GATE 2022 PH – Question 60

Correct answer: 3

Explanation

For a $p$ electron, $l=1$ and $s=\tfrac12$, and the spin-orbit interaction is $V_{so}\,\mathbf l\cdot\mathbf s$ with
$$\mathbf l\cdot\mathbf s=\frac{\hbar^2}{2}\left[j(j+1)-l(l+1)-s(s+1)\right].$$

• **$j=\tfrac32$ ($p_{3/2}$):** $\mathbf l\cdot\mathbf s=\dfrac{\hbar^2}2\left[\tfrac{15}4-2-\tfrac34\right]=\dfrac{\hbar^2}{2}$.
• **$j=\tfrac12$ ($p_{1/2}$):** $\mathbf l\cdot\mathbf s=\dfrac{\hbar^2}2\left[\tfrac34-2-\tfrac34\right]=-\hbar^2$.

**Separation of the doublet:**
$$\Delta E=V_{so}\left(\frac{\hbar^2}{2}+\hbar^2\right)=\frac{3}{2}V_{so}\hbar^2=3\times\left(V_{so}\frac{\hbar^2}{2}\right).$$

In units of $V_{so}\hbar^2/2$ the separation is **3**.