角運動量とラプラシアン

${\bm L}=-i{\bm R}\times\nabla$

$$\begin{array}{ll} L^2&=-({\bm R}\times\nabla)\cdot({\bm R}\... ...=-R^2\bigtriangleup+R^2\partial_R^2+2R\partial_R\\ \end{array}$$ (2.18)

$$\bigtriangleup=\frac{\partial^2}{\partial R^2} +\frac 2R\frac{\partial}{\partial R}-\frac 1{R^2}L^2\\$$ (2.19)

(2.17)より極座標では

$$L^2= -\frac 1{\sin\theta}\frac\partial{\partial\theta} \s... ...heta} -\frac 1{\sin^2\theta}\frac{\partial^2}{\partial\phi^2}$$ (2.20)