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{{short description|Basis used to express spherical tensors}}
{{redirect|Spherical tensor|the concept related to operators|tensor operator}}
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In [[pure mathematics|pure]] and [[applied mathematics]], particularly [[quantum mechanics]] and [[computer graphics]] and their applications, a '''spherical basis''' is the [[basis (linear algebra)|basis]] used to express '''[[Tensor operator|spherical tensors]]'''.{{definition needed|date=April 2020}}  The spherical basis closely relates to the description of [[angular momentum]] in quantum mechanics and [[spherical harmonic]] functions.

While [[spherical polar coordinates]] are one [[orthogonal coordinates|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 use [[complex number]]s.

==In three dimensions==

A vector '''A''' in 3D [[Euclidean space]] {{math|'''R'''<sup>3</sup>}} can be expressed in the familiar [[Cartesian coordinate system]] in the [[standard basis]] '''e'''<sub>''x''</sub>, '''e'''<sub>''y''</sub>, '''e'''<sub>''z''</sub>, and [[Coordinate vector|coordinates]] ''A<sub>x</sub>'', ''A<sub>y</sub>'', ''A<sub>z</sub>'':

{{NumBlk|:|<math>\mathbf{A} = A_x \mathbf{e}_x + A_y \mathbf{e}_y + A_z \mathbf{e}_z </math>|{{EquationRef|1}}}}

or any other [[coordinate system]] with associated [[basis (linear algebra)|basis]] set of vectors. From this extend the scalars to allow multiplication by complex numbers, so that we are now working in <math>\mathbb{C}^3</math> rather than <math>\mathbb{R}^3</math>.

===Basis definition===

In the spherical bases denoted '''e'''<sub>+</sub>, '''e'''<sub>−</sub>, '''e'''<sub>0</sub>, and associated coordinates with respect to this basis, denoted ''A''<sub>+</sub>, ''A''<sub>−</sub>, ''A''<sub>0</sub>, the vector '''A''' is:

{{NumBlk|:|<math>\mathbf{A} = A_+ \mathbf{e}_{+} + A_{-} \mathbf{e}_{-} + A_0 \mathbf{e}_0 </math>|{{EquationRef|2}}}}

where the spherical basis vectors can be defined in terms of the Cartesian basis using [[complex number|complex]]-valued coefficients in the ''xy'' plane:<ref>{{cite book|title=Angular Momentum|author=W.J. Thompson|year=2008|publisher=John Wiley & Sons|page=311|url=https://books.google.com/books?id=0NMjkQnQN6oC&q=spherical+basis+tensor&pg=PA311|isbn=9783527617838}}</ref>

{{NumBlk|:|<math>\begin{align}
\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{align} \quad \rightleftharpoons \quad \mathbf{e}_\pm = \mp\frac{1}{\sqrt{2}}\left(\mathbf{e}_x \pm i\mathbf{e}_y\right)\,</math>|{{EquationRef|3A}}}}

in which <math>i</math> denotes the [[imaginary unit]], and one normal to the plane in the ''z'' direction:

:<math>\mathbf{e}_0 = \mathbf{e}_z </math>

The inverse relations are:

{{NumBlk|:|<math>\begin{align}\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_{-}} \\
\mathbf{e}_z & = \mathbf{e}_0
\end{align}</math>|{{EquationRef|3B}}}}

===Commutator definition===
While giving a basis in a 3-dimensional space is a valid definition for a spherical tensor, it only covers the case for when the rank <math>k</math> is 1. For higher ranks, one may use either the commutator, or rotation definition of a spherical tensor. The commutator definition is given below, any operator <math>T_q^{(k)}</math> that satisfies the following relations is a spherical tensor:
<math display="block">[J_\pm, T_q^{(k)}] = \hbar \sqrt{(k\mp q)(k\pm q+1)}T_{q\pm 1}^{(k)}</math>
<math display="block">[J_z, T_q^{(k)}] = \hbar q T_q^{(k)}</math>

===Rotation definition===

Analogously to how the [[spherical harmonics]] transform under a rotation, a general spherical tensor  transforms as follows,  when the states transform under the [[unitary operator|unitary]] [[Wigner D-matrix]] <math>\mathcal{D}(R)</math>, where {{mvar|R}} is a (3×3 rotation) group element in [[SO(3)]]. That is, these matrices represent the rotation group elements. With the help of its [[Lie algebra]], one can show these two definitions are equivalent. 
:<math>\mathcal{D}(R)T_q^{(k)}\mathcal{D}^{\dagger}(R) = \sum_{q' = -k}^k T_{q'}^{(k)} \mathcal{D}_{q' q}^{(k)}</math>

{{see also| Wigner D-matrix}}

===Coordinate vectors===

{{main|Coordinate vector}}

For the spherical basis, the [[Coordinate vector|coordinates]] are complex-valued numbers ''A''<sub>+</sub>, ''A''<sub>0</sub>, ''A''<sub>−</sub>, and can be found by substitution of ({{EquationNote|3B}}) into ({{EquationNote|1}}), or directly calculated from the [[dot product#Complex vectors|inner product]] {{langle}}, {{rangle}} ({{EquationNote|5}}):

