# 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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### The $L^2$ operator not returning the expected value (2) when applied on the (2,1,0) Hydrogen wave function [closed]

Let the Hydrogen wave function for the state $n=2, l=1, m=0$ be described as: $$\psi_{210} =cos(\sigma )*f(r)$$ Since the squared angular momentum eigenvalue is $L^2=l(l+1)$, I would expect it to be 2 ...
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### Series Solution of Laplace Equation in Spherical Coordinates

I was recently Studying Griffiths Electrodynamics after a long time and there I saw the Laplace equation. Because it was my second time going over Griffiths so I thought maybe I should try to derive ...
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### Question about the "axis of evil"in cosmology and the Doppler effect due to the solar system's motion

The cosmic microwave background (CMB) can be described by its anisotropies in a direction $\hat{n}$ in the celestial sphere $$\delta T(\hat{n})=\frac{ T(\hat{n})-\bar{T}}{\bar{T}}$$ where $\bar{T}$ ...
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### Selection of spherical harmonics for the orbital calculation of electrons

As shown in this answer, the calculation of the electron probability in atoms is based on Spherical Harmonics and the resulting orbit probabilities usually have symmetry axes on the Cartesian ...
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### Difference between Mutlipole moments calculated with normal integration and with Wigner-$D$ matrices

Im learning about Wigner-$D$ matrices and the applications to spherical harmonics. Now I wanted to test my knowledge, but i failed miserably :( (worked on this the whole day). So here is my problem: I ...
1 vote
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### Scalar spherical harmonics in $S_n$

In the Kaluza klein reduction we can "decompose" the spacetime $M_n$ as $M_n = M_4 \otimes K_d$, in which $K_d$ is a compact spacetime. So, functions like a scalar $\phi(x,y)$ can be ...
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### Boundary Conditions for Spherical Harmonics Problem

I have encountered a problem with electrostatics potentials. The problem is given as follows: A sphere of radius $𝑎$ has the potential $\Phi(a,\theta, \phi)$ at the boundary. Obtain expressions of ...
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### Square Integrability of spherical symmetric wave

In my class we were discussing some wave equation for a spherical symmetric wave $u(t,r)$ and my professor investigated the behaviour of the solutions asymptotics $r \rightarrow \infty$. The solution ...
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### Tangential component of the "electric" vector spherical harmonic $N_{lm} = \nabla \times (h X_{lm})$?

So I want to solve the problem of an electromagnetic plane wave scattering by a sphere myself, and one of the crucial steps is to solve the scattering of a converging TE- or TM-polarized vector ...
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### Can someone identify this vector representation of $\rm SO(3)$ in terms of multi-variable polynomials?

I am trying to get some deeper intuition for the representations of $\rm SO(3)$ and how they combine with each other, and I ran into an odd object that I'm hoping that folks here might help me ...
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1 vote
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### Multipole expansion and spectral decomposition

We can always write a Hermitian operator in the form of its spectral decomposition: $\hat{A}=\sum_i \lambda_i | \chi_i(\boldsymbol{r})\rangle\langle\chi_i(\boldsymbol{r}')|$ where $\lambda_i$ are the ...
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### Spherical coordinates using ladder operators [closed]

In an exercise I am dealing with, where I consider a system resembling a 3D quantum harmonic oscillator, it says that the transition to spherical coordinates can now be made in the operator language. ...
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### Why is the square of the magnitude of a spherical harmonic function related to the zenith angle $θ$?

My intuition tells me that the wave function corresponding to a spherically symmetric potential should only depend on r (of course, intuition can be misleading), otherwise changing the orientation of ...
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### Calculating minimum $l$ energy in central potential problems using this generalization of the variational theorem?

The variational theorem talks about giving an upper bound on the lowest eigenvalue of a given Hermitian operator, and there is a simple generalization if we allow ourselves to constrain the space of ...
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### How to calculate angular velocity from forces on spherical pendulum head?

I have a bunch of forces that add up the force vector $(a_x, a_y, a_z)$ which is applied to the head of a spherical pendulum with given angles and angular velocities $(\phi, \theta, \phi', \theta')$ ...
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### Spherical Harmonics and the Salpeter Eq $i\hbar\frac{\partial\Psi}{\partial t}=\left(E_0\sqrt{1+L_0^2\Delta}+\frac{kZ}{r}\right)\Psi(r,\theta,\phi,t)$
Recall the spinless Salpeter equation $$i\hbar \frac{\partial \Psi}{\partial t} = \left(E_0\sqrt{1-L_0^2 \Delta}+\frac{kZ}{r}\right) \Psi(r, \theta, \phi, t)$$ where $E_0 = mc^2$, \$L_0 = \frac{\hbar}...