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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Spherical Harmonics and the Salpeter Eq $i\hbar\frac{\partial\Psi}{\partial t}=\left(E_0\sqrt{1+L_0^2\Delta}+\frac{kZ}{r}\right)\Psi(r,\theta,\phi,t)$
Recall the spinless Salpeter equation
$$
i\hbar \frac{\partial \Psi}{\partial t} = \left(E_0\sqrt{1-L_0^2 \Delta}+\frac{kZ}{r}\right) \Psi(r, \theta, \phi, t)
$$
where $E_0 = mc^2$, $L_0 = \frac{\hbar}...
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Nodal circles on spherical harmonics [closed]
Homework question:
For the spherical harmonic with $l=2$, $m=0$ , at what angle theta relative to the polar axis is the nodal circle in the northern hemisphere?
My attempted answer:
I know that I need ...
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How to estimate $d$-wave nuclear scattering cross section?
I would like to estimate the $\ell=2$ ($d$-wave) contribution to elastic scattering cross section (or scattering length) for collisions such as $n$-$n$ and $n$-$p$ at low energy (a few MeV or less). I ...
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What does the imaginary component of the spherical harmonics tell us about the hydrogen atom orbital?
I just went through a derivation of the spherical harmonics. I totally get how graphing the real part of the spherical harmonics create the shapes of orbitals, but I'm not sure what the imaginary part ...
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How to prove that spherical and Cartesian $l$th multipole moments have the same number of independent components?
This is Problem 4.3 from Jackson Classical Eletrodynamics (3rd edition).
I have searched online about this problem but have not found any satisfactory solutions.
In the problem, Jackson says that $q_{...
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What is the probability distribution the angular component of the electron in the hydrogen atom?
The angular component of the electron in a hydrogen atom is the family of spherical harmonic functions, Y(θ,Φ). I have seen the angular function probability distribution graphically represented as the ...
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Is all information of the quantum numbers $m\ell$ lost in a superposition?
I have learned that to create real spherical harmonics, a superposition of degenerate spherical harmonic eigenfunctions can be created using ±ml. For example, the px and py (l = 1) spherical harmonics ...
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Scattered spherical wave does not satisfy the Schroedinger equation?
I am a beginner in the study of quantum mechanics so I apologize if my question turns out to be naive.
My problem is in the very beginning of the theory of 3-dimensional scattering: we assume to have ...
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Wavefunction with two different values at same point
Consider a particle on sphere. Its Hamiltonian in spherical polar coordinates is given by -
$-\frac{\hbar^2}{2mr^2}\Big(\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\sin\theta\frac{\partial}{\...
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Question on the bounds for finding Fourier coefficients
In Griffit's E&M, when solving Laplace's equation for the potential, he uses the "Fourier trick" on Legendre polynomials, where
my question is, why are the bounds from -1 to 1?
because ...
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Index position of spherical harmonic function and tensorial properties
The spherical harmonics functions are denoted as $Y_l^m$ or $Y_{lm} $in https://en.wikipedia.org/wiki/Spherical_harmonics
e.g.,
\begin{align}
Y_{lm} &=
\dfrac{i}{\sqrt{2}} \left(Y_\ell^{m} - (-1)^...
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Connection between Laplace's equation and hydrogenic electron Schrodinger equation
Consider Laplace's equation:
$\nabla ^2 V = 0$
This holds for an electric potential $V$ in a region of space where no charges are present. This includes a Coulomb potential of a hydrogen nucleus (...
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Why are power spectrum plots $l^2 P(l)$ instead of just $P(l)$?
Why is it typically plotted $l^2 P(l)$, or $l(l+1) P(l)$, vs $l$ instead of just $P(l)$ in power spectrum plots?
For example, we can see it in this plot found in Introduction to Gravitational Lensing ...
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Orthogonality condition for spherical harmonics
The 3ed of Jackson E&M states the following as the orthogonality relationship for the spherical harmonics:
$$\int_{0}^{2\pi}d\phi\int_0^{\pi}\text{sin}\theta d\theta \space Y^*_{l'm'}Y_{lm}=\...
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Multipole moments of line charge
I am supposed to calculate the multipole moments of a line charge with total charge Q spread from $z=-a$ to $z=a$ on the z-axis. I know that each multipole moment is given by:
$$
q_{l,m}=\int_{\mathbb{...
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Checking if a state is an eigenstate of $L^2$ and $L_z$ without performing calculations
If I have a given state $\psi$ which is a linear combination of spherical harmonics, and I am asked if its an eigenstate of $L^2$ and $L_z$, is there a way to do it without using the eigenvalue ...
