# 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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### 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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### 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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### 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 ...