Editing Spherical harmonics (section)

==Conventions==

===Orthogonality and normalization===
{{Disputed section|Condon-Shortley phase|date = December 2017}}

Several different normalizations are in common use for the Laplace spherical harmonic functions <math>S^2 \to \Complex</math>. Throughout the section, we use the standard convention that for <math>m>0</math> (see [[associated Legendre polynomials]])
<math display="block">P_\ell ^{-m} = (-1)^m \frac{(\ell-m)!}{(\ell+m)!} P_\ell ^{m}</math>
which is the natural normalization given by Rodrigues' formula.
[[File:Plot of the spherical harmonic Y l^m(theta,phi) with n=2 and m=1 and phi=pi in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D.svg|alt=Plot of the spherical harmonic Y l^m(theta,phi) with n=2 and m=1 and phi=pi in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D|thumb|Plot of the spherical harmonic <math>Y_\ell^m(\theta,\varphi)</math> with <math>\ell=2</math> and <math>m=1</math> and <math>\varphi=\pi</math> in the complex plane from <math>-2-2i</math> to <math>2+2i</math> with colors created with Mathematica 13.1 function ComplexPlot3D]]
In [[acoustics]],<ref>{{Cite book|title=Fourier acoustics : sound radiation and nearfield acoustical holography |last=Williams |first=Earl G. |date=1999|publisher=Academic Press|isbn=0080506909|location=San Diego, Calif.|oclc=181010993}}</ref> the Laplace spherical harmonics are generally defined as (this is the convention used in this article)
<math display="block"> Y_\ell^m( \theta , \varphi ) = \sqrt{\frac{(2\ell+1)}{4\pi} \frac{(\ell-m)!}{(\ell+m)!}} \, P_\ell^m ( \cos{\theta} ) \, e^{i m \varphi } </math>
while in [[quantum mechanics]]:<ref>{{cite book |last=Messiah |first=Albert |title=Quantum mechanics : two volumes bound as one |publisher=Dover |year=1999 |isbn=0486409244 |edition=Two vol. bound as one, unabridged reprint |location=Mineola, NY |pages=520–523}}</ref><ref>{{cite book|author1=Claude Cohen-Tannoudji | author2=Bernard Diu | author3=Franck Laloë |translator=Susan Reid Hemley |display-translators=etal | title=Quantum mechanics | year=1996 | publisher=Wiley | location=Wiley-Interscience | isbn=9780471569527}}</ref>
<math display="block"> Y_\ell^m( \theta , \varphi ) = (-1)^m \sqrt{\frac{(2\ell+1)}{4\pi}\frac{(\ell-m)!}{(\ell+m)!}} \, P_{\ell}^m ( \cos{\theta} ) \, e^{i m \varphi } </math>

where <math>P_{\ell}^{m}</math> are associated Legendre polynomials without the Condon–Shortley phase (to avoid counting the phase twice).

In both definitions, the spherical harmonics are orthonormal
<math display="block">\int_{\theta=0}^\pi\int_{\varphi=0}^{2\pi}Y_\ell^m \, Y_{\ell'}^{m'}{}^* \, d\Omega=\delta_{\ell\ell'}\, \delta_{mm'},</math>
where {{math|''δ''<sub>''ij''</sub>}} is the [[Kronecker delta]] and {{math|1=''d''Ω = sin(''θ'') ''dφ'' ''dθ''}}. This normalization is used in quantum mechanics because it ensures that probability is normalized, i.e.,
<math display="block">\int{|Y_\ell^m|^2 d\Omega} = 1.</math>

The disciplines of [[geodesy]]<ref name="Blakely 1995 113">{{cite book | last = Blakely | first = Richard | title = Potential theory in gravity and magnetic applications | url = https://archive.org/details/potentialtheoryg00blak_461 | url-access = limited | publisher = Cambridge University Press | location = Cambridge England New York | year = 1995 | isbn = 978-0521415088 | page=[https://archive.org/details/potentialtheoryg00blak_461/page/n132 113] }}</ref> and spectral analysis use

