Questions tagged [spherical-harmonics]
Questions on spherical harmonics, a set of basis functions that satisfy an orthogonality relation over the sphere.
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Legendre addition theorem in $2$ dimensions
We know the addition theorem for Legendre polynomials in spherical coordinates is
$$P_\ell(\cos\gamma)=\dfrac{4\pi}{2\ell +1}\sum_{m=-\ell}^\ell\mathrm Y_{\ell m}(\theta_1,\phi_1)\,\mathrm Y_{\ell m}^\...
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Higher Dimensional Spherical Harmonics in Cartesian Form
Are there any tables in the literature or computer software for computing higher dimensional spherical harmonics in Cartesian form, like this Wikipedia article, which lists them for three dimensions.
...
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The Helmholtz equation for the spherical harmonics with delta functions
In three dimensions, the Green’s function for the Helmholtz equation with a radiating point source
$$
(\nabla^{2}+k_{0}^{2})g(\textbf{r},\textbf{r}')=\delta(\textbf{r}-\textbf{r}')
$$
is
$$
g(\textbf{...
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Need verification of coefficients in Spherical Harmonic
I made up an example which lacks azimuthal symmetry and exists on the boundary and outside of the sphere:
$$\nabla^2 u(r,\theta,\phi) = 0; 0<\theta<\pi,0<\phi<2\pi \\ u(a,\theta,\phi) = a^...
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How to linearly combine the components of $\ell=2$ Cartesian tensor to ensure that the transformation matrix is orthogonal?
A Cartesian tensor of rank 2 should can be decomposed into 3 irreducible part:
$$X=\frac{1}{3}tr(X)I + \frac{1}{2}(X-X^T) + \frac{1}{2}(X+X^T-\frac{2}{3}tr(X)I)$$
and $S=\frac{1}{2}(X+X^T-\frac{2}{3}...
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Any reference about these topics (waves on the sphere)?
At the end of the section 1.9.2 in the book A Panoramic View of Riemannian Geometry, it has the following text about waves on the sphere:
"We will not say much now about spherical harmonics ... ...
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Sum of Spherical Harmonics and Rotational Invariance
PROBLEM
Suppose that
$$
\sum_{m=-l}^{l} c_m Y_m^l(\theta, \phi) Y_m^l(\theta', \phi')^*
$$
is rotationally invariant, then how can we show that the $c_m$'s must be all equal?
ATTEMPT AT A SOLUTION
I ...
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calculation about the differential cross section of the metastable state in resonance scattering
I met a problem in the book by John R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions. In the question 13.4 (page:258), I need to integrate:
$$
\int^{\infty}_0r^2dr|\psi(\...
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Algebraic approach to spherical harmonics
I am interested in an algebraic approach to the following theorem:
Theorem. Consider the sphere $S^{n-1} \subseteq \Bbb{R}^n$ and for each $k=0,1,2,\ldots$, the space $H^k$ consisting of homogeneous
...
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Product of 3 or more Spherical Harmonics
The product of two spherical harmonics can be written as the sum of spherical harmonics with coefficients related to Wigner 3j matrices (ref eqn 16) :
$$Y_{l_1m_1}(\theta,\phi)Y_{l_2m_2}(\theta,\phi) =...
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How to plot spherical harmonics? [closed]
Let me start by saying that I am only interested in the mathematical aspect of the thing.
I would like to plot just for the fun of it the spherical harmonics that are used to plot the electronic ...
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Sum of a binomial relation confusion
I'm having trouble in understanding how to use the sum in the relation below:
$$
{}_sP_{jm}(\cos\theta)
= \frac{(j+m)!}{\!\sqrt{(j+s)!(j-s)!}}
\biggl(\!\sin{\frac{\theta}{2}}\biggr)^{\!2j} \,
\sum_{...
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Spherical multipole moments after flipping an axis
I have an interior spherical multipole expansion (as in Modern Electrodynamics by Andrew Zangwill):
$$f(\textbf{r}):=\sum_{l=0}^{\infty} \sum_{m=-l}^{l} B_{lm} r^{l} Y_{lm}^* $$
with spherical ...
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Expansion of $\frac{\text{erfc}({|\vec{r} - \vec{r'}|})}{|\vec{r} - \vec{r'}|}$ in spherical harmonics?
How can I derive the spherical harmonic expansion coefficients for the function $$ \frac{\text{erfc}({|\mathbf{r} - \mathbf{r'}|})}{{|\mathbf{r} - \mathbf{r'}|}} $$ by expressing it as $$f(\theta, \...
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Help with an integral involving associated Legendre functions?
In the last few days I've come across this nice integral involving the associated Legendre functions
$$
{\large\int_{-1}^{1}} \frac{P_{\ell}^m\left(u\right) P_{\lambda}^{m}\left(u\right)}{\sqrt{1 - u^...
