Editing Spherical harmonics (section)

==Laplace's spherical harmonics==
[[File:Rotating spherical harmonics.gif|thumb|right|Real (Laplace) spherical harmonics <math>Y_{\ell m}</math> for <math>\ell=0,\dots,4</math> (top to bottom) and <math>m=0,\dots,\ell</math> (left to right). Zonal, sectoral, and tesseral harmonics are depicted along the left-most column, the main diagonal, and elsewhere, respectively. (The negative order harmonics <math>Y_{\ell(-m)}</math> would be shown rotated about the ''z'' axis by <math>90^\circ/m</math> with respect to the positive order ones.)]]

[[File:Sphericalfunctions.svg|thumb|500px|Alternative picture for the real spherical harmonics <math>Y_{\ell m}</math>.]]

[[Laplace's equation]] imposes that the [[Laplacian]] of a scalar field {{math|''f''}} is zero. (Here the scalar field is understood to be complex, i.e. to correspond to a (smooth) function <math>f:\R^3 \to \Complex</math>.) In [[Spherical coordinate system|spherical coordinates]] this is:<ref>The approach to spherical harmonics taken here is found in {{harv|Courant|Hilbert|1962|loc=§V.8, §VII.5}}.</ref>

<math display="block"> \nabla^2 f
 = \frac{1}{r^2} \frac{\partial}{\partial r}\left(r^2 \frac{\partial f}{\partial r}\right)
 + \frac{1}{r^2 \sin\theta} \frac{\partial}{\partial \theta}\left(\sin\theta \frac{\partial f}{\partial \theta}\right)
 + \frac{1}{r^2 \sin^2\theta} \frac{\partial^2 f}{\partial \varphi^2} = 0.</math>

Consider the problem of finding solutions of the form {{math|1=''f''(''r'', ''θ'', ''φ'') = ''R''(''r'') ''Y''(''θ'', ''φ'')}}. By [[Separation of variables#pde|separation of variables]], two differential equations result by imposing Laplace's equation:
<math display="block">\frac{1}{R}\frac{d}{dr}\left(r^2\frac{dR}{dr}\right) = \lambda,\qquad \frac{1}{Y}\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta \frac{\partial Y}{\partial\theta}\right) + \frac{1}{Y}\frac{1}{\sin^2\theta}\frac{\partial^2Y}{\partial\varphi^2} = -\lambda.</math>
The second equation can be simplified under the assumption that {{math|''Y''}} has the form {{math|1=''Y''(''θ'', ''φ'') = Θ(''θ'') Φ(''φ'')}}. Applying separation of variables again to the second equation gives way to the pair of differential equations

<math display="block">\frac{1}{\Phi} \frac{d^2 \Phi}{d\varphi^2} = -m^2</math>
<math display="block">\lambda\sin^2\theta + \frac{\sin\theta}{\Theta} \frac{d}{d\theta} \left(\sin\theta \frac{d\Theta}{d\theta}\right) = m^2</math>

for some number {{math|''m''}}. A priori, {{math|''m''}} is a complex constant, but because {{math|Φ}} must be a [[periodic function]] whose period evenly divides {{math|2''π''}}, {{math|''m''}} is necessarily an integer and {{math|Φ}} is a linear combination of the complex exponentials {{math|''e''<sup>± ''imφ''</sup>}}. The solution function {{math|''Y''(''θ'', ''φ'')}} is regular at the poles of the sphere, where {{math|1=''θ'' = 0, ''π''}}. Imposing this regularity in the solution {{math|Θ}} of the second equation at the boundary points of the domain is a [[Sturm–Liouville problem]] that forces the parameter {{math|''λ''}} to be of the form {{math|1=''λ'' = ''ℓ'' (''ℓ'' + 1)}} for some non-negative integer with {{math|''ℓ'' ≥ {{!}}''m''{{!}}}}; this is also explained [[#Orbital angular momentum|below]] in terms of the [[angular momentum operator|orbital angular momentum]]. Furthermore, a change of variables {{math|1=''t'' = cos ''θ''}} transforms this equation into the [[associated Legendre function|Legendre equation]], whose solution is a multiple of the [[associated Legendre polynomial]] {{math|''P{{su|b=ℓ|p=m}}''(cos ''θ'')}} . Finally, the equation for {{math|''R''}} has solutions of the form {{math|1=''R''(''r'') = ''A r{{i sup|ℓ}}'' + ''B r''{{i sup|−''ℓ'' − 1}}}}; requiring the solution to be regular throughout {{math|'''R'''<sup>3</sup>}} forces {{math|1=''B'' = 0}}.<ref>Physical applications often take the solution that vanishes at infinity, making {{math|1=''A'' = 0}}. This does not affect the angular portion of the spherical harmonics.</ref>

