Editing Spherical harmonics (section)

===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]].