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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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### How do you apply the antisymmetrization operator?

I have an expression like, $Y^{L M_L}_{l_1 l_2}(\Omega_1, \Omega_2) = \sum_{m_1 m_2} \langle l_1 m_1l_2m_2|L m_L\rangle Y_{l_1m_1}(\Omega_1) Y_{l_2m_2}(\Omega_2)$ , as the angular part of a two ...
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### Plasmon modes of cylinder metalic particle

Solving Laplace equation gives plasmon modes of spherical metalic particle radius $R$, plasma frequency $\omega_p$. Famous result is $l = 0, 1, 2, ...$ $$\omega_l = \sqrt{\frac{l}{2l+1}} \omega_p$$ ...
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### If there are eigenstates of $L_z$ in a degenerate subspace, are there also eigenstates of $L^2$?

The question arises from an exercise but tackles deeper understanding of angular momentum operators. Suppose we have a 2D harmonic oscillator and an infinite square well in the third dimension: \...
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### Quantum mechanics angular momentum spherical tensor components

In Sakurai Quantum Mechanics, problem 3.25b we imagine $J_z^2$ as the component of a tensor with components $T_{ij} = J_iJ_j$. $J_z^2 = \frac{1}{3}\pmb{J}^2 + (J_z^2 - \frac{1}{3}\pmb{J}^2)$ The ...
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How to calculate expectation value of $\langle x\rangle, \langle y\rangle,\langle z\rangle$ for the general $\psi_{nlm}$ state? $x$ has $\sin(\theta)\cos(\phi)$ angular part which can be expressed as $... 0answers 32 views ### How to determine the best-fit magnetic dipole Gauss coefficient in a shifted coordinate system? Introduction A planetary magnetic field$\vec{B}$can be described outside of the planet using Gauss coefficients$g_n^m$and$h_n^m$and a spherical harmonic expansion: $$\vec{B} \;=\; -\vec{\... 1answer 21 views ### 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 ... 2answers 127 views ### Action Of Time-Reversal Operator On Spherical Harmonics Given some spherical harmonic of the form \textbf{Y}_l^m = (i)^lY_l^m Where Y_l^m is a standard spherical harmonic, I would like to find the action of the time-reversal operator T. My attempt ... 0answers 55 views ### How does the hydrogen atom actually “look like”? [duplicate] When deriving the solutions for the "simple" quantum mechanical hydrogen problem, one normally uses spherical coordinates (r,\theta,\phi), since the problem has rotational symmetry. The solution has ... 2answers 479 views ### In a spherically symmetric central potential why do we look for eigenfunctions of the angular momentum operator? In finding the solutions to the wave equation for a spherically symmetric potential V(r), we look for the eigenfunctions of \hat L^2 and \hat L_z operators. However, what is the reasoning behind ... 3answers 118 views ### Quantization of electrons' angular momentum in atoms and molecules It is known that the Schrödinger's equation of the electron's wave function in atoms can be solved analitically only when a single electron is present (the "hydrogenlike atom"). In that case, the ... 3answers 237 views ### Systematic expansion of e^{i\vec{k}\cdot\vec{r}} in atomic physics in terms of Legendre polynomials and identifying different l terms In the context of light-matter interaction one often makes the approximation e^{i\vec{k}\cdot\vec{r}}\approx 1. Keeping higher order terms in e^{i\vec{k}\cdot\vec{r}} give magnetic dipole, ... 2answers 107 views ### Directional Eigenket? I'm reading Sakurai's Modern Quantum Mechanics. In page 99, there is a concept of Directional Eigenket. It doesn't have a definition, or any properties about it. Because the angular dependence is ... 2answers 105 views ### Linear combination of 2 spherical harmonic functions The task is to form 2 linear combinations out of the 2 given spherical harmonic functions. I dont understand why the resultant wave function has to be multiplied with the constant 1/sqrt(2)? 0answers 84 views ### Can spherical harmonics be used in relativity equations? I have a neutral pion of mass m_{\pi}, and it decays into two photons. In it's reference frame the decay is isotropic. One of the photons has a helicity of +\hbar and the other -\hbar. In ... 4answers 265 views ### What are the boundary conditions for the Hydrogen Atom which cause the polar power series to need to terminate? I am trying to solve the Hydrogen Atom, and I am stuck in the polar equation. To simplify, I have taken the special case in which m=0 to get the Legendre Equation:$$(1-x^2)P''(x)-2xP'(x)+AP(x)(... 1answer 83 views ### matrix elements of$\hat{z}$operator under the angular momentum basis I found a quite challenge quantum mechanics problem in a preparation sample test for a midterm. The sample test does not have a solution, so it is bothering. The question reads as follows: ... 1answer 67 views ### How to evaluate the matrix element of coulomb repulsion term between electrons in an atom suing spherical harmonics multipole expansion? This is a lecture notes take from the following link on numerical calculation of atomic physics:http://www.phys.ubbcluj.ro/~lnagy/pdf/1curs.pdf I am trying to evaluate the two electron matrix element ... 