Editing Spherical harmonics (section)

==Algebraic properties==

=== Addition theorem ===
A mathematical result of considerable interest and use is called the ''addition theorem'' for spherical harmonics. Given two vectors {{math|'''r'''}} and {{math|'''r′'''}}, with spherical coordinates <math>(r,\theta,\varphi)</math> and <math>(r, \theta ', \varphi ')</math>, respectively, the angle <math>\gamma</math> between them is given by the relation
<math display="block">\cos\gamma = \cos\theta'\cos\theta + \sin\theta\sin\theta' \cos(\varphi-\varphi')</math>
in which the role of the trigonometric functions appearing on the right-hand side is played by the spherical harmonics and that of the left-hand side is played by the [[Legendre polynomial]]s.

The ''addition theorem'' states<ref>{{cite book|last1=Edmonds|first1=A. R.|title=Angular Momentum In Quantum Mechanics| year=1996 | url=https://archive.org/details/angularmomentumq00edmo|url-access=limited|publisher=Princeton University Press|page=[https://archive.org/details/angularmomentumq00edmo/page/n68 63]}}</ref>
{{NumBlk||<math display="block">
P_\ell( \mathbf{x}\cdot\mathbf{y} ) = \frac{4\pi}{2\ell+1}\sum_{m=-\ell}^\ell Y_{\ell}^m(\mathbf{y}) \, Y_{\ell}^m{}^*(\mathbf{x})
\quad \forall \, \ell \in \N_0 \;
\forall\, \mathbf{x}, \mathbf{y} \in \R^3 \colon \; \| \mathbf{x} \|_2 = \| \mathbf{y} \|_2 = 1 \,,
</math>|{{EquationRef|1}}}}
where {{math|''P''<sub>''ℓ''</sub>}} is the [[Legendre polynomial]] of degree {{mvar|ℓ}}. This expression is valid for both real and complex harmonics.<ref>This is valid for any orthonormal basis of spherical harmonics of degree {{mvar|ℓ}}. For unit power harmonics it is necessary to remove the factor of {{math|4''π''}}.</ref> The result can be proven analytically, using the properties of the [[Poisson kernel]] in the unit ball, or geometrically by applying a rotation to the vector '''y''' so that it points along the ''z''-axis, and then directly calculating the right-hand side.<ref>{{harvnb|Whittaker|Watson|1927|p=395}}</ref>

In particular, when {{math|1='''x''' = '''y'''}}, this gives Unsöld's theorem<ref>{{harvnb|Unsöld|1927}}</ref>
<math display="block">\sum_{m=-\ell}^\ell Y_{\ell}^m{}^*(\mathbf{x}) \, Y_{\ell}^m(\mathbf{x}) = \frac{2\ell + 1}{4\pi}</math>
which generalizes the identity {{math|1=cos<sup>2</sup>''θ'' + sin<sup>2</sup>''θ'' = 1}} to two dimensions.

In the expansion ({{EquationNote|1}}), the left-hand side <math>P_{\ell} (\mathbf{x} \cdot \mathbf{y})</math> is a constant multiple of the degree {{mvar|ℓ}} [[zonal spherical harmonic]]. From this perspective, one has the following generalization to higher dimensions. Let {{math|''Y''<sub>''j''</sub>}} be an arbitrary orthonormal basis of the space {{math|'''H'''<sub>''ℓ''</sub>}} of degree {{mvar|ℓ}} spherical harmonics on the {{mvar|n}}-sphere. Then <math>Z^{(\ell)}_{\mathbf{x}}</math>, the degree {{mvar|ℓ}} zonal harmonic corresponding to the unit vector {{mvar|x}}, decomposes as<ref>{{harvnb|Stein|Weiss|1971|loc=§IV.2}}</ref>
{{NumBlk||<math display="block">Z^{(\ell)}_{\mathbf{x}}({\mathbf{y}}) = \sum_{j=1}^{\dim(\mathbf{H}_\ell)}\overline{Y_j({\mathbf{x}})}\,Y_j({\mathbf{y}})</math>|{{EquationRef|2}}}}

Furthermore, the zonal harmonic <math>Z^{(\ell)}_{\mathbf{x}}({\mathbf{y}})</math> is given as a constant multiple of the appropriate [[Gegenbauer polynomial]]:
{{NumBlk||<math display="block">Z^{(\ell)}_{\mathbf{x}}({\mathbf{y}}) = C_\ell^{((n-2)/2)}({\mathbf{x}}\cdot {\mathbf{y}})</math>|{{EquationRef|3}}}}
Combining ({{EquationNote|2}}) and ({{EquationNote|3}}) gives ({{EquationNote|1}}) in dimension {{math|1=''n'' = 2}} when {{math|'''x'''}} and {{math|'''y'''}} are represented in spherical coordinates. Finally, evaluating at {{math|1='''x''' = '''y'''}} gives the functional identity
<math display="block">\frac{\dim \mathbf{H}_\ell}{\omega_{n-1}} = \sum_{j=1}^{\dim(\mathbf{H}_\ell)}|Y_j({\mathbf{x}})|^2</math>
where {{math|''ω''<sub>''n''−1</sub>}} is the volume of the (''n''−1)-sphere.

=== Contraction rule ===
Another useful identity expresses the product of two spherical harmonics as a sum over spherical harmonics<ref>{{cite book | last1=Brink|first1=D. M.|last2=Satchler|first2=G. R.|title=Angular Momentum|publisher=Oxford University Press|page=146}}</ref>
<math display="block"> Y_a^{\alpha}\left(\theta,\varphi\right)Y_b^{\beta}\left(\theta,\varphi\right) = \sqrt{\frac{\left(2a+1\right) \left(2b+1\right)}{4\pi}}\sum_{c=0}^{\infty}\sum_{\gamma=-c}^{c}\left(-1\right)^{\gamma}\sqrt{2c+1}\begin{pmatrix}
a & b & c\\
\alpha & \beta & -\gamma
\end{pmatrix} \begin{pmatrix}
a & b & c\\
0 & 0 & 0
\end{pmatrix} Y_c^{\gamma}\left(\theta,\varphi\right). </math>
Many of the terms in this sum are trivially zero. The values of <math> c </math> and <math>\gamma</math> that result in non-zero terms in this sum are determined by the selection rules for the [[3j-symbol]]s.

=== Clebsch–Gordan coefficients ===
{{Main|Clebsch–Gordan coefficients}}
The Clebsch–Gordan coefficients are the coefficients appearing in the expansion of the product of two spherical harmonics in terms of spherical harmonics themselves. A variety of techniques are available for doing essentially the same calculation, including the Wigner [[3-jm symbol]], the [[Racah coefficients]], and the [[Slater integrals]]. Abstractly, the Clebsch–Gordan coefficients express the [[tensor product]] of two [[irreducible representation]]s of the [[rotation group SO(3)|rotation group]] as a sum of irreducible representations: suitably normalized, the coefficients are then the multiplicities.