Here the solution was assumed to have the special form {{math|1=''Y''(''θ'', ''φ'') = Θ(''θ'') Φ(''φ'')}}. For a given value of {{math|''ℓ''}}, there are {{math|2''ℓ'' + 1}} independent solutions of this form, one for each integer {{math|''m''}} with {{math|−''ℓ'' ≤ ''m'' ≤ ''ℓ''}}. These angular solutions <math>Y_{\ell}^m : S^2 \to \Complex</math> are a product of [[trigonometric function]]s, here represented as a [[Euler's formula|complex exponential]], and associated Legendre polynomials:

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

which fulfill
<math display="block"> r^2\nabla^2 Y_\ell^m (\theta, \varphi ) = -\ell (\ell + 1 ) Y_\ell^m (\theta, \varphi ).</math>

Here <math>Y_{\ell}^m:S^2 \to \Complex</math> is called a '''spherical harmonic function of degree {{math|''ℓ''}} and order {{math|''m''}}''', <math>P_{\ell}^m:[-1,1]\to \R</math> is an [[associated Legendre polynomial]], {{math|''N''}} is a normalization constant,<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Spherical Harmonic |url=https://mathworld.wolfram.com/ |access-date=2023-05-10 |website=mathworld.wolfram.com |language=en}}</ref> and {{math|''θ''}} and {{math|''φ''}} represent colatitude and longitude, respectively. In particular, the [[colatitude]] {{math|''θ''}}, or polar angle, ranges from {{math|''0''}} at the North Pole, to {{math|''π''/2}} at the Equator, to {{math|''π''}} at the South Pole, and the [[longitude]] {{math|''φ''}}, or [[azimuth]], may assume all values with {{math|0 ≤ ''φ'' < 2''π''}}. For a fixed integer {{math|''ℓ''}}, every solution {{math|''Y''(''θ'', ''φ'')}}, <math>Y: S^2 \to \Complex</math>, of the eigenvalue problem
<math display="block"> r^2\nabla^2 Y = -\ell (\ell + 1 ) Y</math>
is a [[linear combination]] of <math>Y_\ell^m : S^2 \to \Complex</math>. In fact, for any such solution, {{math|''r<sup>ℓ</sup> Y''(''θ'', ''φ'')}} is the expression in spherical coordinates of a [[homogeneous polynomial]] <math>\R^3 \to \Complex</math> that is harmonic (see [[#Higher dimensions|below]]), and so counting dimensions shows that there are {{math|2''ℓ'' + 1}} linearly independent such polynomials.

The general solution <math>f:\R^3 \to \Complex</math> to [[Laplace's equation]] <math>\Delta f = 0</math> in a ball centered at the origin is a [[linear combination]] of the spherical harmonic functions multiplied by the appropriate scale factor {{math|''r<sup>ℓ</sup>''}},

<math display="block"> f(r, \theta, \varphi) = \sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m r^\ell Y_\ell^m (\theta, \varphi ), </math>

where the <math>f_{\ell}^m \in \Complex</math> are constants and the factors {{math|''r<sup>ℓ</sup> Y<sub>ℓ</sub><sup>m</sup>''}} are known as (''regular'') [[solid harmonics]] <math>\R^3 \to \Complex</math>. Such an expansion is valid in the [[Ball (mathematics)|ball]]

<math display="block"> r < R = \frac{1}{\limsup_{\ell\to\infty} |f_\ell^m|^{{1}/{\ell}}}.</math>

For <math> r > R</math>, the solid harmonics with negative powers of <math>r</math> (the ''irregular'' [[solid harmonics]] <math>\R^3 \setminus \{ \mathbf{0} \} \to \Complex</math>) are chosen instead. In that case, one needs to expand the solution of known regions in [[Laurent series]] (about <math>r=\infty</math>), instead of the [[Taylor series]] (about <math>r = 0</math>) used above, to match the terms and find series expansion coefficients <math>f^m_\ell \in \Complex</math>.

===Orbital angular momentum===
In quantum mechanics, Laplace's spherical harmonics are understood in terms of the [[angular momentum operator|orbital angular momentum]]<ref>{{harvnb|Edmonds|1957|loc=§2.5}}</ref>
<math display="block">\mathbf{L} = -i\hbar (\mathbf{x}\times \mathbf{\nabla}) = L_x\mathbf{i} + L_y\mathbf{j}+L_z\mathbf{k}.</math>
The {{math|''ħ''}} is conventional in quantum mechanics; it is convenient to work in units in which {{math|1=''ħ'' = 1}}. The spherical harmonics are eigenfunctions of the square of the orbital angular momentum
<math display="block">\begin{align}
\mathbf{L}^2 &= -r^2\nabla^2 + \left(r\frac{\partial}{\partial r}+1\right)r\frac{\partial}{\partial r}\\
&= -\frac{1}{\sin\theta} \frac{\partial}{\partial \theta}\sin\theta \frac{\partial}{\partial \theta} - \frac{1}{\sin^2\theta} \frac{\partial^2}{\partial \varphi^2}.
\end{align}</math>
Laplace's spherical harmonics are the joint eigenfunctions of the square of the orbital angular momentum and the generator of rotations about the azimuthal axis:
<math display="block">\begin{align}
L_z &= -i\left(x\frac{\partial}{\partial y} - y\frac{\partial}{\partial x}\right)\\
&=-i\frac{\partial}{\partial\varphi}.
\end{align}</math>

