Editing Spherical harmonics (section)

== Spherical harmonics in Cartesian form ==

The complex spherical harmonics <math>Y_\ell^m</math> give rise to the [[solid harmonics]] by extending from <math>S^2</math> to all of <math>\R^3</math> as a [[homogeneous function]] of degree <math>\ell</math>, i.e. setting
<math display="block">R_\ell^m(v) := \|v\|^\ell Y_\ell^m\left(\frac{v}{\|v\|}\right)</math>
It turns out that <math>R_\ell^m</math> is basis of the space of harmonic and [[homogeneous polynomial]]s of degree <math>\ell</math>. More specifically, it is the (unique up to normalization) [[Gelfand-Tsetlin-basis]] of this representation of the rotational group <math>SO(3)</math> and an [[Solid harmonics#Complex form|explicit formula]] for <math>R_\ell^m</math> in cartesian coordinates can be derived from that fact.

===The Herglotz generating function===

If the quantum mechanical convention is adopted for the <math>Y_{\ell}^m: S^2 \to \Complex</math>, then
<math display="block">
e^{v{\mathbf a}\cdot{\mathbf r}}
= \sum_{\ell=0}^{\infty} \sum_{m = -\ell}^{\ell}
 \sqrt{\frac{4\pi}{2\ell +1}}
 \frac{r^{\ell} v^{\ell} {\lambda^m}}{\sqrt{(\ell +m)!(\ell-m)!}} Y_{\ell}^m (\mathbf{r}/r).
</math>
Here, <math>\mathbf r</math> is the vector with components <math>(x, y, z) \in \R^3</math>, <math>r = |\mathbf{r}|</math>, and
<math display="block">
{\mathbf a}
 = {\mathbf{\hat z}}
 - \frac{\lambda}{2}\left({\mathbf{\hat x}} + i {\mathbf{\hat y}}\right)
 + \frac{1}{2\lambda}\left({\mathbf{\hat x}} - i {\mathbf{\hat y}}\right).
</math>
<math>\mathbf a </math>     is a vector with complex coordinates:

<math>\mathbf a =[\frac{1}{2}(\frac{1}{\lambda}-\lambda),-\frac{i}{2 }(\frac{1}{\lambda} +\lambda),1       ] .</math>

The essential property of <math>\mathbf a</math> is that it is null:
<math display="block">\mathbf a \cdot \mathbf a = 0.</math>

It suffices to take <math>v</math> and <math>\lambda</math> as real parameters.
In naming this generating function after [[Gustav Herglotz|Herglotz]], we follow {{harvnb|Courant|Hilbert|1962|loc= §VII.7}}, who credit unpublished notes by him for its discovery.

Essentially all the properties of the spherical harmonics can be derived from this generating function.<ref>See, e.g., Appendix A of Garg, A., Classical Electrodynamics in a Nutshell (Princeton University Press, 2012).</ref> An immediate benefit of this definition is that if the vector <math>\mathbf r</math> is replaced by the quantum mechanical spin vector operator <math>\mathbf J</math>, such that <math>\mathcal{Y}_{\ell}^m({\mathbf J})</math> is the operator analogue of the [[Solid harmonics|solid harmonic]] <math>r^{\ell}Y_{\ell}^m (\mathbf{r}/r)</math>,<ref>{{Citation| last1=Li|first1=Feifei| last2=Braun|first2=Carol| last3=Garg|first3=Anupam|title=The Weyl-Wigner-Moyal Formalism for Spin|journal=Europhysics Letters|year=2013|volume=102|issue=6| page=60006|doi=10.1209/0295-5075/102/60006|arxiv=1210.4075|bibcode=2013EL....10260006L|s2cid=119610178}}</ref> one obtains a generating function for a standardized set of [[spherical tensor operator]]s, <math>\mathcal{Y}_{\ell}^m({\mathbf J})</math>:

<math display="block">
e^{v{\mathbf a}\cdot{\mathbf J}}
= \sum_{\ell=0}^{\infty} \sum_{m = -\ell}^{\ell}
 \sqrt{\frac{4\pi}{2\ell +1}}
 \frac{v^{\ell} {\lambda^m}}{\sqrt{(\ell +m)!(\ell-m)!}}
 {\mathcal Y}_{\ell}^m({\mathbf J}).
</math>

The parallelism of the two definitions ensures that the <math>\mathcal{Y}_{\ell}^m</math>'s transform under rotations (see below) in the same way as the <math>Y_{\ell}^m</math>'s, which in turn guarantees that they are spherical tensor operators, <math>T^{(k)}_q</math>, with <math>k = {\ell}</math> and <math>q = m</math>, obeying all the properties of such operators, such as the [[Clebsch-Gordan coefficients|Clebsch-Gordan]] composition theorem, and the [[Wigner-Eckart theorem]]. They are, moreover, a standardized set with a fixed scale or normalization.

