Ethan's Notes

    Angular Momentum Operator

    #Physics

    $\displaystyle \hat{L}\equiv \hat{L}{x}^{2}+\hat{L}{y}^{2}+\hat{L}_{z}^{2}$

    • Definition of total angular momentum of a particle

    $\displaystyle \hat{L}{i}=\varepsilon{ijk}(r_{j}p_{k}-r_{k}p_{j})$

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

    $\displaystyle \hat{L}^{2}f_{l}^{m}=(\hat{L}{\pm }\hat{L}{\mp }+\hat{L}{z}^{2}\mp \hbar \hat{L}{z})f^{m}{l}=\hbar ^{2}l(l+1)f{l}^{m}$

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

    $\displaystyle \hat{L}{z}f^{m}{l}=\hbar mf^{m}_{l}$

    • Eigenvalues of $\displaystyle L_{z}$ operator

    $\displaystyle \hat{L}_{z}=-i\hbar \frac{\partial }{\partial \phi}$

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