Questions tagged [spherical-harmonics]

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21
votes
3answers
10k views

Integral of the product of three spherical harmonics

Does anyone know how to derive the following identity for the integral of the product of three spherical harmonics?: \begin{align}\int_0^{2\pi}\int_0^\pi Y_{l_1}^{m_1}(\theta,\phi)Y_{l_2}^{m_2}(\...
8
votes
1answer
1k views

Why do we need the Condon-Shortley phase in spherical harmonics?

I'm confused with different definitions of spherical harmonics: $$Y_{lm}(\theta,\phi) = (-1)^m \left( \frac{(2l+1)(l-m)!}{4\pi(1+m)!} \right)^{1/2} P_{lm} (\cos\theta) e^{im\phi}$$ For example here ...
4
votes
2answers
1k views

Why does $\ell=0$ correspond to spherically symmetric solutions for the spherical harmonics?

In quantum mechanics why do states with $\ell=0$ in the Hydrogen atom correspond to spherically symmetric spherical harmonics?
8
votes
1answer
580 views

Eigenvalues of spherical harmonics in $d$ dimensions

I'm working on the Schrodinger equation for a hydrogen atom in a $d$-dimensional space, so I'm interested in the possible eigenvalues of the angular momentum part of the $d$-dimensional Laplace ...
2
votes
2answers
1k views

Irreducible form of Spherical tensor operators

In the section on spherical tensors in Sakurai, he introduces the idea of going from Cartesian tensors to irreducible spherical tensors. He states the following: A spherical harmonic can be written ...
3
votes
1answer
1k views

Klein Gordon equation in Schwarzschild spacetime (spherical harmonic mode expansion)

My Question: In his GR text, Robert Wald claims that solutions, $\phi$, to the Klein-Gordon equation, $\nabla_a\nabla^a \phi = m^2 \phi$, in Schwarzschild spacetime can be expanded in modes of the ...
3
votes
1answer
5k views

Hydrogen wave function in momentum space

We can seperate the wave function of an hydrogen atom in a radial and an angle part: $$ \phi_{n,l,m} (\mathbf{r}) = R_{n,l,m}(r) Y_{l,m}(\vartheta,\varphi) \, , $$ where $Y_{l,m}$ are the spherical ...
3
votes
0answers
38 views

Do relativistically-contracted electron states have the same energy and angular momentum values?

I've been reading that electron bound states are defined by four quantum numbers, $n$, $l$, $m_l$, and $m_s$, respectively the principal quantum number, the azimuthal quantum number, the magnetic ...
1
vote
1answer
444 views

Can a standing wave exist on a spherical surface?

I've often seen the DeBroglie wave illustrated by a two dimensional surface as a standing wave, but then the 'electron cloud' surrounding an atom is hardly two dimensional and furthermore held to the ...
1
vote
2answers
372 views

Is there a relation between the Legendre generator function and Spherical Harmonics for a Potential?

Recently I had to solve a simple problem in which I had a sphere of radius $R$ with a constant potential (but with different sign), on both of the hemispheres, and I was asked to get the electrostatic ...