e) Construction des harmoniques sphériques

$\displaystyle L_z\,Y^m_l(\theta,\varphi)$

$\displaystyle =$

$\displaystyle \frac{\hbar}{i}\,\frac{\partial}{\partial\varphi}\,Y^m_l(\theta,\varphi) ~=~ m\hbar\,Y^m_l(\theta,\varphi)$

$\displaystyle Y^m_l(\theta,\varphi)$

$\displaystyle =$

$\displaystyle f^m_l(\theta)\,e^{im\varphi}$

$\displaystyle \vec{L}^{\,2}\,Y^0_l(\theta,\varphi)$

$\displaystyle =$

$\displaystyle \vec{L}^{\,2}\,f_l(\theta) ~=~ \frac{-\hbar^2}{\sin\theta}\,\partial_\theta(\sin\theta\,\partial_\theta)\,f_l(\theta) ~=~ l(l+1)\hbar^2$

$\displaystyle u$	$\displaystyle =$	$\displaystyle \cos\theta$

$\displaystyle \partial_\theta$	$\displaystyle =$	$\displaystyle -\sin\theta\,\partial_u$

$\displaystyle \partial^2_{\theta^2}$	$\displaystyle =$	$\displaystyle -\cos\theta\,\partial_u + \sin^2\theta\,\partial^2_{u^2}$

$\displaystyle \left[ (1-u^2)\, \frac{d^2}{du^2} -2u\,\frac{d}{du} + l(l+1)\right]\,f_l(u)$

$\displaystyle =$

$\displaystyle f_l(u)$

$\displaystyle =$

$\displaystyle Y^0_l(\theta) ~=~ \sqrt{\frac{2l+1}{4\pi}}\,P_l(\cos\theta)$

$P_l(\cos\theta)$ désignant le $\mathrm{i}^\mathrm{\grave{e}me}$ polynôme de Legendre :

$\displaystyle P_0$

$\displaystyle =$

$\displaystyle 1~~~~~~~~~~~~~~~~~~~~~~~~P_1 ~=~ u~~~~~~~~~~~~~~~~~~~~~~~~P_2~=~ \frac{1}{2}\,(3u^2-1)$

$\displaystyle P_3$

$\displaystyle =$

$\displaystyle \frac{1}{2}\,(5u^3-3u)~~~~~~~~~~~~~~~~~~~~~~~~ P_4~=~\frac{1}{8}\,(35u^4-30u^2+3)$

La fonction harmonique sphérique la plus générale est solution de l'équation aux dérivées partielles suivante :

$\displaystyle \vec{L}^{\,2}\,Y^m_l(\theta,\varphi)$	$\displaystyle =$	$\displaystyle -\hbar^2\,\left[ \frac{1}{\sin\theta}\,\partial_\theta\,(\sin\the... ...+ \frac{1}{\sin^2\theta}\,\partial^2_{\varphi^2} \right]\,Y^m_l(\theta,\varphi)$

	$\displaystyle =$	$\displaystyle l(l+1)\,\hbar^2\,Y^m_l(\theta,\varphi)$