Question

(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?

Short answer

1. The magnitude of the orbital angular momentum in a state with$\mathcal{l}=3$ is $3.65\times {10}^{-34}J.s$.
2. The magnitude of its largest projection on an imposed z-axis is $3.16\times {10}^{-34}J.s$.

Step-by-step solution

The given data

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

Understanding the concept of orbital angular momentum

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:

• The magnitude of the orbital angular momentum,

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

• The magnitude of the z-component of the angular momentum,

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

a) Calculation of the magnitude of orbital angular momentum

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 $3.65\times {10}^{-34}J.s$.

b) Calculation of the magnitude of the largest projection of the component

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 $3.16\times {10}^{-34}J.s$.