# Questions tagged [spherical-harmonics]

Questions on spherical harmonics, a set of basis functions that satisfy an orthogonality relation over the sphere.

310 questions
Filter by
Sorted by
Tagged with
69 views

### Prove a relation involving the Laplace–Beltrami operator and spherical harmonics

Let $P_ℓ$ denote the space of complex-valued homogeneous polynomials of degree $ℓ$ in $n$ real variables, here considered as functions $\mathbb {R} ^{n}\to \mathbb {C}$. Let $A_ℓ$ ...
87 views

### Prove the orthogonal decomposition of the space of spherical harmonics

Let $\mathbb{S}^{n}$ denote the $n$-dimensional unit sphere in $\mathbb{R}^{n+1}$ endowed with the standard metric. For $k=0,1, \cdots$, denote by $\mathscr{H}_{k}^{n}$ the space of spherical ...
27 views

### Spherical harmonics orthogonality

I've been struggling with this integral $$\int_0^{2\pi}\int_0^{\pi} \sin(\theta) e^{-i\phi} Y_{l m}(\theta,\phi) Y^*_{l'm'} (\theta,\phi) \mathrm d\theta \mathrm d\phi$$ I've tried to use the ...
1 vote
10 views

40 views

84 views

### Do Spherical harmonics have continuous extensions to the entire sphere?

This article contains the following formula for the spherical harmonics: $$Y_l^m(\theta,\phi) = \sqrt{\frac{(2l+1)}{4\pi}\frac{(l-m)!}{(l+m)!}} P_l^m(\cos \theta) e^{im\phi}$$ Now let $S$ be the unit ...
1 vote
82 views

### Spherical harmonics as orthonormal basis in quantum mechanics

In this article https://mrtrix.readthedocs.io/en/dev/concepts/spherical_harmonics.html the following statement is given: Spherical harmonics are special functions defined on the surface of a sphere. ...
15 views

### Spherical Harmonics Relation

I have an expression like $$(\mathbf{q+K})Y^{m_{\ell}}_{\ell}(\widehat{q+K})$$ inside of an integral over $q$and it would be a lot easier to perform the integral if I only had spherical harmonics ...
1 vote
16 views

### Spherical harmonics defined on $S^2$ vs defined on $SO(3)$

I'm reading the following paragraph of these lecture notes (pp 35): I'm trying to understand the difference in defining spherical harmonics as functions on $S^2$ or $SO(3)$. From what I've read we ...
47 views

### Trying to understand the relationship between Hydrogen atom, spherical harmonics and central field force in quantum mechanics

I have a problem understanding three arguments in quantum mechanic: When we talk about a particle in a central field we have this kind of Hamiltonian: $$H=\frac{p^2}{2m}+V(r)$$ if we use spherical ...
49 views

### A question about the laplacian of homogeneous functions.

I'm reading the following document trying to understand the basics of spherical harmonics. Now, I understand that if we deal with homogeneous functions it's convenient to represent them in polar ...
70 views

### Harmonic polynomials property

The textbook i am currently reading states that: If $a_1^2+ \cdots + a_n^2 = 0$ and $a_i \in \mathbb{C}$ then the polynomial $$f(x) = (a_1x_1 + \cdots a_nx_n)^m$$ is a harmonic polynomial and ...
21 views

### Spherical harmonics and dimension of SO(n-1) invariant subspace

Let $K \cong SO(n-1)$ the isotropy group for $e_n$. Let $H_m$ be space of homogeneous polynomials, and $Y_m$ the space of spherical harmonics. H_m^k denote the subspace of $K$ invariant polynomials in ...
Let's say I have a combination of spherical harmonics e.g. $$Y_{\text{total}} = aY_{4}^{2} + bY_{6}^{2} + cY_{8}^{2}$$ Is there a way to find what is the weight of each spherical harmonic via ...