Angular Momentum Operator

$\hat{L}\equiv {\hat{L}}_x^2+{\hat{L}}_y^2+{\hat{L}}_z^2$

• Definition of total angular momentum of a particle

${\hat{L}}_i={\epsilon }_{ijk}(r_j{p}_k-r_k{p}_j)$

• Happens because $\vec{L}=\vec{r}\times \vec{p}$

${\hat{L}}^2{f}_l^m=({\hat{L}}_{\pm }{\hat{L}}_{\mp }+{\hat{L}}_z^2\mp \hslash {\hat{L}}_z)f_l^m={\hslash }^{2}l(l+1)f_l^m$

• Eigenvalues of ${\hat{L}}^2$
• $l=\frac{k}{2}:k\in \mathbb{N}+\left\{0\right\}$
• $m\in \mathbb{N}:[-l,l]$

${\hat{L}}_z{f}_l^m=\hslash m{f}_l^m$

• Eigenvalues of $L_z$ operator

${\hat{L}}_z=-i\hslash \frac{\partial }{\partial \varphi }$

${\hat{L}}^2=-{\hslash }^2\left[\frac{1}{\sin \theta }\frac{\partial }{\partial \theta }\left(\sin \theta \frac{\partial }{\partial \theta }\right)+\frac{1}{{\sin }^2\theta }\frac{{\partial }^2}{\partial {\varphi }^2}\right]$

• The eigenfunction of $L^2$ is just the Spherical Harmonics