GATE 2023 PH – Question 49
A particle has $\psi(x,y,z)=Nz e^{-\alpha(x^2+y^2+z^2)}$ with positive alpha. Which are the eigenvalues of $L^2,L_z$? Given $Y_0^0=1/\sqrt{4\pi}$, $Y_1^0=\sqrt{3/(4\pi)}\cos\theta$ and $Y_1^{\pm1}=\mp\sqrt{3/(8\pi)}\sin\theta e^{\pm i\phi}$.
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Correct answer: (C) $2\hbar^2$ and 0
Explanation
The wave function is $\psi=N\,z\,e^{-\alpha(x^2+y^2+z^2)}=N\,r\cos\theta\,e^{-\alpha r^2}$.
**Angular part.** The exponential and the factor $r$ are radial, so the angular dependence is $\cos\theta$. From the given harmonics, $Y_1^0=\sqrt{\dfrac3{4\pi}}\cos\theta$, so the angular part is proportional to $Y_1^0$: $l=1$, $m=0$.
**Eigenvalues:**
- $L^2=l(l+1)\hbar^2=2\hbar^2$,
- $L_z=m\hbar=0$.
Answer **$2\hbar^2$ and $0$** (option C).