4.6 Kugelflächenfunktionen (Spherical harmonics)

Diese bilden ein vollständiges Orthogonalsystem in zwei Dimensionen. Kugelflächenfunktionen bilden die Eigenfunktionen der Drehimpulsoperatoren $\vec{L}^{2}_{}$ und L_z. ^4.5

$\displaystyle \int$ d $\displaystyle \Omega$ = $\displaystyle \int_{0}^{2\pi}$ d $\displaystyle \varphi$ $\displaystyle \int_{0}^{\pi}$ sin( $\displaystyle \theta$ )d $\displaystyle \theta$ = $\displaystyle \int_{0}^{2\pi}$ d $\displaystyle \varphi$ $\displaystyle \int_{-1}^{+1}$ d (cos( $\displaystyle \theta$ )

(4.35)

$\displaystyle \int$ d $\displaystyle \Omega$ $\displaystyle \overline{Y_l^m(\theta, \varphi)}$ Y_l'^m'( $\displaystyle \theta{^\prime}$ , $\displaystyle \varphi{^\prime}$ ) = $\displaystyle \delta_{l,l'}^{}$ $\displaystyle \delta_{m,m'}^{}$

(4.36)

$\displaystyle \sum_{l=0}^{\infty}$ $\displaystyle \sum_{m=-l}^{l}$ Y_l^m( $\displaystyle \theta$ , $\displaystyle \varphi$ ) $\displaystyle \overline{Y_l^m(\theta', \varphi')}$ = $\displaystyle \delta$ ( $\displaystyle \varphi$ - $\displaystyle \varphi{^\prime}$ ) $\displaystyle \delta$ (cos( $\displaystyle \theta$ ) - cos( $\displaystyle \theta{^\prime}$ ))

(4.37)

Y_l^m( $\displaystyle \theta$ , $\displaystyle \varphi$ ) = $\displaystyle \sqrt{{{(2l+1)(l-\vert m\vert)!} \over {4\pi(l+\vert m\vert)!}}}$ e^imP_l^mcos( $\displaystyle \theta$ )

(4.38)

$\displaystyle {P_l^m(cos(\theta)) = \varepsilon_m sin^{\vert m\vert}(\theta) {\left(d \over {d(\cos(\theta)}\right)}^m P_l(\cos(\theta))}$
		mit $\displaystyle \varepsilon_{m}^{}$ = $\displaystyle \left\{\vphantom{ \begin{array}{ll} (-1)^m & x\leq 0 \\ 1 & x<0 \\ \end{array}}\right.$ $\displaystyle \begin{array}{ll} (-1)^m & x\leq 0 \\ 1 & x<0 \\ \end{array}$ $\displaystyle \left.\vphantom{ \begin{array}{ll} (-1)^m & x\leq 0 \\ 1 & x<0 \\ \end{array}}\right.$	(4.39)

Y_l^-m( $\displaystyle \theta$ , $\displaystyle \varphi$ ) = (- 1)^m $\displaystyle \overline{Y_l^m(\theta, \varphi)}$

(4.40)

Y_l⁰( $\displaystyle \theta$ , $\displaystyle \varphi$ ) = $\displaystyle \sqrt{{{2l+1} \over {4\pi}}}$ P_l(cos( $\displaystyle \theta$ ))

(4.41)

Y_l^m(0, $\displaystyle \varphi$ )=
$\displaystyle {\delta_{m,0}\sqrt{{{2l+1}\over {4\pi}}} \left({{\partial^2} \ove... ...artial^2}\over {\partial \varphi^2}} + l(l+1)\right)\;\;Y_l^m(\theta, \varphi)}$
	=	0

cos( $\displaystyle \theta$ )Y_l^m = $\displaystyle \sqrt{{{(l+m)(l-m)} \over {(2l-1)(2l+1)}}}$ Y_{l - 1}^m + $\displaystyle \sqrt{{{(l+m+1)(l-m+1)} \over {(2l-1)(2l+3)}}}$ Y_{l + 1}^m

(4.42)

-sin( $\displaystyle \theta$ )eⁱY_l^m=
$\displaystyle {\mp \sqrt{{{(l\mp m)(l\mp m-1)} \over {(2l-1)(2l+1)}}} Y_{l-1}^{m\pm 1} \pm \sqrt{{{(l\pm m+2)(l\pm m+1)} \over {(2l+1)(2l+3)}}} Y_{l+1}^{m\pm 1}}$