Angular Momentum Questions

1. Find the eigen states of angular moments vector's component $L_z$ of a spherically symmetric system [Mains 2001]

2. Solve the eigen value equation $L^{2}Y(\theta,\Phi)=\lambda h^2Y(\theta,\Phi)$ And obtain the eigen value of $L^2$.

3. Using the definition $\vec{L}= \vec{r} \times \vec{p}$ of the orbital angular momentum operator, evaluate $[L_x,L_y]$ [Mains 2013]

4. Write the commutation relations of angular momentum operator $\hat{L_x}$, $\hat{L_y}$ and $\hat{L_z}$ and calculate the commutators $[\hat{L_+},\hat{L_z}]$ and $[\hat{L_-},\hat{L_z}]$. $$$$ Show that $\hat{L_+}\ket{l,m} = \sqrt{l(l+1) - m(m+1)}\ket{l,m+1}$, where $\ket{l,m}$ is the state with definite values for $L^2$ and $L_z$. [Mains 2008]

5. Using the commutation relations $$ [x,p_x] = [y, p_y] = [z,p_z] = i\hbar,$$ deduce the commutation relation between the components of angular momentum operator $L$. $$ \begin{matrix} [L_x,L_y] = i \hbar L_z \\ [L_y,L_z] = i \hbar L_x \\ [L_z,L_x] = i \hbar L_y \end{matrix} $$ [Mains 2014]

6. Express the Cartesian components of the angular momentum L in operator form. Show that $\left[L^2,{\mathrm{\ }L}_z\right]=0$. What is the significance of this commutation relation?