# Questions tagged [spherical-harmonics]

The tag has no usage guidance.

10 questions
Filter by
Sorted by
Tagged with
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}(\...
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 ...
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?
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...