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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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Normalizing the spherical coordinate wavefunction
I try to normalize the following wave function
$$\psi=C_{n l} e^{-\rho / 2} L_{1}^{2 l+1} Y_{l m}$$
Using the normalization condition
$$ 1 = |C_{nl}|^2 \int_{0}^{\infty} e^{-\rho} \rho^2 \{L_{1}^{2l+1}...
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Equivariance and Wigner $D$ matrices for Spherical harmonic rotations
I am trying to understand equivariance in machine learning, specially as discussed in the following paper. Claim is that equivariance is when Group symmetry operation, such as rotation, commutes with ...
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Spherical harmonics How do they span eigenspaces?
When trying to find common eigenstates of $L^2$ and $L_z$, we find the eigenstate $Y_m^l (\theta, \phi)$
My question is, if $m_1$ and $\lambda_1 = l_1(l_1+1)$ both have multiplicity $3$, then there is ...
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Normalize the wave function with spherical harmonics [closed]
I have this wave function:
$$\Psi=C e^{-\rho / 2} \rho^{l} L_{1}^{3} Y_{l, m} $$
To normalize the function I have tried to express the polynomial in the function as follows. If:
$$L_{1}^{3}(x)=(-1)^{1}...
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How to find expansion of slightly modified Coulomb's potential?
From here I know that,
${{\frac {1}{|\mathbf {r}_1 -\mathbf {r}_2|}}=\sum _{\ell =0}^{\infty }{\frac {4\pi }{2\ell +1}}\sum _{m=-\ell }^{\ell }(-1)^{m}{\frac {r_1^{\ell }}{r_2^{\ell +1}}}Y_{\ell }^{-m}...
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In order to solve for the states of a spherically symmetric parabolic potential do we need to use cartesian and cylindrical coordinates?
In the general case a spherically symmetric potential, the Time Independent Schrodinger Equation is separable in spherical coordinates but not in cartesian, or cylindrical coordinate as in general $V(...
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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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Angular Integral of with Spherical Harmonics and Cross Product
I have an integral involving spherical harmonics and a cross product. It reads
$$
\int d^3k d^3Q \phi^{*}_{L'}(k+Q)Y^{*}_{L',M'_{L'}}(\widehat{k+Q})C^{J'M'_{J'}}_{L'S'}\mathbf{S}\cdot \mathbf{(k\times ...
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Eigenstates of $L^2$ and $L_z$ no $r$ dependence Why? [duplicate]
Does anybody know why the eigenstates common to the casimir operator $L^2$ and the $z$-axis angular momentum $L_z$ have no $r$ dependence (they are written $\psi_{m,l} (\theta, \phi)$)?
Is it because $...
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Bohr radius: does the mean distance between the proton et the electron in $^1H$ equals to $a_0$ or $\frac{3}{2}a_0$?
According to Wikipedia, the Bohr radius (already described here) is:
a physical constant, approximately equal to the most probable distance between the nucleus and the electron in a hydrogen atom in ...
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Spherical harmonics with 2 cosmological probes : considering $a_{\ell m, photo}$ as a constant and $\hat{a}_{\ell m, spectro}$ as an estimator
I am in cosmological context where the survey on which I am working has 2 probes : a photometric galaxy clustering ($GC_{ph}$) probe and a spectroscopic galaxy clustering ($GC_{sp}$ probe).
We use an ...
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Spherical harmonics expansion: from scalars to tensors
It is well known that a scalar field on the unit sphere can be expanded in spherical harmonics, see e.g. this. I am wondering if there is a related concept for vector fields and, in general, for any ...
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Why is there a non-zero probability density of finding an $l=0$ electron at the origin of a Hydrogen-like atom?
A well known result for the $l=0$ hydrogenic functions is that
$$\psi_{nlm_l}=R_{nl}(r)Y_{lm_l}$$
$$|\psi_{n00}|^2=\frac{Z^3}{\pi a_0^3n^3}$$
where $R_{nl}$ and $Y_{lm_l}$ are the radial function and ...
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Spherical harmonics integral
I've been struggling with this integral
$$
\int_0^{2\pi}\int_0^{\pi} \sin\theta~ e^{-i\phi} Y_{l m}(\theta,\phi) Y^*_{l'm'}(\theta,\phi) ~d\theta ~d\phi
$$
I've tried to use the definition of the ...
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Properties of Vector Spherical Harmonics
In section 5.3.2 of the book Advanced Classical Electromagnetism by Robert Wald, in deriving the multipole expansion for the retarded solution of electromagnetic field in presence of charge-current ...
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Working with decomposition of fields
I'm trying to follow a text I found online.
