GATE 2026 CY – Question 65
An electron has orbital angular momentum magnitude $\sqrt{12}\hbar$. The minimum angle between L and its z-component in degrees (one decimal place) is _____
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Correct answer: 29.85 to 30.15
Explanation
The magnitude of the orbital angular momentum is $|L|=\sqrt{l(l+1)}\,\hbar$.
**Find $l$:**
$$\sqrt{l(l+1)}=\sqrt{12}\;\Rightarrow\;l(l+1)=12\;\Rightarrow\;l=3 .$$
**Angle with the $z$ axis.** The $z$-component is $L_z=m\hbar$, so
$$\cos\theta=\frac{L_z}{|L|}=\frac{m}{\sqrt{12}} .$$
The **smallest** angle corresponds to the largest $m$, which is $m=l=3$:
$$\cos\theta=\frac3{\sqrt{12}}=\frac{3}{3.4641}=0.8660\;\Rightarrow\;\theta=\mathbf{30.0^\circ}.$$