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GATE 2024 PH – Question 37

Atomic and Molecular Physics · spin-orbit interaction: L-S and j-j coupling schemes · 2 marks · Multiple choice

The spin–orbit Hamiltonian is $H^{\prime}=-k\mathbf L\cdot\mathbf S$. The splitting between $^2p_{3/2}$ and $^2p_{1/2}$ is

  1. $k\hbar^2/2$
  2. $3k\hbar^2/2$
  3. $3k\hbar^2/4$
  4. $2k\hbar^2$

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Correct answer: (B) $3k\hbar^2/2$

Explanation

**Spin-orbit coupling.** $H^\prime=-k\,\mathbf L\cdot\mathbf S$. With $\mathbf J=\mathbf L+\mathbf S$, we have $J^2=L^2+S^2+2\mathbf L\cdot\mathbf S$, so
$$\mathbf L\cdot\mathbf S=\frac{\hbar^2}{2}\big[j(j+1)-l(l+1)-s(s+1)\big].$$

For a $p$ electron, $l=1$ and $s=\tfrac12$:
- $j=\tfrac32$: $\mathbf L\cdot\mathbf S=\dfrac{\hbar^2}{2}\left(\dfrac{15}{4}-2-\dfrac34\right)=\dfrac{\hbar^2}{2}$
- $j=\tfrac12$: $\mathbf L\cdot\mathbf S=\dfrac{\hbar^2}{2}\left(\dfrac34-2-\dfrac34\right)=-\hbar^2$

**Splitting** between the two levels:
$$|\Delta E|=k\left|\frac{\hbar^2}{2}-(-\hbar^2)\right|=\frac{3k\hbar^2}{2}\quad(\text{option B}).$$