\mathbf {a} }}=\mathbf {e} _{i}{\hat {a}}_{i}\,,\quad {\hat {\mathbf {b} }}=\mathbf {e} _{j}{\hat {b}}_{j}$

into a rank 0 tensor (scalar), a rank 1 tensor (an antisymmetric tensor), and a rank 2 tensor (a symmetric tensor with zero trace):

$\mathbf {T} =\mathbf {T} ^{(1)}+\mathbf {T} ^{(2)}+\mathbf {T} ^{(3)}$

where the first term

$T_{ij}^{(1)}={\frac {{\hat {a}}_{k}{\hat {b}}_{k}}{3}}\delta _{ij}$

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

$T_{ij}^{(2)}={\frac {1}{2}}[{\hat {a}}_{i}{\hat {b}}_{j}-{\hat {a}}_{j}{\hat {b}}_{i}]={\hat {a}}_{[i}{\hat {b}}_{j]}$

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

$T_{ij}^{(3)}={\frac {1}{2}}({\hat {a}}_{i}{\hat {b}}_{j}+{\hat {a}}_{j}{\hat {b}}_{i})-{\frac {{\hat {a}}_{k}{\hat {b}}_{k}}{3}}\delta _{ij}={\hat {a}}_{(i}{\hat {b}}_{j)}-T_{ij}^{(1)}$

includes five independent components. Throughout, δ_ij 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 2ℓ + 1 for ℓ = 0, 1, 2, the same as the ranks for each tensor. Each of the irreducible representations T⁽¹⁾, T⁽²⁾ ... transform like angular momentum eigenstates according to the number of independent components.

Consider the case where a is the position vector x and b the momentum vector p for a particle. The reduction of T into the three parts above will contain terms with orbital angular momentum and action.

Spherical harmonic functions and Spherical tensors

The spherical basis is closely related to the algebra of orbital angular momentum in quantum mechanics, and the eigenstates |l, m⟩ of the orbital angular momentum operator L are spherical harmonics:

$Y_{\ell }^{m}(\theta ,\varphi )=\langle \theta ,\phi |m,\ell \rangle ={\sqrt {{(2\ell +1) \over 4\pi }{(\ell -m)! \over (\ell +m)!}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\phi }$

where P_ℓ^m is an associated Legendre polynomial. The formalism of spherical harmonics which have wide applications in applied mathematics; solutions of Laplace's equation on a 3d sphere are also spherical harmonics.

A spherical harmonic state rotates according to:

${U(R)}^{\dagger }T_{k}^{q'}U(R)=\sum _{q}D_{qq'}^{(k)}T_{k}^{q}$

A spherical tensor T_k^q of rank k is defined to rotate according to:

${U(R)}^{\dagger }T_{k}^{q'}U(R)=\sum _{q}D_{qq'}^{(k)}T_{k}^{q}$

where q = ... k, k − 1, ..., −k + 1, −k. For spherical tensors, k and q are analogous labels for ℓ and m respectively.

from which the commutation relations can be derived:

$\left[J_{\pm },T_{k}^{q}\right]=\pm \hbar {\sqrt {(k\mp q)(k\pm q+1)}}T_{k}^{q\pm 1}$

$\left[J_{z},T_{k}^{q}\right]=\hbar qT_{k}^{q}$

A spherical harmonic state can be constructed out of the Clebsch–Gordan coefficients:

similarly for the spherical tensor:

Orbital angular momentum and spherical harmonics

Orbital angular momentum operators have the ladder operators:

$L_{\pm }=L_{x}\pm iL_{y}=\pm (\pm L_{x}+iL_{y})$

which raise or lower the orbital magnetic quantum number m by one unit, m takes values −ℓ, −ℓ + 1, ... ℓ − 1, ℓ. (Some authors place a factor of 1/2 on the left hand side of the equation). This has almost exactly the same form as the spherical basis, aside from constant multiplicative factors.

The angular momentum operators satisfy the commutation rules:

$\left[L_{x},L_{y}\right]=i\hbar L_{z}$

and cyclic permutations of x, y, z components.

Spherical tensor operators and quantum spin

See also: Spin-weighted spherical harmonics and spin spherical harmonics

Spherical tensors are formed from algebraic combinations of the spin matrices S_x, S_y, S_z, for a spin system with total quantum number j = ℓ + s (and ℓ = 0). Tensor operators can be constructed from two perspectives.

One way is to specify how spherical tensors transform under a physical rotation - a group theoretical definition. A rotated spin 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 spin eigenkets under a rotation.

Another related procedure requires that the spherical tensors satisfy certain commutation relations with respect to the basic operators S_x, S_y, S_z - a differential or algebraic definition, and arises from the behavior of tensor operators under infinitesimally small rotations.

Since S_x, S_y, S_z are usually introduced algebraically through the angular momentum commutation rules

$\left[S_{x},S_{y}\right]=i\hbar S_{z}$

and cyclic permutations of x, y, z components. The commutation approach is a popular way of introducing spherical tensors. Additional physical and practical motivations for utilizing spherical tensors include:

• aid in the calculation of matrix elements when rotational symmetry is present.

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 NMR and EPR, spherical tensor operators are employed to express the quantum dynamics of 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.

See also

• Wigner–Eckart theorem
• Tensor operator

References

Notes

• P.T. Callaghan (2011). Translational Dynamics and Magnetic Resonance:Principles of Pulsed Gradient Spin Echo NMR. Oxford University Press. ISBN 0-191-621-048.

• V.V. Balashov, A.N. Grum-Grzhimailo, N.M. Kabachnik (2000). Polarization and Correlation Phenomena in Atomic Collisions: A Practical Theory Course. Springer. {{cite book}}: line feed character in |title= at position 61 (help)CS1 maint: multiple names: authors list (link)

• J. A. Tuszynski (1990). Spherical Tensor Operators: Tables of Matrix Elements and Symmetries. World Scientific. ISBN 981-0202-830.