{{NumBlk|:|<math> 
\begin{align}
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{align} \quad \rightleftharpoons \quad A_\pm = \left\langle \mathbf{e}_\pm, \mathbf{A} \right\rangle = \frac{1}{\sqrt{2}} \left(\mp A_x + iA_y \right) </math>|{{EquationRef|4A}}}}

:<math> A_0 = \left\langle \mathbf{e}_0, \mathbf{A}  \right\rangle = \left\langle \mathbf{e}_z, \mathbf{A}  \right\rangle = A_z </math>

with inverse relations:

{{NumBlk|:|<math>\begin{align} 
A_x & = - \frac{1}{\sqrt{2}} A_+ + \frac{1}{\sqrt{2}} A_{-} \\
A_y & = - \frac{i}{\sqrt{2}} A_+ - \frac{i}{\sqrt{2}} A_{-} \\
A_z & = A_0
\end{align} </math>|{{EquationRef|4B}}}}

In general, for two vectors with complex coefficients in the same real-valued orthonormal basis '''e'''<sub>''i''</sub>, with the property '''e'''<sub>''i''</sub>·'''e'''<sub>''j''</sub> = ''δ<sub>ij</sub>'', the [[dot product#Complex vectors|inner product]] is:

{{NumBlk|:|<math>\left\langle \mathbf{a} , \mathbf{b} \right\rangle = \mathbf{a} \cdot \mathbf{b}^\star = \sum_j a_j b_j^\star </math>|{{EquationRef|5}}}}

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

==Properties (three dimensions)==

===Orthonormality===

The spherical basis is an [[orthonormal basis]], since the [[dot product#Complex vectors|inner product]] {{langle}}, {{rangle}} ({{EquationNote|5}}) of every pair vanishes meaning the basis vectors are all mutually [[orthogonality|orthogonal]]:

:<math>\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 </math>

and each basis vector is a [[unit vector]]:

:<math> \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 </math>

hence the need for the normalizing factors of <math>1/\!\sqrt{2}</math>.

===Change of basis matrix===

{{see also|change of basis}}

The defining relations ({{EquationNote|3A}}) can be summarized by a [[transformation matrix]] '''U''':

:<math>\begin{pmatrix}
\mathbf{e}_+ \\
\mathbf{e}_{-} \\
\mathbf{e}_0
\end{pmatrix} = \mathbf{U}\begin{pmatrix}
\mathbf{e}_x \\
\mathbf{e}_y \\
\mathbf{e}_z
\end{pmatrix} \,,\quad \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}\,,
</math>

with inverse:

:<math>\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} \,,\quad \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}\,.</math>

It can be seen that '''U''' is a [[unitary matrix]], in other words its [[Hermitian conjugate]] '''U'''<sup>†</sup> ([[complex conjugate]] and [[matrix transpose]]) is also the [[inverse matrix]]  '''U'''<sup>−1</sup>.

For the coordinates:

:<math>\begin{pmatrix}
A_+ \\
A_{-} \\
A_0
\end{pmatrix} =\mathbf{U}^\mathrm{*} \begin{pmatrix}
A_x \\
A_y \\
A_z
\end{pmatrix} \,,\quad \mathbf{U}^\mathrm{*} = \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}\,,
</math>

and inverse:

:<math>\begin{pmatrix}
A_x \\
A_y \\
A_z
\end{pmatrix} = (\mathbf{U}^\mathrm{*})^{-1} \begin{pmatrix}
A_+ \\
A_{-} \\
A_0
\end{pmatrix} \,,\quad (\mathbf{U}^\mathrm{*})^{-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}\,.
</math>

===Cross products===

Taking [[cross product]]s of the spherical basis vectors, we find an obvious relation:

:<math>\mathbf{e}_{q}\times\mathbf{e}_{q} = \boldsymbol{0}</math>

where ''q'' is a placeholder for +, −, 0, and two less obvious relations:

:<math>\mathbf{e}_{\pm}\times\mathbf{e}_{\mp} = \pm i \mathbf{e}_0</math>
:<math>\mathbf{e}_{\pm}\times\mathbf{e}_{0} = \pm i \mathbf{e}_{\pm}</math>

===Inner product in the spherical basis===

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

:<math>\left\langle \mathbf{A} , \mathbf{B} \right\rangle = A_+ B_+^\star + A_{-}B_{-}^\star + A_0 B_0^\star </math>

==See also==

*[[Wigner–Eckart theorem]]
*[[Wigner D matrix]]
*[[3D rotation group]]

==References==

{{reflist}}

=== General ===

*{{cite book|title=Introductory Nuclear Physics|author=S. S. M. Wong|publisher=John Wiley & Sons|isbn=978-35-276-179-13|edition=2nd|year=2008|url=https://books.google.com/books?id=mAr0uwfBLwIC&q=angular+momentum+operators+when+expressed+in+the+spherical+basis&pg=PA69}}

==External links==

{{Tensors}}

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