1) Show that it is properly normalized and show that is is an Eigenfunction of the angular momentum operator
1) Show that it is properly normalized and show that is is an Eigenfunction of the angular momentum operator
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![The equation given is for the spherical harmonic with quantum numbers \(\ell = 3\) and \(m\ell = \pm 3\):
\[
\sqrt{\frac{35}{64\pi}} \sin^3 \theta \, e^{\pm 3i\phi}
\]
**Task:**
1) Show that this spherical harmonic is properly normalized and demonstrate that it is an eigenfunction of the angular momentum operator:
\[
\hat{L}^2 = -\hbar^2 \left[ \csc \theta \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial}{\partial \theta} \right) + \csc^2 \theta \frac{\partial^2}{\partial \phi^2} \right]
\]
Verify this with the expected eigenvalue.
**Details:**
- **Normalization:** Ensure that the spherical harmonic function satisfies the normalization condition typically required for functions representing quantum states over the unit sphere.
- **Angular Momentum Operator:** The operator \(\hat{L}^2\) acts on the spherical harmonic. As an eigenfunction, the spherical harmonic should return itself multiplied by a constant (the eigenvalue) after this operation.
- **Eigenvalue:** The expected eigenvalue for \(\hat{L}^2\) acting on this spherical harmonic is \(\ell(\ell+1)\hbar^2\), where \(\ell = 3\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F831851de-f673-41f5-9f36-80ab878b6fd6%2F1a091653-b4ec-4d10-be55-57424b19ffa5%2Fbac7qdp_processed.png&w=3840&q=75)
Transcribed Image Text:The equation given is for the spherical harmonic with quantum numbers \(\ell = 3\) and \(m\ell = \pm 3\):
\[
\sqrt{\frac{35}{64\pi}} \sin^3 \theta \, e^{\pm 3i\phi}
\]
**Task:**
1) Show that this spherical harmonic is properly normalized and demonstrate that it is an eigenfunction of the angular momentum operator:
\[
\hat{L}^2 = -\hbar^2 \left[ \csc \theta \frac{\partial}{\partial \theta} \left( \sin \theta \frac{\partial}{\partial \theta} \right) + \csc^2 \theta \frac{\partial^2}{\partial \phi^2} \right]
\]
Verify this with the expected eigenvalue.
**Details:**
- **Normalization:** Ensure that the spherical harmonic function satisfies the normalization condition typically required for functions representing quantum states over the unit sphere.
- **Angular Momentum Operator:** The operator \(\hat{L}^2\) acts on the spherical harmonic. As an eigenfunction, the spherical harmonic should return itself multiplied by a constant (the eigenvalue) after this operation.
- **Eigenvalue:** The expected eigenvalue for \(\hat{L}^2\) acting on this spherical harmonic is \(\ell(\ell+1)\hbar^2\), where \(\ell = 3\).
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