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Questions tagged [spherical-harmonics]

Questions on spherical harmonics, a set of basis functions that satisfy an orthogonality relation over the sphere.

If we wish to apply the Laplacian to a polynomial, we apply it to each term of a homogeneous polynomial (noting that the Laplacian is linear).

A homogeneous harmonic polynomial is a spherical harmonic.

A spherical harmonic is a restriction to the unit sphere of homogeneous harmonic polynomials of degree $n.$

A function on the sphere is harmonic.

You can recover it if you know the spherical harmonics.

Fourier series on the $n-$dimensional sphere are in terms of spherical harmonics.

354 questions
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Spherical harmonics for dummies

Adding to the for dummies. The real spherical harmonics are orthonormal basis functions on the surface of a sphere. I'd like to fully understand that sentence and what it means. Still grappling with Orthonormal basis functions (I believe this is…
bobobobo
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What is the relationship between spherical harmonics and the schrodinger equation?

spherical harmonics (below image) Schrödinger equation(below image) What is the relationship between spherical harmonics and the schrodinger equation?
User3910
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What is the correct formula for $ n $-dimensional spherical harmonics?

In the Wikipedia article, the formula for $ n $-dimensional spherical harmonics is given as $$ Y_{\ell_1, ..., \ell_{n-1}}( \theta_1, \dots \theta_{n-1} ) = \frac{1}{\sqrt{2\pi}} e^{i \ell_1 \theta_1} \prod_{j = 2}^{n-1} {}_j…
joy
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What is spherical harmonics for the two dimensional case?

We know that in general spherical harmonics of a unit vector $\hat{\mathbf{r}}$ is $Y_l^m(\hat{\mathbf{r}})=Y_l^m(\theta,\phi)$. I am interested to know what happens to this sperical harmonics if the dimension of the problem is changed to two…
titanium
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Hartree potential (Coulomb) in spherical symmetry (expansion of spherical harmonics)

I have a question regarding the Hartree potential in spherical symmetry. Specifically, The Hartree potential reads: $$V_H(\mathbf{r})=\int\frac{n(\mathbf{r}^\prime)}{|\mathbf{r}-\mathbf{r}^\prime|}d\mathbf{r}^\prime$$ where $n(\mathbf{r})$ is the…
Hui Zhang
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Integral of the product of spherical harmonics and derivatives of spherical harmonics

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_\theta…
Fluid
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Separation of variables and complex numbers

I began with the Laplace's equation in the context of spherical harmonics. From wikipedia, one reads. So far I have followed, but in the sequel is stated that $m \in \Bbb{R}$ since $\Phi$ is periodic. Then assume $Y(\theta,\varphi)$ is regular at…
Conrado Costa
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Four product of sphrerical harmonics

I encounter four product of sphrerical harmonic problems and found it has this equation $Y_{l_1}^{m_1}(\theta,\phi)Y_{l_2}^{m_2}(\theta,\phi) =$ $\displaystyle\sum\limits_{l,m}\sqrt{\frac{(2l_1+1)(2l_2+1)(2l+1)}{4\pi}} \left( {\begin{array}{ccc}…
Motohide
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Overlap integral of hyper spherical harmonics

Is there a simple way to compute the overlap integral of three hyperspherical harmonics on the three-sphere? To be more precise, is there a closed form expression for $$ \int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{\pi} \sin^2\theta \ \sin\chi\ Y_{\ell_1\…
user12588
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Do Spherical harmonics have continuous extensions to the entire sphere?

This article contains the following formula for the spherical harmonics: $$Y_l^m(\theta,\phi) = \sqrt{\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}} P_l^m(\cos \theta) e^{im\phi}$$ Now let $S$ be the unit sphere in three dimensions. Clearly, the…
Filippo
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Spherical Harmonics - Inner product

If $\mathcal{P}$ is space of polynomials in $n$ variables with complex coefficients. Let $\mathcal{P_m}$ be the subspace of homogeneous polynomials of degree m. How would i show that for two polynomials $$p(x) = \sum_{\alpha} a_\alpha x^\alpha, q(x)…
user622405
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Decompose a product into angular harmonics

The function $f$ of two 3-dimensional unit vectors $\hat r_1$ and $\hat r_2$ only depends on the angle from one vector to another, so we can write $f(\hat r_1, \hat r_2)=g(\theta)$, where $\theta$ is the angle from $\hat r_1$ to $\hat r_2$. We can…
Mr. Gentleman
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Spherical Harmonics Expansion of Analytic Function derivation

Someone has led me to understand that the following spherical harmonic expansion of an analytic function is completely general, however I am having trouble seeing how one would derive it. As far as I understand the following formula is the general…
falsovuoto
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completeness of derivatives of spherical harmonics

Any function $f(\theta,\phi)$ which obeys: $\int_0^{2\pi} \int_0^{\pi} \sin \theta |f(\theta,\phi)|^2 d\phi d\theta < \infty$ can be written in terms of spherical harmonics $Y_{lm}(\theta,\phi)$ as $f(\theta,\phi) = \sum_{l=0}^{\infty}…
D_J_S
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Spherical harmonics popularity.

I know that any periodic function of it's two arguments can be expended on spherical harmonics, and in here they claim The development in spherical harmonics is the equivalent, applied to the angular functions, of the development in Fourier…
lakehal
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