Questions tagged [spherical-harmonics]

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EM Field Quantization in Spherical polars

Is it possible to quantize the electromagnetic field in spherical polar coordinates instead of cartesian ones? Such that creation and annihilation operators correspond to harmonic oscillator modes ...
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68 views

In General Relativity, can I represent a Tetrad/Frame field in terms of ladder operators?

I've been interested in expressing the metric tensor $g$ in terms of it's harmonic expansions. In particular I'm interested in writing the tetrad/frame-fields in terms of such expansions. For ...
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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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185 views

Conventions in defining spherical harmonics and associated Legendre polynomials

Relevant Background Spherical harmonics are defined with several different conventions: the definition used in quantum mechanics according to Wikipedia is $Y_l^{\,m}(\theta,\phi) = (-1)^m\sqrt{\frac{...
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Question on Kleinman-Bylander pseudopotentials

When Kleinman-Bylander pseudopotentials are used the Hamiltonian operator is given by $$\hat{H} = -\frac{1}{2}\nabla^2+V_{\textrm{local}}+\delta \hat{V}_{\textrm{NL}}$$ where $$\hat{V}_{\textrm{NL}} = ...
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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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85 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 ...
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191 views

Can you write the Einstein equation as an eigenvalue equation for analytical solutions (to local problems) on non-flat spacetimes

In reading about various local solutions to Einstein's field equation it is easy to forget that they almost all assume a flat background spacetime (at least asymptotically). Considering this made me ...
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Spherical Formulation of Quantum Mechanics

I always wondered, during my QM courses, if we don't explore enough of the freedom that the Lagrangian and Hamiltonian Classical Dynamics give us. Classically, we can always make canonical ...
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406 views

Meaning of different Stokes' coefficients in harmonic expansion of gravitational potential

I know that geophysical quantities, such as gravitational potential, can be written in terms of spherical harmonics: $$W(r,\phi,\lambda;t) = \frac{G m_e}{r} + \frac{G m_e}{r} \sum_{l=2}^{N_{max}}\left(...
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177 views

Is there an intuitive interpretation of the shape of the angular momentum eigenstate?

I was watching a MIT lecture video on angular momentum eigenstate. Toward the end of the lecture, the professor had shown some plots of the first few spherical harmonics, in an attempt to explain ...
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The product of three spherical harmonics in higher dimension

As I see, e.g., in this question, a closed formula for the following integral \begin{equation} \int_{\mathbb{S}^N} Y_{\ell_1\ldots \ell_N}\,Y_{\ell'_1\ldots\ell'_N}\,Y_{\ell''_1\ldots \ell''_N}\, d^N ...
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508 views

Degeneracy, spherical harmonics

In a 3D oscillator, the energy levels are known to be $(n_x + n_y + n_z + \frac{3}{2})\hbar \omega = (n + \frac{3}{2})\hbar \omega$. Say for $n = 1$, any of the $n$'s can be $1$ and the rest are $0$. ...
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368 views

One more relation with spherical spinors

Let's have the spherical spinors: $$ \mathbf {Y}_{j, m, l = j \pm \frac{1}{2}} = \frac{1}{\sqrt{2l + 1}}\begin{pmatrix} \pm \sqrt{l \pm m +\frac{1}{2}}Y_{l, m - \frac{1}{2}} \\ \sqrt{l \mp m +\frac{1}{...
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1answer
23 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 ...
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131 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_\...
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47 views

Angular power spectrum: Calculating bias from N weighted events

I'm interested in calculating the angular power spectrum $C_{l,N,\omega}$ of $N$ weighted (weight $\omega_i$ for event $i$) events from a full sky map with distribution $C_l$? Interesting quantities ...
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195 views

Units of angular power spectrum

I am not sure whether this question might be better suited for crossvalidated or stackoverflow, but I will give it a try here: I have a map of the full sky in in the healpix format, units are $\rm ...
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138 views

conducting hollow sphere in magnetic monopole

if a hollow copper sphere(or any conducting hollow sphere) is connected to dc at points diametrical and a magnetic monopole is right at the center of the sphere then will there be any movement of the ...
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129 views

Rewriting $\langle {\bf k} \vert E,l,m \rangle$ as $\langle {\bf k} \vert ~k,l,m \rangle$ Spherical Harmonics

From Sakurai eq. 6.4.21a we have that $$\langle {\bf k} \vert E,l,m \rangle=\frac{\hbar}{\sqrt{M k}}\delta\left(E-\frac{\hbar^2 k^2 }{2M}\right) Y_l^m({\bf\hat k}),$$ where $M$ is the mass of the ...
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564 views

Total angular momentum in multielectron atoms

I have some confusion about orbitals in multielectron atoms. Let's say we consider an atom (Lithium, for example, $1s^2\, 2p^1$) and that the state of the last electron is [n=2, l=1, ml=0, s=1/2, ms=...
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47 views

Why in Landau the spherical harmonics had a multiple of $i^l$?

Quote Landau The Classical Theory of Fields Page 100 Equation 41.11 $$\displaystyle Y_{lm}(\theta,\phi)=(-1)^m i^l \sqrt{\frac{2l+1}{4\pi} \frac{(l-m)!}{(l+m)!} } P_l^{|m|}(\cos\theta) e^{im\phi} ...
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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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61 views

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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60 views

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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69 views

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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39 views

Matrix representation in angular momentum basis

I'm trying to find a way to verify that the following expansion is valid for any potential, including noncentral ones, $$ \langle \textbf{k}' |V|\textbf{k}\rangle = \frac2\pi\sum_{lm} V_l (k', k) Y_{...
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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{\...
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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 ...
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47 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, ...
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2answers
578 views

Multipole expansion of the electromagnetic field

In Jackson's Classical Electrodynamics, section 9.7, he develops the multipole expansion of the electromagnetic fields in terms of the vector spherical harmonics and the spherical Bessel and Hankel ...
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123 views

How to find an action of $(\hat {\sigma} \cdot \hat {\mathbf L} )$ on spherical spinors?

Let's have the spherical spinors $\psi_{j, m, l = j \pm \frac{1}{2}}$, $$ Y_{j, m, l = j \pm \frac{1}{2}} = \frac{1}{\sqrt{2l + 1}}\begin{pmatrix} \pm \sqrt{l \pm m +\frac{1}{2}}Y_{l, m - \frac{1}{2}} ...
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379 views

Perturbation of a Hydrogen Atom in a Quadrupole Field

Question: A hydrogen atom is located in a quadrupole field, which gives it a perturbation $$H_1=A(x^2-y^2)$$ where $A$ is some constant. Calculate the quantity $...