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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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The expectation of $L^2_x$

In some problems, we use $$\langle L^2_x\rangle=\frac{1}{2}(\langle L^2\rangle-\langle L^2_ z\rangle)$$ But in other problems, we use $$\langle L^2_x\rangle=\frac{1}{4}\langle[L^2_+ + L^2_- +2(L^2-L^...
Suhail Sarwar's user avatar
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Rotation of spherical harmonics

I have a question about the rotation of spherical harmonics. In Wikipedia it is mentioned that if we make a rotation in 3D space: $R\vec{r}=\vec{r}'$,then the Spherical Harmonics can be written as a ...
Thanos Athanasopoulos's user avatar
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Magnetic parity and electric parity parts of solutions?

I'm currently reading the paper Conserved charges of the extended Bondi-Metzner-Sachs algebra by Flanagan and Nichols. In equation (2.15), the solution $$Y^A = D^A\chi + \epsilon^{AB}D_B\kappa$$ is ...
CactusSnow's user avatar
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Why do electrons fill the orbitals instead of lin.combination?

In a multi electron atom we start by filling the states 1s, 2s, 2p etc. For the 2p state, we have 2px,2py,2pz, and we fill each one with an electron. Since the 2p state is degenerate, I do not ...
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Wigner-Eckart theorem in classical physics?

The Wigner-Eckart theorem is a useful result in quantum physics and its many applications. Most presentations of this material in books on QM and online lecture notes seem to be variations on the same ...
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Decomposition of $(x \pm i y) \, Y_{l m}$ and $z \, Y_{l m}$ on spherical harmonics

Using the various algebraic properties of the associated Legendre polynomials $P_l^m(u)$ and of the spherical harmonics $Y_{l m}(\theta, \varphi)$, I was able to decompose the following expressions, ...
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Free particle in spherical coordinates

I'm trying to solve the very simple equation: $$-\frac{\hbar^2}{2m}\nabla^2 \psi = E\psi$$ but in polar coordinates. I used separation of variables to find out that my wave function is of the form: $$\...
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The $L^2$ operator not returning the expected value (2) when applied on the (2,1,0) Hydrogen wave function [closed]

Let the Hydrogen wave function for the state $n=2, l=1, m=0$ be described as: $$\psi_{210} =cos(\sigma )*f(r)$$ Since the squared angular momentum eigenvalue is $L^2=l(l+1)$, I would expect it to be 2 ...
Sergio Prats's user avatar
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Series Solution of Laplace Equation in Spherical Coordinates

I was recently Studying Griffiths Electrodynamics after a long time and there I saw the Laplace equation. Because it was my second time going over Griffiths so I thought maybe I should try to derive ...
Charu _Bamble's user avatar
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Question about the "axis of evil"in cosmology and the Doppler effect due to the solar system's motion

The cosmic microwave background (CMB) can be described by its anisotropies in a direction $\hat{n}$ in the celestial sphere $$ \delta T(\hat{n})=\frac{ T(\hat{n})-\bar{T}}{\bar{T}} $$ where $\bar{T}$ ...
P. C. Spaniel's user avatar
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Selection of spherical harmonics for the orbital calculation of electrons

As shown in this answer, the calculation of the electron probability in atoms is based on Spherical Harmonics and the resulting orbit probabilities usually have symmetry axes on the Cartesian ...
HolgerFiedler's user avatar
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Difference between Mutlipole moments calculated with normal integration and with Wigner-$D$ matrices

Im learning about Wigner-$D$ matrices and the applications to spherical harmonics. Now I wanted to test my knowledge, but i failed miserably :( (worked on this the whole day). So here is my problem: I ...
Stefan283's user avatar
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Scalar spherical harmonics in $S_n$

In the Kaluza klein reduction we can "decompose" the spacetime $M_n$ as $M_n = M_4 \otimes K_d$, in which $K_d$ is a compact spacetime. So, functions like a scalar $\phi(x,y)$ can be ...
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Boundary Conditions for Spherical Harmonics Problem

I have encountered a problem with electrostatics potentials. The problem is given as follows: A sphere of radius $𝑎$ has the potential $\Phi(a,\theta, \phi)$ at the boundary. Obtain expressions of ...
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Square Integrability of spherical symmetric wave

In my class we were discussing some wave equation for a spherical symmetric wave $u(t,r)$ and my professor investigated the behaviour of the solutions asymptotics $r \rightarrow \infty$. The solution ...
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Angular momentum completeness relation

Can anyone tell me if the angular momentum completeness relation is given by $$ \sum_{l=0}^{\infty} \sum_{m=-l}^{l} (2l + 1) |l,m\rangle\langle l,m| = I $$ or $$ \sum_{l=0}^{\infty} \sum_{m=-l}^{l} |l,...
Dr. user44690's user avatar
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Proving spherical harmonics' spherical symmetry and invariance under rotations?

