Questions tagged [spherical-harmonics]

Special functions defined on the surface of a sphere, often employed in solving partial differential equations. They form a complete set of orthogonal functions and thus an orthonormal basis.

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Magnetic potential in spherical coordinates

In a current-free region, we have $\nabla \times \mathbf{B} = 0$, allowing us to write $\mathbf{B} = -\nabla V$ for some scalar function $V$. We also have $\nabla \cdot \mathbf{B} = 0$, meaning that ...
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Normalizing the spherical coordinate wavefunction

I try to normalize the following wave function $$\psi=C_{n l} e^{-\rho / 2} L_{1}^{2 l+1} Y_{l m}$$ Using the normalization condition $$ 1 = |C_{nl}|^2 \int_{0}^{\infty} e^{-\rho} \rho^2 \{L_{1}^{2l+1}...
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Equivariance and Wigner $D$ matrices for Spherical harmonic rotations

I am trying to understand equivariance in machine learning, specially as discussed in the following paper. Claim is that equivariance is when Group symmetry operation, such as rotation, commutes with ...
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Spherical harmonics How do they span eigenspaces?

When trying to find common eigenstates of $L^2$ and $L_z$, we find the eigenstate $Y_m^l (\theta, \phi)$ My question is, if $m_1$ and $\lambda_1 = l_1(l_1+1)$ both have multiplicity $3$, then there is ...
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Normalize the wave function with spherical harmonics [closed]

I have this wave function: $$\Psi=C e^{-\rho / 2} \rho^{l} L_{1}^{3} Y_{l, m} $$ To normalize the function I have tried to express the polynomial in the function as follows. If: $$L_{1}^{3}(x)=(-1)^{1}...
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How to find expansion of slightly modified Coulomb's potential?

From here I know that, ${{\frac {1}{|\mathbf {r}_1 -\mathbf {r}_2|}}=\sum _{\ell =0}^{\infty }{\frac {4\pi }{2\ell +1}}\sum _{m=-\ell }^{\ell }(-1)^{m}{\frac {r_1^{\ell }}{r_2^{\ell +1}}}Y_{\ell }^{-m}...
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In order to solve for the states of a spherically symmetric parabolic potential do we need to use cartesian and cylindrical coordinates?

In the general case a spherically symmetric potential, the Time Independent Schrodinger Equation is separable in spherical coordinates but not in cartesian, or cylindrical coordinate as in general $V(...
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On the completeness of solutions of a quantum particle on the surface of a sphere

The Hamiltonian of a particle of mass $m$ on the surface of a sphere of radius $R$ is $$H=\frac{L^2}{2mR^2}$$ where $L$ is the angular momentum operator. I want to solve the TISE $\hat{H}\psi=E\psi$ ...
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Approximation of Spherical Bessel function [closed]

I am currently studying the CMB power spectrum from a numerical approach (easier than the analytical approach). In a Mathematica notebook that I am following, they work with spherical Bessel functions ...
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Angular Integral of with Spherical Harmonics and Cross Product

I have an integral involving spherical harmonics and a cross product. It reads $$ \int d^3k d^3Q \phi^{*}_{L'}(k+Q)Y^{*}_{L',M'_{L'}}(\widehat{k+Q})C^{J'M'_{J'}}_{L'S'}\mathbf{S}\cdot \mathbf{(k\times ...
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Eigenstates of $L^2$ and $L_z$ no $r$ dependence Why? [duplicate]

Does anybody know why the eigenstates common to the casimir operator $L^2$ and the $z$-axis angular momentum $L_z$ have no $r$ dependence (they are written $\psi_{m,l} (\theta, \phi)$)? Is it because $...
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Bohr radius: does the mean distance between the proton et the electron in $^1H$ equals to $a_0$ or $\frac{3}{2}a_0$?

According to Wikipedia, the Bohr radius (already described here) is: a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in ...
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Spherical harmonics with 2 cosmological probes : considering $a_{\ell m, photo}$ as a constant and $\hat{a}_{\ell m, spectro}$ as an estimator

I am in cosmological context where the survey on which I am working has 2 probes : a photometric galaxy clustering ($GC_{ph}$) probe and a spectroscopic galaxy clustering ($GC_{sp}$ probe). We use an ...
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Spherical harmonics expansion: from scalars to tensors

It is well known that a scalar field on the unit sphere can be expanded in spherical harmonics, see e.g. this. I am wondering if there is a related concept for vector fields and, in general, for any ...
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Why is there a non-zero probability density of finding an $l=0$ electron at the origin of a Hydrogen-like atom?

