GATE 2024 PH – Question 48
For a one-electron atom, $H_0=p^2/(2m)-V(r)$, with Coulomb V. Add $H^{\prime}=\frac1{2m^2c^2r}\frac{dV}{dr}\mathbf L\cdot\mathbf S$. Which statement(s) is/are true?
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Correct answer: (A) $H^{\prime}$ commutes with $L^2$; (C) At fixed n,l there are $2(2l+1)$ degenerate eigenstates of $H_0$; (D) $H_0,L^2,S^2,L_z,S_z$ have simultaneous eigenstates
Explanation
For a one-electron atom, $H_0=\dfrac{p^2}{2m}+V(r)$ is spherically symmetric and spin-independent. The spin-orbit perturbation is $H^\prime\propto f(r)\,\mathbf L\cdot\mathbf S$.
- **A. $H^\prime$ commutes with $L^2$.** True. $\mathbf L\cdot\mathbf S=L_xS_x+L_yS_y+L_zS_z$ and each $L_i$ commutes with $L^2$. ✓
- **B. $H^\prime$ commutes with $L_z$ and $S_z$.** False. $\mathbf L\cdot\mathbf S$ commutes with $J_z=L_z+S_z$, but not with $L_z$ or $S_z$ separately, because of the $L_xS_x+L_yS_y$ terms.
- **C. At fixed $n$ and $l$ there are $2(2l+1)$ degenerate eigenstates of $H_0$.** True: $2l+1$ values of $m_l$ and 2 of $m_s$. ✓
- **D. $H_0,L^2,S^2,L_z,S_z$ have simultaneous eigenstates.** True: $H_0$ commutes with all the others, and all of them commute with one another. ✓
Answer **A, C and D**.