2answers 76 views ### Musical Instrument that Exhibits Spherical Harmonics A guitar string exhibits standing wave patterns when its struck, some superposition of sines and cosines, a drum head exhibits a superposition of Bessel functions when its struck. Is there any ... 2answers 164 views ### Definition of the spherical harmonics do not agree I Griffiths' Introduction to quantum mechanics, the spherical harmonics are defined as $$Y_l^m(\theta,\phi) = \epsilon\sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}}e^{im\phi} P_l^m(\cos \theta)$$ ... 1answer 153 views ### Probability of finding a particle in the solid angle$d\Omega$at$\theta$and$\phi$[closed] For a spinless particle with the wavefunction \begin{equation} \psi(x,y,z)= K(x+y+2z)\exp(-\alpha r) \end{equation} with$r=\sqrt{x^2+y^2+z^2}$and K and$\alpha$are real constants. I have to ... 2answers 232 views ### What does the spherical-harmonic notation$Y^{m}_l(\hat{\textbf{r}})$mean, and how does it relate to the usual$Y^m_l(\theta, \varphi)$? By using the plane wave expansion, the decomposition of stationary harmonic plane wave into partial waves can be given by $$e^{i\textbf{k}\cdot\textbf{r}} = e^{ikz} = e^{ikr\cos\theta} = \sum^{\infty}... 1answer 80 views ### Vector Spherical Harmonics and total angular momentum In their book Akhiezer et al. give a definition of vector spherical harmonics (p.18 of Russian Edition) as$$\pmb{Y}_{j\ell m}(\pmb \Omega) = \sum_{m' \lambda} \langle \ell m' 1\lambda| jm \rangle Y_{... 2answers 55 views ### Putting the ket$|l,m_x \rangle$in terms of the ket$|l, m_z \rangle$Could someone guide me in my thought process of this problem? I don’t know if I’m thinking about it the right way. The problem is the following: I have a system which possible states are generated ... 1answer 124 views ### Orthonormal Basis integration and Kronecker delta Given that this integral I'm trying to solve is $$\frac{2}{\pi}\sum^{\infty}_{l=0}\sum^{l}_{m=-l}\int_{r=0}^{\infty}\int_{k=0}^{\infty} R_{nl}(r)b_{lm}(k)j_{l}(kr)k^2 r^2 \int_{\theta = 0}^{\pi}\int_{\... 0answers 37 views ### Do relativistically-contracted electron states have the same energy and angular momentum values? I've been reading that electron bound states are defined by four quantum numbers, n, l, m_l, and m_s, respectively the principal quantum number, the azimuthal quantum number, the magnetic ... 3answers 148 views ### Why are there three p-orbitals? This question is specifically about Schrödinger quantum mechanics, but if an answer in some other mode would illuminate it could be acceptable, as demonstrating a physical or mathematical reason for ... 1answer 189 views ### What are spin-2 spherical harmonics and why are they needed? A function f(\theta,\phi) (with \theta,\phi\in \mathbb{S}^2) can be expanded in terms of spherical harmonics Y_{l.m}(\theta,\phi). Recently, in this Particle Data Group review titled Cosmic ... 1answer 130 views ### Simplifying CMB correlation function with spherical harmonics I originally asked this on the physics Stack Exchange site, but perhaps it could be more easily answered here. Given the definition of the correlation function for CMB temperature fluctuations as$$ ... 2answers 360 views ### 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 $$Ψ_{... 0answers 44 views ### Hydrogen Atom, polar equation eigenfunctions In my textbook, Quantum Mechanics by David McIntyre on page 235, the solutions to the polar equation resulting from the separation of variables to the hydrogen atom are the eigenstates: The book ... 0answers 45 views ### What is a force multipole? In a recent talk about physics and mechanics inside the cell, I heard such terms as 'force monopole' and 'force dipole'. What do such terms mean? Are they talking about the angular distributions, ... 1answer 131 views ### Increase in energy with increasing orbital quantum number If we consider the problem in a spherically symmetric potential, we can see that with an increase in the orbital quantum number, the energy state in the spectrum increases. This is observed both in ... 5answers 482 views ### What is the geometry of DeBroglie standing waves? I asked a similar question here . But never received a complete answer. So I've made the question more specific to DeBroglie waves. So from what I've read DeBroglie waves are indeed standing waves ... 0answers 123 views ### Integral of spherical harmonics and their derivatives What is the Integral of the product of spherical harmonics and derivatives of spherical harmonics? More precisely, I am looking for$$\int_\Omega d\Omega\, Y_{l}^m Y_{l^\prime}^{m^\prime} \partial_\... 2answers 456 views ### Compact Formula for Wigner D-matrix I have done an elementary calculation that appears to be giving me a simple (intuitive) formula for an arbitrary Wigner D-matrix. I can't seem to find this formula anywhere. In the following,$\...
Let's take a direction eigenket $|{\bf\hat{n}}\rangle$ in 3-dimensional space oriented with angles $\theta\in\left[0,\pi\right]$ and $\phi\in\left[0,2\pi\right]$ in spherical coordinates. Next take ...