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# Questions tagged [spherical-harmonics]

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### What is the unit of the Angular Power Spectrum $C_\ell$ of Galaxy Clustering and Weak Lensing?

If the matter power spectrum $P_m(k,z)$ is measured in $h/Mpc^3$ and $k$ is measured in $h/Mpc$, what about its Fourier Transform $C_\ell$ and the conjugate vector to $k$, $\ell$? Thank you so much
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### What happened to the factor of $\pi$ in this question?

$\\$ I was going through the answer to this problem, when I noticed that a factor of $\pi$ in the denominator disappeared and a factor of 4 appeared in the numerator when the author started ...
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### Eigenfunctions in Spherically Symmetric Well

I am looking at a problem that has a potential $$V(r) = \begin{cases} 0 & a<r<b\\ \infty & \text{elsewhere} \end{cases}$$ This is a modification of the infinite spherical well ...
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### Neumann boundary condition in spherical coordinates

I'm trying to solve heat equation $$\nabla^2 u = \frac{1}{k}\frac{\partial u}{\partial t}$$ in the region $$a \leq r \leq b, \ \ \ \ 0 \leq \varphi \leq 2\pi, \ \ \ \ 0 \leq \theta \leq \theta_0$$ ...
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### What does it mean to normalize a combination of Spherical Harmonics?

Using the following as an example: Show that the combinations $$-\frac{1}{\sqrt{2}}\left(Y_{11}-Y_{1-1}\right)\quad\text{&}\quad\frac{i}{\sqrt{2}}\left(Y_{11}+Y_{1-1}\right)$$ are real and ...
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### Depolarization factors of a drude metal plasmonic spheroid

A practice question asks for the depolarization factors $L_{i=x,y,z}$ of a plasmonic spheroid made of a drude metal having the same resonance as the SPP resonance frequency. The answer turns out to ...
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### Quick way to compute $\langle n^{'}l^{'}m^{'}|r^k|nlm \rangle$, $k \in I$; $|nlm\rangle$ is $H$ atom eigenfunction [closed]

I want to compute quickly (using maybe some scaling arguments) $\langle n^{'}l^{'}m^{'}|r^k|nlm\rangle$, where $k \in I$. $|nlm \rangle$ is the eigenfunction of the Hydrogen atom ($H$). Example: ...
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### Do relativistically-contracted electron states have the same energy and angular momentum values?

I've been reading that electron bound states are defined by four quantum numbers, $n$, $l$, $m_l$, and $m_s$, respectively the principal quantum number, the azimuthal quantum number, the magnetic ...
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### Why are there three $p$-orbitals?

This question is specifically about Schrödinger quantum mechanics, but if an answer in some other mode would illuminate it could be acceptable, as demonstrating a physical or mathematical reason for ...
### Expectation Value $\langle \frac{1}{r^2} \rangle$ using Hellmann–Feynman theorem
Suppose we have the hydrogen atom$$H ~=~ \frac{p_r ^2}{2m} + \frac{L^2}{2mr^2} -\frac{e^2}{r} \,.$$And have solved the Schrodinger equation finding $$E_n = - \frac{me^4}{2 \hbar ^2 n^2}$$ and  Ψ_{...
### Eigenvalues of spherical harmonics in $d$ dimensions
I'm working on the Schrodinger equation for a hydrogen atom in a $d$-dimensional space, so I'm interested in the possible eigenvalues of the angular momentum part of the $d$-dimensional Laplace ...