<math display="block"> Y_\ell^m( \theta , \varphi ) = \sqrt{{(2\ell+1) }\frac{(\ell-m)!}{(\ell+m)!}} \, P_\ell^m ( \cos{\theta} )\, e^{i m \varphi } </math>

which possess unit power

<math display="block">\frac{1}{4 \pi} \int_{\theta=0}^\pi\int_{\varphi=0}^{2\pi}Y_\ell^m \, Y_{\ell'}^{m'}{}^* d\Omega=\delta_{\ell\ell'}\, \delta_{mm'}.</math>

The [[Magnetism|magnetics]]<ref name="Blakely 1995 113"/> community, in contrast, uses Schmidt semi-normalized harmonics

<math display="block"> Y_\ell^m( \theta , \varphi ) = \sqrt{\frac{(\ell-m)!}{(\ell+m)!}} \, P_\ell^m ( \cos{\theta} ) \, e^{i m \varphi }</math>

which have the normalization

<math display="block"> \int_{\theta=0}^\pi\int_{\varphi=0}^{2\pi}Y_\ell^m \, Y_{\ell'}^{m'}{}^*d\Omega = \frac{4 \pi}{(2 \ell + 1)} \delta_{\ell\ell'}\, \delta_{mm'}.</math>

In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after [[Giulio Racah]].

It can be shown that all of the above normalized spherical harmonic functions satisfy

<math display="block">Y_\ell^{m}{}^* (\theta, \varphi) = (-1)^{-m} Y_\ell^{-m} (\theta, \varphi),</math>

where the superscript {{math|*}} denotes complex conjugation. Alternatively, this equation follows from the relation of the spherical harmonic functions with the [[Wigner D-matrix#Relation to spherical harmonics and Legendre polynomials|Wigner D-matrix]].

===Condon–Shortley phase===
One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of <math>(-1)^m</math>, commonly referred to as the [[Edward Condon|Condon]]–Shortley phase in the quantum mechanical literature. In the quantum mechanics community, it is common practice to either include this [[phase factor]] in the definition of the [[associated Legendre polynomials]], or to append it to the definition of the spherical harmonic functions. There is no requirement to use the Condon–Shortley phase in the definition of the spherical harmonic functions, but including it can simplify some quantum mechanical operations, especially the application of [[Ladder operator|raising and lowering operators]]. The geodesy<ref>Heiskanen and Moritz, Physical Geodesy, 1967, eq. 1-62</ref> and magnetics communities never include the Condon–Shortley phase factor in their definitions of the spherical harmonic functions nor in the ones of the associated Legendre polynomials.<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Condon-Shortley Phase |url=https://mathworld.wolfram.com/Condon-ShortleyPhase.html |access-date=2022-11-02 |website=mathworld.wolfram.com |language=en}}</ref>

