The spherical harmonic function Y, m(θ, φ) with appropriate values of and m, is an eigenfunction of Lx²+Ly²… · ABC26PH0319

Subject: Physical Chemistry · Chapter: Atomic Structure · Topic: Atomic Structure – General · Exam: GATE 2005-2021 · Marks: 2 · Difficulty: Medium

The spherical harmonic function $\mathrm{Y}_{\ell, \mathrm{m}}(\theta, \phi)$ with appropriate values of $\ell$ and m , is an eigenfunction of $\hat{\mathrm{L}}_{\mathrm{x}}^{2}+\hat{\mathrm{L}}_{\mathrm{y}}^{2}$ operator. The corresponding eigenvalue is
(a)$\left(\ell(\ell+1)-\mathrm{m}^{2}\right) \hbar^{2}$
(b)$\left(\ell(\ell+1)+\mathrm{m}^{2}\right) \hbar^{2}$
(c)$\ell(\ell+1) \hbar^{2}$
(d)$\mathrm{m}^{2} \hbar^{2}$
Answer
Answer: A ✓ checked by 4AB · confidence high
Explanation
Lₓ² + L_y² = L² − L_z², eigenvalue [l(l+1) − m²]ħ² (a).

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