{{see also|Spherical basis}}

===Separated Cartesian form===
The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of <math>z</math> and another of <math>x</math> and <math>y</math>, as follows (Condon–Shortley phase):
<math display="block">
r^\ell\,
\begin{pmatrix} Y_\ell^{m} \\ Y_\ell^{-m} \end{pmatrix}
=
\left[\frac{2\ell+1}{4\pi}\right]^{1/2} \bar{\Pi}^m_\ell(z)
\begin{pmatrix} \left(-1\right)^m (A_m + i B_m) \\ (A_m - i B_m) \end{pmatrix} ,
\qquad m > 0.
</math>
and for {{math|1=''m'' = 0}}:
<math display="block">r^\ell\,Y_\ell^{0} \equiv \sqrt{\frac{2\ell+1}{4\pi}} \bar{\Pi}^0_\ell .</math>
Here
<math display="block">A_m(x,y) = \sum_{p=0}^m \binom{m}{p} x^p y^{m-p} \cos \left((m-p) \frac{\pi}{2}\right),</math>
<math display="block">B_m(x,y) = \sum_{p=0}^m \binom{m}{p} x^p y^{m-p} \sin \left((m-p) \frac{\pi}{2}\right),</math>
and
<math display="block">
\bar{\Pi}^m_\ell(z)
= \left[\frac{(\ell-m)!}{(\ell+m)!}\right]^{1/2}
\sum_{k=0}^{\left \lfloor (\ell-m)/2\right \rfloor}
 (-1)^k 2^{-\ell} \binom{\ell}{k}\binom{2\ell-2k}{\ell} \frac{(\ell-2k)!}{(\ell-2k-m)!}
\; r^{2k}\; z^{\ell-2k-m}.
</math>
For <math>m = 0</math> this reduces to
<math display="block">
\bar{\Pi}^0_\ell(z)
= \sum_{k=0}^{\left \lfloor \ell/2\right \rfloor}
 (-1)^k 2^{-\ell} \binom{\ell}{k}\binom{2\ell-2k}{\ell} \; r^{2k}\; z^{\ell-2k}.
</math>

The factor <math>\bar{\Pi}_\ell^m(z)</math> is essentially the associated Legendre polynomial <math>P_\ell^m(\cos\theta)</math>, and the factors <math>(A_m \pm i B_m)</math> are essentially <math>e^{\pm i m\varphi}</math>.

====Examples====

Using the expressions for <math>\bar{\Pi}_\ell^m(z)</math>, <math>A_m(x,y)</math>, and <math>B_m(x,y)</math> listed explicitly above we obtain:
<math display="block">
 Y^1_3
= - \frac{1}{r^3} \left[\tfrac{7}{4\pi}\cdot \tfrac{3}{16} \right]^{1/2} \left(5z^2-r^2\right) \left(x+iy\right)
= - \left[\tfrac{7}{4\pi}\cdot \tfrac{3}{16}\right]^{1/2} \left(5\cos^2\theta-1\right) \left(\sin\theta e^{i\varphi}\right)
</math>

<math display="block">
Y^{-2}_4 = \frac{1}{r^4} \left[\tfrac{9}{4\pi}\cdot\tfrac{5}{32}\right]^{1/2} \left(7z^2-r^2\right) \left(x-iy\right)^2
= \left[\tfrac{9}{4\pi}\cdot\tfrac{5}{32}\right]^{1/2} \left(7 \cos^2\theta -1\right) \left(\sin^2\theta e^{-2 i \varphi}\right)
</math>
It may be verified that this agrees with the function listed [[Table of spherical harmonics#ℓ = 3|here]] and [[Table of spherical harmonics#ℓ = 4|here]].

====Real forms====

Using the equations above to form the real spherical harmonics, it is seen that for <math>m>0</math> only the <math>A_m</math> terms (cosines) are included, and for <math>m<0</math> only the <math>B_m</math> terms (sines) are included:

<math display="block">
r^\ell\,
\begin{pmatrix} Y_{\ell m} \\ Y_{\ell -m} \end{pmatrix}
=
\sqrt{\frac{2\ell+1}{2\pi}} \bar{\Pi}^m_\ell(z)
\begin{pmatrix} A_m \\ B_m \end{pmatrix} ,
\qquad m > 0.
</math>
and for ''m'' = 0:
<math display="block">
r^\ell\,Y_{\ell 0} \equiv \sqrt{\frac{2\ell+1}{4\pi}}
\bar{\Pi}^0_\ell .
</math>