• L. Castellani, J. Wess (1996). Quantum Groups and Their Applications in Physics: Varenna on Lake Como, Villa Monastero, 28 June-8 July 1994. Società Italiana di Fisica, IOS. ISBN 905-199-24-75. {{cite book}}: line feed character in |title= at position 50 (help)

• Introduction to the Graphical Theory of Angular Momentum. Springer. 2009. ISBN 364-203-11-96.

• A. R. Edmonds (1996). Angular Momentum in Quantum Mechanics (2nd ed.). Princeton University Press. ISBN 0-691-025-894.

• L.J. Mueller (2011). "Tensors and rotations in NMR". Wiley Periodicals. doi:10.1002/cmr.a.20224.

• M.S. Anwar (2004). "Spherical Tensor Operators in NMR" (PDF).

• P. Callaghan (1993). Principles of nuclear magnetic resonance microscopy. Oxford University Press. p. 56-57. ISBN 0-198-539-975.

Further reading

Spherical harmonics

• G.W.F. Drake (2006). Springer Handbook of Atomic, Molecular, and Optical Physics (2nd ed.). Springer. p. 57. ISBN 0-3872-6308-X.

• F.A. Dahlen, J. Tromp (1998). Theoretical global seismology (2nd ed.). Princeton University Press. p. appendix C. ISBN 0-69100-1243.

• D.O. Thompson, D.E. Chimenti (1997). Review of Progress in Quantitative Nondestructive Evaluation. Review Of Progress In Quantitative Nondestructive Evaluation. Vol. 16. Springer. p. 1708. ISBN 0-3064-55978.

• H. Paetz, G. Schieck (2011). Nuclear Physics with Polarized Particles. Lecture Notes in Physics. Vol. 842. Springer. p. 31. ISBN 364-224-225-1.

• V. Devanathan (1999). Angular Momentum Techniques in Quantum Mechanics. Fundamental Theories of Physics. Vol. 108. Springer. p. 34, 61. ISBN 0-7923-5866-X.

• V.D. Kleiman, R.N. Zare (1998). "5". A Companion to Angular Momentum. John Wiley & Sons. p. 112. ISBN 0-4711-9249-X.

Angular momentum and spin

• "Vectors and Tensors in Spherical Basis". Angular Momentum Techniques in Quantum Mechanics. Fundamental Theories of Physics. Vol. 108. 2002. p. 24-33. doi:10.1007/0-306-47123-X_3. ISBN 978-0-306-47123-0 (online). {{cite book}}: Check |isbn= value: invalid character (help)

• K.T. Hecht (2000). Quantum mechanics. Graduate texts in contemporary physics. Springer. ISBN 0-387-989-196.

Condensed matter physics

• J.A. Mettes, J.B. Keith, R.B. McClurg (2002). "Molecular Crystal Global Phase Diagrams:I Method of Construction" (PDF).{{cite news}}: CS1 maint: multiple names: authors list (link)

• B.Henderson, R.H. Bartram (2005). Crystal-Field Engineering of Solid-State Laser Materials. Cambridge Studies in Modern Optics. Vol. 25. Cambridge University Press. p. 49. ISBN 0-52101-8013.

• E. U. Condon, Halis Odabasi (1980). Atomic Structure. CUP Archive. ISBN 0-5212-98938.

• K.D. Bonin, V.V. Kresin (2008). "2". In Melinda J. Duer (ed.). Solid State NMR Spectroscopy: Principles and Applications. John Wiley & Sons. p. 113. ISBN 0-4709-9938-1.

• K.D. Bonin, V.V. Kresin (1997). "2". Electric - Dipole Polarizabilities of Atoms, Molecules and Clusters. World Scientific. p. 14-15. ISBN 981-022-493-1.

• A.E. McDermott, T.Polenova (2012). Solid State NMR Studies of Biopolymers. EMR handbooks. John Wiley & Sons. p. 42. ISBN 111-858-889-4.

Magnetic resonance

• L.J. Mueller (2011). "Tensors and rotations in NMR". Wiley Periodicals. doi:10.1002/cmr.a.20224.

• M.S. Anwar (2004). "Spherical Tensor Operators in NMR" (PDF).

• P. Callaghan (1993). Principles of nuclear magnetic resonance microscopy. Oxford University Press. p. 56-57. ISBN 0-198-539-975.

Image processing

• M. Reisert, H. Burkhardt (2009). S. Aja-Fernández (ed.). Tensors in Image Processing and Computer Vision. Springer. ISBN 184-8822-995.

• D.H. Laidlaw, J. Weickert (2009). Visualization and Processing of Tensor Fields: Advances and Perspectives. Mathematics and Visualization. Springer. ISBN 354-088-378-9.

External links

• (2012) Clebsch-Gordon coefficients and the tensor spherical harmonics
• The tensor spherical harmonics
• (2010) Irreducible Tensor Operators and the Wigner-Eckart Theorem
• Tensor operators
• M. Fowler (2008), Tensor Operators
• Tensor_Operators
• (2009) Tensor Operators and the Wigner Eckart Theorem
• The Wigner-Eckart theorem
• (2004) Rotational Transformations and Spherical Tensor Operators
• Tensor operators
• Evaluation of the matrix elements for radiative transitions
• C. Mueller (2009) Tensor Operators and the Wigner-Eckart Theorem
• D.K. Ghosh, (2013) Angular Momentum - III : Wigner- Eckart Theorem
• B. Baragiola (2002) Tensor Operators
• Spherical Tensors