These operators commute, and are [[Densely defined operator|densely defined]] [[self-adjoint operator]]s on the [[Lp space#Weighted Lp spaces|weighted]] [[Hilbert space]] of functions ''f'' square-integrable with respect to the [[normal distribution]] as the weight function on '''R'''<sup>3</sup>:
<math display="block">\frac{1}{(2\pi)^{3/2}}\int_{\R^3} |f(x)|^2 e^{-|x|^2/2}\,dx < \infty.</math>
Furthermore, '''L'''<sup>2</sup> is a [[positive operator]].

If {{math|''Y''}} is a joint eigenfunction of {{math|'''L'''<sup>2</sup>}} and {{math|''L''<sub>''z''</sub>}}, then by definition
<math display="block">\begin{align}
\mathbf{L}^2Y &= \lambda Y\\
L_zY &= mY
\end{align}</math>
for some real numbers ''m'' and ''λ''. Here ''m'' must in fact be an integer, for ''Y'' must be periodic in the coordinate ''φ'' with period a number that evenly divides 2''π''. Furthermore, since
<math display="block">\mathbf{L}^2 = L_x^2 + L_y^2 + L_z^2</math>
and each of ''L''<sub>''x''</sub>, ''L''<sub>''y''</sub>, ''L''<sub>''z''</sub> are self-adjoint, it follows that {{math|''λ'' ≥ ''m''<sup>2</sup>}}.

Denote this joint eigenspace by {{math|''E''<sub>''λ'',''m''</sub>}}, and define the [[raising and lowering operators]] by
<math display="block">\begin{align}
L_+ &= L_x + iL_y\\
L_- &= L_x - iL_y
\end{align}</math>
Then {{math|''L''<sub>+</sub>}} and {{math|''L''<sub>−</sub>}} commute with {{math|'''L'''<sup>2</sup>}}, and the Lie algebra generated by {{math|''L''<sub>+</sub>}}, {{math|''L''<sub>−</sub>}}, {{math|''L''<sub>''z''</sub>}} is the [[special linear Lie algebra]] of order 2, <math>\mathfrak{sl}_2(\Complex)</math>, with commutation relations
<math display="block">[L_z,L_+] = L_+,\quad [L_z,L_-] = -L_-, \quad [L_+,L_-] = 2L_z.</math>
Thus {{math|''L''<sub>+</sub> : ''E''<sub>''λ'',''m''</sub> → ''E''<sub>''λ'',''m''+1</sub>}} (it is a "raising operator") and {{math|''L''<sub>−</sub> : ''E''<sub>''λ'',''m''</sub> → ''E''<sub>''λ'',''m''−1</sub>}} (it is a "lowering operator"). In particular, {{math|1=''L''{{su|b=+|p=''k''}} : ''E''<sub>''λ'',''m''</sub> → ''E''<sub>''λ'',''m''+''k''</sub>}} must be zero for ''k'' sufficiently large, because the inequality {{mvar|''λ'' ≥ ''m''<sup>2</sup>}} must hold in each of the nontrivial joint eigenspaces. Let {{mvar|''Y'' ∈ ''E''<sub>''λ'',''m''</sub>}} be a nonzero joint eigenfunction, and let {{mvar|k}} be the least integer such that
<math display="block">L_+^kY = 0.</math>
Then, since
<math display="block">L_-L_+ = \mathbf{L}^2 - L_z^2 - L_z</math>
it follows that
<math display="block">0 = L_-L_+^k Y = (\lambda - (m+k)^2-(m+k))Y.</math>
Thus {{math|1=''λ'' = ''ℓ''(''ℓ'' + 1)}} for the positive integer {{math|1=''ℓ'' = ''m'' + ''k''}}.

The foregoing has been all worked out in the spherical coordinate representation, <math>\langle \theta, \varphi| l m\rangle = Y_l^m (\theta, \varphi)</math> but may be expressed more abstractly in the complete, orthonormal [[Angular momentum operator#Orbital angular momentum in spherical coordinates|spherical ket basis]].