GATE 2023 PH – Question 33
In the vector model of angular momentum applied to atoms, what is the minimum angle in degrees (in integer) made by the orbital angular momentum vector and the positive z axis for a 2p electron?
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Correct answer: 45
Explanation
For a 2p electron the quantum numbers are $n=2$, $l=1$. In the vector model the orbital angular momentum has magnitude $\sqrt{l(l+1)}\,\hbar=\sqrt2\,\hbar$, and its $z$-component is $m_l\hbar$ with $m_l=-1,0,+1$.
The angle with the positive $z$ axis is given by $\cos\theta=\dfrac{m_l}{\sqrt{l(l+1)}}$. The **minimum** angle corresponds to the largest $m_l=+1$:
$$\cos\theta=\frac1{\sqrt2}\;\Rightarrow\;\theta=\mathbf{45^\circ}.$$