Answer
$n\geq7$, $\mathcal{l}=6$, and $ m_{\mathcal{l}}=2$.
Work Step by Step
Use equation 28–3 to find $\mathcal{l}$.
$$L=\sqrt{\mathcal{l}(\mathcal{l}+1)}\hbar$$
$$\mathcal{l}(\mathcal{l}+1)=\frac{L^2}{\hbar^2}=\frac{(6.84\times10^{-34}J \cdot s)^2}{(1.055\times10^{-34}J \cdot s)^2}=42$$
Since $\mathcal{l}$ is a positive integer, we see that $\mathcal{l}=6$.
The possible values of $\mathcal{l}$ are from 0 to (n-1), so $n\geq7$.
Use the equation right after 28-3 to find $m_{\mathcal{l}}$.
$$L_z= m_{\mathcal{l}}\hbar$$
$$ m_{\mathcal{l}}=\frac{L_z}{\hbar}=\frac{2.11\times10^{-34}J \cdot s}{1.055\times10^{-34}J \cdot s}=2$$
In summary, $n\geq7$, $\mathcal{l}=6$, and $ m_{\mathcal{l}}=2$.
