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making use of that 1² = Î? + ½ (εÎ_ + Î_Î4) prove that + (T|Z²|a) = -ħ² What are the eigenfunctions of [²? 1 8² sin²0 26² 1 + sint (sind)] (a) Ə de

making use of that 1² = Î? + ½ (εÎ_ + Î_Î4) prove that + (T|Z²|a) = -ħ² What are the eigenfunctions of [²? 1 8² sin²0 26² 1 + sint (sind)] (a) Ə de

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Here is the transcribed text from the image, suitable for an educational website: --- ### Making Use of That $\hat{L}^2 = \hat{L}_z^2 + \frac{1}{2} (\hat{L}_+ \hat{L}_- + \hat{L}_- \hat{L}_+)$, Prove That \[ \langle \vec{x} | \hat{L}^2 | \alpha \rangle = -\hbar^2 \left[ \frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial \phi^2} + \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial}{\partial \theta} \right) \right] \langle \vec{x} | \alpha \rangle \] ### What Are the Eigenfunctions of $\hat{L}^2$? --- This transcription includes mathematical notations formatted using LaTeX, a widely-used typesetting system for scientific and mathematical documents. The text presents a problem in quantum mechanics involving the angular momentum operator. Students are required to demonstrate a proof using given expressions and to identify the eigenfunctions of the angular momentum squared operator $\hat{L}^2$.
Transcribed Image Text:Here is the transcribed text from the image, suitable for an educational website: --- ### Making Use of That $\hat{L}^2 = \hat{L}_z^2 + \frac{1}{2} (\hat{L}_+ \hat{L}_- + \hat{L}_- \hat{L}_+)$, Prove That \[ \langle \vec{x} | \hat{L}^2 | \alpha \rangle = -\hbar^2 \left[ \frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial \phi^2} + \frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial}{\partial \theta} \right) \right] \langle \vec{x} | \alpha \rangle \] ### What Are the Eigenfunctions of $\hat{L}^2$? --- This transcription includes mathematical notations formatted using LaTeX, a widely-used typesetting system for scientific and mathematical documents. The text presents a problem in quantum mechanics involving the angular momentum operator. Students are required to demonstrate a proof using given expressions and to identify the eigenfunctions of the angular momentum squared operator $\hat{L}^2$.
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