GATE 2026 CY – Question 28
The correct statement(s) about spherical harmonics $Y_l^m$ is(are)
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Correct answer: (B) They are eigenfunctions of $\hat L^2$; (C) They are eigenfunctions of $\hat L_z$; (D) $Y_1^1$ and $Y_1^{-1}$ are degenerate
Explanation
The spherical harmonics $Y_l^m(\theta,\phi)$ are the angular parts of the solutions for a particle on a sphere.
- **A. All are complex functions.** False. For $m=0$, $Y_l^0$ is real (it contains no $e^{im\phi}$ factor).
- **B. Eigenfunctions of $\hat L^2$.** True: $\hat L^2Y_l^m=l(l+1)\hbar^2Y_l^m$. ✓
- **C. Eigenfunctions of $\hat L_z$.** True: $\hat L_zY_l^m=m\hbar Y_l^m$. ✓
- **D. $Y_1^1$ and $Y_1^{-1}$ are degenerate.** True. For a central (spherically symmetric) potential, the energy depends on $l$ but not on $m$, so states with the same $l$ and different $m$ have the same energy. ✓
Answer **B, C and D**.