A well known result for the $l=0$ hydrogenic functions is that $$\psi_{nlm_l}=R_{nl}(r)Y_{lm_l}$$ $$|\psi_{n00}|^2=\frac{Z^3}{\pi a_0^3n^3}$$ where $R_{nl}$ and $Y_{lm_l}$ are the radial function and ...
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Spherical harmonics integral

I've been struggling with this integral $$ \int_0^{2\pi}\int_0^{\pi} \sin\theta~ e^{-i\phi} Y_{l m}(\theta,\phi) Y^*_{l'm'}(\theta,\phi) ~d\theta ~d\phi $$ I've tried to use the definition of the ...
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Properties of Vector Spherical Harmonics

In section 5.3.2 of the book Advanced Classical Electromagnetism by Robert Wald, in deriving the multipole expansion for the retarded solution of electromagnetic field in presence of charge-current ...
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Working with decomposition of fields

I'm trying to follow a text I found online. The author decomposes EM fields such $$ \mathbf{E} = \sum_{lm}\left(f_l(r) \mathbf{Y}_{lm} - i \frac{l(l+1)}{r} g_l(r) \mathbf{\Psi}_{lm} - i\left(\frac{d ...
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How to convert from plus and cross polarization modes ($h_{+}$, $h_{×}$) to spin-weighted spherical harmonic $h_{lm}$?

I was wondering if there is a method to express the $h_{+}$ & $h_{×}$ polarization modes to spin-weighted spherical harmonic $h_{lm}$. I ask this in the context of gravitational waves. We see that ...
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Parity of the vector spherical harmonics?

The vector spherical harmonics can be defined as $$\textbf{Y}_{j, \ell, 1}^m(\theta, \phi) = \sum_{m_{\ell}=-\ell}^{+\ell} \sum_{m_s=-1}^{+1}C_{\ell, m_{\ell}, 1, m_s}^{j, m_j} Y_{\ell, m_{\ell}}(\...
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Spheroidal (elliptical) waves: a boundary problem

I'm trying to solve a boundary problem. I have known EM waves, which "bounce" of a charged, rotating oblate spheroid (ellipsoid). Now the formulation of the boundary problem is not so hard, ...
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Particle on a sphere, it's solution having extra $(-1)^m$ term

To solve a particle on a sphere problem in quantum mechanics we get the below equation :$\left[\frac{1}{\sin \theta} \frac{d}{d \theta}\left(\sin \theta \frac{d}{d \theta}\right)-\frac{m^{2}}{\sin ^{2}...
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How to make notation like $Y_{l m_{l}}(\theta, \phi)\chi_{m_s}$ more rigorous as a tensor product?

Sometimes in quantum mechanics we come across notation like $Y_{l m_{l}}(\theta, \phi)\chi_{sm_s}$ where $Y_{lm_l}$ is a spherical harmonic representing the spatial part of some particle wavefunction ...
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Spin, Orbital and Total Angular Momentum For Classical Vector Fields

I ask this question motivated by trying to understand vector spherical harmonics and find or come up with an elegant abstract derivation of their form. Suppose we have a 3D field of 3D vectors: $\...
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Inner products with spherical harmonics in quantum mechanics

Let $|l,m\rangle$ be a simultaneous eigenstate of operators $L^2$ and $L_z$ and we want to calculate $\langle l,m|\cos(\theta)|l,m'\rangle$ where $\theta$ is the angle $[0,\pi]$. It is true that in ...
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Angular momentum measurements on a system of non-identical particles

There is a system of two non-identical particles that are in the state, at time $ t=0$ $$ \psi(\vec{r}_1, \vec{r}_2) = N |\vec{r}_1-\vec{r}_2|^2 e^{-\alpha > (r_1^2+r_2^2)} $$ where $\alpha>0$ ...
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Inverting a spherical harmonic expansion

I am referring to this paper by Srednicki, page 4, equations 14-17. Equation 14 is a Hamiltonian, which is to be written as $H = \sum_{lm} H_{lm}$. The $H_{lm}$ are given in equation (17).I am trying ...
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Central symmetric field in Landau book

In Landau-Lifshitz Quantum Mechanics at page 102 when talking of central symmetric field I read the following sentence: The angular momentum is conserved during motion in a centrally symmetric field. ...
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Algorithm to generate cartesian names for real spherical harmonics

Spherical Harmonics are used to describe atomic orbitals. For a physicist the complex spherical harmonics, indexed by $l, m$ with $-l \le m \le l$ are more natural because atomic states with spatial ...
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Momenta basis for a quantum system with configuration space $\mathbb{R}\mathrm{P}^2$

Let $\mathcal{X}$ be the configuration space of a (quantum) system. When $\mathcal{X} = S^1 \simeq \mathrm{SO}(2)$, a momenta basis is $$\{ |\ell\rangle : \ell \in \mathbb{Z} \}.$$ When $\mathcal{X} ...
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Spherical harmonics and Bra-Ket notation

Let's assume i have a wave function $$\psi = N(x+y+z)e^{-r^2/a^2} $$ with $N$ and $a$ some constants. This function can be written as a sum of the spherical harmonix $Y_{1,0}$, $Y_{1,1}$ and $Y_{1,-1}$...
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Spherical harmonics visualization as oscillations on sphere

I‘ve watched this video where they visualize the spherical harmonics as oscillations. Since the spherical harmonics determine the spatial shape of the orbitals of the hydrogen atom their time ...
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Potential of a dielectric sphere and point charge

I've seen in Jackson's 4.9 problem and also in a similar example given in the book (pag. 157) they use the azimuthal symmetry solution for Laplace equation which gives the ordinary Legendre ...
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How do we obtain radial part and angular part with spherical harmonics?