===Real form===
A real basis of spherical harmonics <math>Y_{\ell m}:S^2 \to \R</math> can be defined in terms of their complex analogues <math>Y_{\ell}^m: S^2 \to \Complex</math> by setting
<math display="block">\begin{align}
Y_{\ell m} &= \begin{cases}
\dfrac{i}{\sqrt{2}} \left(Y_\ell^{m} - (-1)^m\, Y_\ell^{-m}\right) & \text{if}\ m < 0\\
Y_\ell^0 & \text{if}\ m=0\\
\dfrac{1}{\sqrt{2}} \left(Y_\ell^{-m} + (-1)^m\, Y_\ell^{m}\right) & \text{if}\ m > 0.
\end{cases}\\
&= \begin{cases}
\dfrac{i}{\sqrt{2}} \left(Y_\ell^{-|m|} - (-1)^{m}\, Y_\ell^{|m|}\right) & \text{if}\ m < 0\\
Y_\ell^0 & \text{if}\ m=0\\
\dfrac{1}{\sqrt{2}} \left(Y_\ell^{-|m|} + (-1)^{m}\, Y_\ell^{|m|}\right) & \text{if}\ m>0.
\end{cases}\\
&= \begin{cases}
\sqrt{2} \, (-1)^m \, \Im [{Y_\ell^{|m|}}] & \text{if}\ m<0\\
Y_\ell^0 & \text{if}\ m=0\\
\sqrt{2} \, (-1)^m \, \Re [{Y_\ell^m}] & \text{if}\ m>0.
\end{cases}
\end{align}
</math>
The Condon–Shortley phase convention is used here for consistency. The corresponding inverse equations defining the complex spherical harmonics <math>Y_{\ell}^m : S^2 \to \Complex</math> in terms of the real spherical harmonics <math>Y_{\ell m}:S^2 \to \R</math> are
<math display="block">
Y_{\ell}^{m} = \begin{cases}
\dfrac{1}{\sqrt{2}} \left(Y_{\ell |m|} - i Y_{\ell,-|m|}\right) & \text{if}\ m<0 \\[4pt]
Y_{\ell 0} &\text{if}\ m=0 \\[4pt]
\dfrac{(-1)^m}{ \sqrt{2}} \left(Y_{\ell |m|} + i Y_{\ell,-|m|}\right) & \text{if}\ m>0.
\end{cases}
</math>

The real spherical harmonics <math>Y_{\ell m}:S^2 \to \R</math> are sometimes known as ''tesseral spherical harmonics''.<ref>{{harvnb|Whittaker|Watson|1927|p=392}}.</ref> These functions have the same orthonormality properties as the complex ones <math>Y_{\ell}^m : S^2 \to \Complex</math> above. The real spherical harmonics <math>Y_{\ell m}</math> with {{math|''m'' > 0}} are said to be of cosine type, and those with {{math|''m'' < 0}} of sine type. The reason for this can be seen by writing the functions in terms of the Legendre polynomials as
<math display="block">
Y_{\ell m} = \begin{cases}
 \left(-1\right)^m\sqrt{2} \sqrt{\dfrac{2\ell+1}{4\pi}\dfrac{(\ell-|m|)!}{(\ell+|m|)!}} \;
 P_\ell^{|m|}(\cos \theta) \ \sin( |m|\varphi )
 &\text{if } m<0
\\[4pt]
 \sqrt{\dfrac{ 2\ell+1}{4\pi}} \ P_\ell^m(\cos \theta)
 & \text{if } m=0
\\[4pt]
 \left(-1\right)^m\sqrt{2} \sqrt{\dfrac{2\ell+1}{4\pi}\dfrac{(\ell-m)!}{(\ell+m)!}} \;
 P_\ell^m(\cos \theta) \ \cos( m\varphi )
 & \text{if } m>0 \,.
\end{cases}
</math>

The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation.

See [[Table of spherical harmonics#Real spherical harmonics|here]] for a list of real spherical harmonics up to and including <math>\ell = 4</math>, which can be seen to be consistent with the output of the equations above.

==== Use in quantum chemistry ====
As is known from the analytic solutions for the hydrogen atom, the eigenfunctions of the angular part of the wave function are spherical harmonics. However, the solutions of the non-relativistic Schrödinger equation without magnetic terms can be made real. This is why the real forms are extensively used in basis functions for quantum chemistry, as the programs don't then need to use complex algebra. Here, the real functions span the same space as the complex ones would.

For example, as can be seen from the [[Table of spherical harmonics#Spherical harmonics|table of spherical harmonics]], the usual {{math|''p''}} functions (<math>\ell = 1</math>) are complex and mix axis directions, but the [[Table of spherical harmonics#Real spherical harmonics|real versions]] are essentially just {{math|''x''}}, {{math|''y''}}, and {{math|''z''}}.