GATE 2022 CY – Question 35
For angular momentum operator $\hat L$ and spherical harmonics $Y_l^m(\theta,\phi)$, $(\hat L_x^2+\hat L_y^2)Y_2^1=n\hbar^2Y_2^1$. The value of n is ______.
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Correct answer: 5 to 5
Explanation
Use the angular-momentum identity $L_x^2+L_y^2=L^2-L_z^2$. A spherical harmonic $Y_l^m$ is a simultaneous eigenfunction of the latter two operators, with eigenvalues $l(l+1)\hbar^2$ and $m^2\hbar^2$. Here $l=2$ and $m=1$, so subtraction gives $[2(3)-1]\hbar^2=5\hbar^2$. Comparing with the requested form gives **n=5**.