GATE 2022 PH – Question 21

Quantum Mechanics · addition of angular momenta · 1 mark · Multiple select

Which pairs of orbital and total angular momentum operators commute?

Show answer and explanation

Correct answer: (A) $L^2,J^2$; (B) $J^2,J_z$; (D) $L_z,J_z$

Explanation

Let $\mathbf J=\mathbf L+\mathbf S$. Check each pair of operators.

**$L^2$ and $J^2$:** $J^2=L^2+S^2+2\mathbf L\cdot\mathbf S$, and $L^2$ commutes with every component of $\mathbf L$ and with $\mathbf S$, so $[L^2,J^2]=0$. ✓
**$J^2$ and $J_z$:** the general rule $[J^2,J_z]=0$. ✓
**$J^2$ and $L_z$:** $J^2$ contains $\mathbf L\cdot\mathbf S=L_xS_x+L_yS_y+L_zS_z$, and $L_z$ does not commute with $L_x$ and $L_y$. So $[J^2,L_z]\neq0$. ✗
**$L_z$ and $J_z$:** $J_z=L_z+S_z$, and $L_z$ commutes with itself and with $S_z$. ✓

Answer **A, B and D**.