Answer
a) $m_{l}$ can be $-1$, $0$ or $1$;
b) $m_{l}$ can be $-3$, $-2$, $-1$, $0$, $1$, $2$ or $3$;
c) $m_{l}$ can be only $0$
Work Step by Step
Magnetic quantum number, $m_{l}$, has to be between $-l$ and $l$, where $l$ is angular quantum number.
a) For $p$ sublevels, $l$ equals $1$. Therefore, magnetic quantum number has to be between $-1$ and $1$, i. e. $-1$, $0$ or $1$.
b) For $f$ sublevels, $l$ equals $3$. Therefore, magnetic quantum number has to be between $-3$ and $3$, i. e. $-3$, $-2$, $-1$, $0$, $1$, $2$ or $3$.
c) If principal quantum number is equal to $3$, angular quantum number can be either $0$, $1$ or $2$. If $l=0$, the only option for magnetic quantum number is $m_{l}=0$, and therefore, this is the only possible value of magnetic quantum number for $all$ the sublevels of the third level.