In a website (sorry, I can't find it anymore) I've read this when talking about spherical harmonics: if we want to determine simultaneous eigenfuntions of $L^2$ and $L_z$ we have to solve this system:...
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Trying to understand the relationship between Hydrogen atom, spherical harmonics and central field force in quantum mechanics

I have a problem understanding three arguments in quantum mechanic: When we talk about a particle in a central field we have this kind of Hamiltonian: $$H=\frac{p^2}{2m}+V(r)$$ if we use spherical ...
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Indices In The Hyperspherical Harmonics

Definition In $d$-dimensional space we have a hyperspherical coordinate system with angles $\theta_1, \theta_2, ..., \theta_{d-2}, \phi$. I am working with the following definition (up to ...
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Taking the mystery out of geopotential coefficients

Geopotential coefficients are usually notated Cnm and Snm (for example, see this IERS Tech Note). My questions (in the order I care about them) are, What are the constraints on n and m? Is n always ...
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$D$-dimensional Coulomb problem, is the generalization from the 3D case supposed to be simple? (Gilmore's Lie algebra, chapter 14, problem 20)

I am solving problem 20 of chapter 14 of Robert Gilmore's Lie groups, physics and geometry: An Introduction for Physicists, Engineers and Chemists, which focuses on the $D$-dimensional Coulomb problem,...
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Can a spherical function with only $r$-dependence be an eigenstate of $L^2$ and $L_z$?

My guess is no, because the equations for $L^2$ and $L_z$ have no $r$-dependence: $$L^2 f_m^\ell ( \theta , \phi ) = \hbar^2 \ell(\ell+1) f_m^\ell ( \theta , \phi )$$ $$L_z f_m^\ell ( \theta , \...
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Spinless Particle wave function [closed]

I've found wave function for spinless particle $$\psi (r, \theta, \varphi)=N(x \pm y + 2z)\:e^{- \alpha r} $$ How do I find mean $\hat{L}^{2}$ and $\hat{L}_{z}$ in this state? I also wanted to ...
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What does a star on a spherical harmonic mean?

So I was studying multipole expansion and the book I am using introduced spherical harmonics. While I could understand the concept of the functions themselves, the book suddenly started putting a “ * “...
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How to normalize a linear combination of spherical harmonics?

I know the formula for normalizing individual spherical harmonics, but do not know how to normalize linear combinations of them Say I have a system $ \alpha (\theta, \phi) = aY_{l_1}^{m_1} + bY_{l_2}^...
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Clebsch-Gordan coefficients and total angular momentum basis question

Suppose I have the total angular momentum basis for two particles, formed by $\{|J,M\rangle\}$, where $J\in |j_{1}-j_{2}|\leq J\leq j_{1}+j_{2}$ and $M=m_{1}+m_{2}$ with $ -J\leq M\leq J$. There is a ...
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Legendre spherical harmonics - Does it make sense to have multipole interval lower than 1?

I introduce in the context of a physical noise $B$ a factor $\Delta\ell$ multipole appearing in the expression of the variance $\operatorname{Var}\left(B\right)$ of $B$. The formula is the following : ...
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Covariant derivative of spherical harmonics

Given is the metric $\gamma_{jk}$ for the surface of a Sphere $S^2$ with $\gamma_{22}=1,\gamma_{23}=\gamma_{32}=0$ and $\gamma_{33}=\sin^2(\theta)$. The coordinates are $x=$($t,r,\theta,\phi$) and $j$ ...
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Rotations of spherical harmonics and Wigner $D$-matrices

I seem to be having trouble understanding how Wigner D-matrices rotate spherical harmonics. I asked this question on the Maths Stack Exchange but decided to cast my net a bit wider and ask the ...
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Simplifying an integration using Legendre expansion

I'm following a derivation for the first and second moments of a conditional probability distribution $f(s; \mathbf{r}, \hat d)$ (see paper here ), where $s$ is the path length that a charged particle ...
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Dirac equation for hydrogen atom

I went over a calculation of the hydrogen wavefunction using Dirac equation (this one) and I am a bit confused by the angular part. The final result for the wavefunction based on that derivation is ...
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Converting $\psi(x,y,z)$ to $f(r)Y_l^m$ and calculating expectation value of the angular momentum operators

I've been trying to solve a particular class of problems, where I'm given a wavefunction $\psi (x,y,z)$ and I'm asked to find the expectation value or eigenvalue related to the angular momentum ...
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How could I calculate the product of two spherical harmonics

For now I have to calculate the product of two spin-weighted spherical harmonics: $ _{s}Y_{lm}\ \times\ _{s}Y_{lm}^{*}$ What’s the result of this product? Maybe it is a spin-weighted 0 spherical ...
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