(a) What is the magnitude of the orbital angular momentum in a state with $\mathcal{l}\mathbf{=}\mathbf{3}$?

(b) What is the magnitude of its largest projection on an imposed zaxis?

The quantum number$\mathcal{l}=3$ is given.

Using the concept of orbital angular momentum for a given value of l, we can get the required magnitude value of the momentum. Again using the formula of the z-component of the angular momentum for the maximum value of the magnetic quantum number, we can get the value of maximum magnitude value of its component.

Formulae:

$\overrightarrow{L}=\sqrt{\mathcal{l}(\mathcal{l}+1})\sout{\mathcal{h}}$ ……(i)

$\overrightarrow{L_z}=m_{\mathcal{l}}\sout{\mathcal{h}}where{m}_{\mathcal{l}}=-\mathcal{l}to+\mathcal{l}$ ………….(ii)

Using the given value$\mathcal{l}=3$ in equation (i), we can get the required magnitude value of the orbital angular momentum as follows:

role="math" localid="1661493371217" $\begin{array}{ll}\overrightarrow{L}=\sqrt{3\left(3+1\right)}\sout{h}& \\ =\sqrt{12}\left(1.055\times {10}^{-34}J.s\right)& \\ =3.65\times {10}^{-34}J.s& \end{array}$

Hence, the value of orbital momentum is $3.65\times {10}^{-34}J.s$.

Using the concept, we can say that the maximum value of the component of the orbital angular momentum is achieved for the value of $m_{\mathcal{l}}=\mathcal{l}$and thus, using $\mathcal{l}=3$in equation (ii), we get the maximum magnitude value of the component as follows:

$\begin{array}{lll}\overrightarrow{L_z}=\mathcal{l}\sout{h}& & \\ =3\left(1.055\times {10}^{-34}J.s\right)& & \\ =3.16\times {10}^{-34}J.s& & \end{array}$

Hence, the value of the largest projection along the z-axis of the orbital angular momentum is $3.16\times {10}^{-34}J.s$.