The author decomposes EM fields such
$$
\mathbf{E} = \sum_{lm}\left(f_l(r) \mathbf{Y}_{lm} - i \frac{l(l+1)}{r} g_l(r) \mathbf{\Psi}_{lm} - i\left(\frac{d ...
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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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Parity of the vector spherical harmonics?
The vector spherical harmonics can be defined as
$$\textbf{Y}_{j, \ell, 1}^m(\theta, \phi) = \sum_{m_{\ell}=-\ell}^{+\ell} \sum_{m_s=-1}^{+1}C_{\ell, m_{\ell}, 1, m_s}^{j, m_j} Y_{\ell, m_{\ell}}(\...
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Spheroidal (elliptical) waves: a boundary problem
I'm trying to solve a boundary problem. I have known EM waves, which "bounce" of a charged, rotating oblate spheroid (ellipsoid).
Now the formulation of the boundary problem is not so hard, ...
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Particle on a sphere, it's solution having extra $(-1)^m$ term
To solve a particle on a sphere problem in quantum mechanics we get the below equation :$\left[\frac{1}{\sin \theta} \frac{d}{d \theta}\left(\sin \theta \frac{d}{d \theta}\right)-\frac{m^{2}}{\sin ^{2}...
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How to make notation like $Y_{l m_{l}}(\theta, \phi)\chi_{m_s}$ more rigorous as a tensor product?
Sometimes in quantum mechanics we come across notation like $Y_{l m_{l}}(\theta, \phi)\chi_{sm_s}$ where $Y_{lm_l}$ is a spherical harmonic representing the spatial part of some particle wavefunction ...
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Spin, Orbital and Total Angular Momentum For Classical Vector Fields
I ask this question motivated by trying to understand vector spherical harmonics and find or come up with an elegant abstract derivation of their form.
Suppose we have a 3D field of 3D vectors: $\...
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Inner products with spherical harmonics in quantum mechanics
Let $|l,m\rangle$ be a simultaneous eigenstate of operators $L^2$ and $L_z$ and we want to calculate $\langle l,m|\cos(\theta)|l,m'\rangle$ where $\theta$ is the angle $[0,\pi]$. It is true that in ...
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Inverting a spherical harmonic expansion
I am referring to this paper by Srednicki, page 4, equations 14-17. Equation 14 is a Hamiltonian, which is to be written as $H = \sum_{lm} H_{lm}$. The $H_{lm}$ are given in equation (17).I am trying ...
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Momenta basis for a quantum system with configuration space $\mathbb{R}\mathrm{P}^2$
Let $\mathcal{X}$ be the configuration space of a (quantum) system.
When $\mathcal{X} = S^1 \simeq \mathrm{SO}(2)$, a momenta basis is $$\{ |\ell\rangle : \ell \in \mathbb{Z} \}.$$
When $\mathcal{X} ...
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Spherical harmonics and Bra-Ket notation
Let's assume i have a wave function $$\psi = N(x+y+z)e^{-r^2/a^2} $$ with $N$ and $a$ some constants. This function can be written as a sum of the spherical harmonix $Y_{1,0}$, $Y_{1,1}$ and $Y_{1,-1}$...
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Spherical harmonics visualization as oscillations on sphere
I‘ve watched this video where they visualize the spherical harmonics as oscillations. Since the spherical harmonics determine the spatial shape of the orbitals of the hydrogen atom their time ...
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How do we obtain radial part and angular part with spherical harmonics?
In a website (sorry, I can't find it anymore) I've read this when talking about spherical harmonics:
if we want to determine simultaneous eigenfuntions of $L^2$ and $L_z$
we have to solve this system:...
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Trying to understand the relationship between Hydrogen atom, spherical harmonics and central field force in quantum mechanics
I have a problem understanding three arguments in quantum mechanic:
When we talk about a particle in a central field we have this kind of Hamiltonian:
$$H=\frac{p^2}{2m}+V(r)$$
if we use spherical ...
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Indices In The Hyperspherical Harmonics
Definition
In $d$-dimensional space we have a hyperspherical coordinate system with angles $\theta_1, \theta_2, ..., \theta_{d-2}, \phi$. I am working with the following definition (up to ...
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Taking the mystery out of geopotential coefficients
Geopotential coefficients are usually notated Cnm and Snm (for example, see this IERS Tech Note).
My questions (in the order I care about them) are,
What are the constraints on n and m? Is n always ...
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$D$-dimensional Coulomb problem, is the generalization from the 3D case supposed to be simple? (Gilmore's Lie algebra, chapter 14, problem 20)
I am solving problem 20 of chapter 14 of Robert Gilmore's Lie groups, physics and geometry: An Introduction for Physicists, Engineers and Chemists, which focuses on the $D$-dimensional Coulomb problem,...
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