Question

What is the angle made by angular momentum vector of an electron with $Z$ -axis?

• A. $cos\theta =l/m$
• B. $cos\theta =\sqrt{\frac{l}{m}}$
• C. $cos\theta =\frac{\sqrt{\left(l \, + \, 1\right)} l}{m}$
• D. $cos\theta =\frac{m}{\sqrt{\left(l \, + \, 1\right)} l}$

Answer 1

Answer: (D)

$Z$ component of the angular momentum $L_{Z} \, =\frac{mh}{2 \pi }$
$L_{Z}=L \, cos\theta =\sqrt{l \, \left(l \, + 1\right)} \, cos\theta \frac{ \, h}{2 \pi }$
$\therefore \sqrt{l \, \left(l \, + 1\right)} \, cos\theta =m$
or $cos\theta =\frac{m}{\sqrt{l \left(l + 1\right)}}$