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.
Questions tagged [spherical-harmonics]
321 questions
24
votes
3 answers
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…
okj
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22
votes
4 answers
How should we think about Spherical Harmonics?
Studying Quantum Mechanics I only thought about Spherical Harmonics $Y_{l,m}(\theta , \phi)$:
$$Y_{l,m}(\theta , \phi)=N_{l,m}P_{l,m}(\theta)e^{im\phi}$$
as the simultaneous eigenfunctions of $L_z$ and $L^2$.
But then I stumbled on these two…
Noumeno
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15
votes
2 answers
Physically unacceptable solutions for the QM angular equation
I'm reading Griffiths's Introduction to Quantum Mechanics 3rd ed textbook [1]. On p.136, the author explains:
But wait! Equation 4.25 (angular equation for the $\theta$-part) is a second-order differential equation: It should have two linearly…
Arete
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15
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2 answers
Plotting the CMB power spectrum - Why $C_\ell \ell (\ell+1)$ rather than only $C_\ell$?
I can't find any convincing answer for the following question :
Why do we always (or often) plot the CMB power spectrum in this way?
I mean the vertical axis is $C_\ell \ell (\ell+1)$ and not only $C_\ell$.
Why?
I know it's because of the scale…
AnSy
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12
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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 they claim, that one can decide whether to include…
ilciavo
- 243
10
votes
4 answers
What determines the shape of electron suborbitals?
The azimuthal quantum number determines the shape of electron distribution around a nucleus ($s$ orbital has spherical distribution, $p$ orbital has dumbell-like distribution and so on).
But what determines the shape of those orbitals? Why doesn't…
RajaKrishnappa
- 230
9
votes
3 answers
Angular orientation of exact solution of the Hydrogen Schrödinger Equation
I am currently trying to get a better understanding of the basis sets used for computational quantum chemistry and as a start read ch.6 on the exact solution of the Schrödinger equation of the hydrogen atom in "Physical Chemistry: A Molecular…
FrameMainTain
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8
votes
2 answers
How to calculate the angular momentum states of isotropic quantum harmonic oscillator?
While trying to calculate the angular momentum states for the first non trivial even and odd states ($N=2$ and $N=3$). When $N=n_x + n_y + n_z$
By solving the radial problem one can see that there 6 states for $N=2$ and 10 states for $N=3$, it stems…
0x90
- 3,456
8
votes
1 answer
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 operator.
Since the angular momentum part corresponds…
casimir
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7
votes
1 answer
Spherical harmonics expansion: from scalars to tensors
Background: A scalar field on the unit sphere can be expanded in spherical harmonics, see this.
This seems related to the multipole expansion for vector fields, but it's not exactly the same concept. Moreover, I know that something called vector…
Quillo
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7
votes
1 answer
Rotations of spherical harmonics and Wigner $D$-matrices
I seem to be having trouble understanding how Wigner D-matrices rotate spherical harmonics. I asked this question on the Maths Stack Exchange but decided to cast my net a bit wider and ask the question here too.
Suppose that we want to understand…
James A
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7
votes
2 answers
Regular solution vs irregular solution
My Quantum Mechanics textbook (John S. Townsend's A Modern Approach to Quantum Mechanics) mentions regular solutions and irregular solutions. It claims that regular solutions (at the origin) to the spherical Bessel equation are called spherical…
Ben Sandeen
- 488
6
votes
1 answer
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 coordinates we can write the Laplacian operator…
Salmon
- 951
6
votes
0 answers
Why can transform the $\rm SU(2)$ spin to $S^2$ space?
Spin lies in $\rm SU(2)$ space, i.e. $S^3$ space, but when we write the spin coherent state:
$$|\Omega(\theta, \phi)\rangle=e^{i S \chi} \sqrt{(2 S) !} \sum_{m} \frac{e^{i m \phi}\left(\cos \frac{\theta}{2}\right)^{S+m}\left(\sin…
Merlin Zhang
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6
votes
1 answer
Coordinate-free general solution to the wave equation
General solutions to the wave equation in $\mathbb R^3$,
$\partial_{tt}\phi(t, \mathbf r) = c^2\Delta \phi(t,\mathbf r)$
can be obtained by first splitting off the time component, e.g. with a Fourier transform or separation of variables, leading to…
mcmayer
- 201