I was told in a lecture the following relationship, where $Y_{\ell}^m$ are the spherical harmonics and the eigenfunctions of the two following operators: $$ L^2 Y_{\ell}^m = \ell(\ell + 1) \hbar^2 Y_{...
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Electric Potential due to an ellipsoid

Recently I came across the following problem: Suppose $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ is an ellipse with surface charge density $\sigma=\sigma_0\sin(\theta)\cos(\phi)$ where $\...
Amit Kumar Basistha's user avatar
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Legendre Polynomials as solutions for Laplace’s equation with azimuthal symmetry

(very amateur) physics student/enthusiast here. While reading Griffith’s textbook on electrodynamics I came across this line about separation of variables solutions for Laplace’s equations, assuming ...
Yoon Henri's user avatar
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Expansion coefficients in multipole expansion as spherical tensors

This is a relatively simple question, but I cannot find a clear answer: Given the multipole expansion of a real scalar function, $$ f(r,\theta,\phi) = \sum_{\lambda\mu} f_{\lambda\mu}(r) Y_{\lambda\mu}...
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Does the Wigner D-matrix suffer from gimbal lock?

Are there specific Euler angles and initial spherical vectors for which the D-matrix loses a degree of freedom, akin to gimbal lock in the conventional Euler rotation matrices? (Of course the ...
BenjaminDSmith's user avatar
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Commutator of raising operator in angular momentum with partial derivative wrt z

While fiddling around with certain commutation relations, i noticed the following relation while using spherical coordinates. What could this relation mean intuitively? Let me know if any information ...
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Function composition and multipole expansion

Assume that I have some function $g(r,\theta,\phi)$ which I have expanded in a multipole series: $$ g(r,\theta,\phi) = \sum_{\ell=0}^\infty \sum_{m=-\ell}^{+\ell} g_{\ell m}(r)\, Y_{\ell m}(\theta,\...
kc9jud's user avatar
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Why is this solution to the angular equation unphysical (spherical harmonics)? [duplicate]

I'm covering the Hydrogen atom wavefunctions $\psi(r,\theta,\phi)$ which are separated into three functions $R(r), \Theta(\theta), \Phi(\phi)$. ((The function $Y(\theta,\phi)$ is what we call a ...
mrGerman's user avatar
2 votes
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55 views

Struggling to prove this identity using $(L_{-})^n$ operators

I'm having difficulties proving the following identity using the lowering operators $(L_{-})^n$ in the context of spherical harmonics: $$ Y_{L}^{-M}(\theta, \phi)=(-1)^{M}Y^{*M}_{L}(\theta, \phi) $$ ...
LPR's user avatar
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Gradient Operator in Vector Spherical Harmonic Basis

The vector spherical harmonic basis (vector generalization to the scalar valued spherical harmonics) is a convenient spectral basis for problems involving vector fields with spherical symmetry (link ...
haricash's user avatar
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Relation beteen harmonic polynomials and spherical harmonics

Considering a spinless particle moving in the 3D space and described by the wavefunction $$\psi(\vec{x})=(x^3+y^3)\frac{e^\frac{-r}{2\lambda}}{r^2}$$ where $\lambda$ is a lengthscale. I want to know ...
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Expansion of interaction potential in terms of spherical harmonics in unconventional superconductivity

I am currently reading the book Introduction to Unconventional Superconductivity by V. P. Mineev and K. V. Samokhin. In Equation (1.6), the interaction potential $V$ is expanded in terms of spherical ...
caz's user avatar
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Paraxial Spherical wave emanating from point source

I'm reading through chapter 5.3.1(Impulse response of a Positive lens), in Goodman's "Fourier Optics"(p.109). An object is placed a distance $z_1$ in front of a lens. If we place a point ...
Sammy Apsel's user avatar
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Multipole expansion of a scalar quantity which is a function of a unit vector in 3D

I have a known scalar quantity $A(\mathbf{\hat{e}})$ which is a function of a unit vector in 3D \begin{equation} \mathbf{\hat{e}}=(e^1,e^2,e^3)=(\sin\theta\cos\phi, \sin\theta\sin\phi,\cos\theta). \...
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Relationship between spherical tensors and 3rd tensor power of cartesian tensors

I have some familiarity with these things from a course I took in Sakurai a few years ago. Cartesian tensors form a 3d irrep of $ \mathrm{SO}(3) $. The angular momentum operators $ J_+,J_-,J_z $ form ...
Ian Gershon Teixeira's user avatar
1 vote
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Spheroidal eigenvalues with shifted boundary conditions

I was studying the spheroidal differential equation in relation to calculating solutions for fields in a general Kerr background metric and, as far as I can tell, the eigenvalues $\lambda$ that enter ...
Marcosko's user avatar
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Cosmology: Covariance between Gaussian distributions for complex spherical harmonics coeficients

In the context of the computation of a variance about $a_{\ell m}$ spherical coefficients of Legendre, I am faced to an issue : There is a term $\langle \text{Re}(a)\text{Im}(a)\rangle$ that appears ...
guizmo133's user avatar
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Poisson noise on $a_{\ell m}$ complex number: real or complex?

In a cosmology context, when I add a centered Poisson noise on $a_{\ell m}$ and I take the definition of a $C_{\ell}$ this way : $C_{\ell}=\dfrac{1}{2\ell+1} \sum_{m=-\ell}^{+\ell} \left(a_{\ell m}+\...
guizmo133's user avatar
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Tangential component of the "electric" vector spherical harmonic $N_{lm} = \nabla \times (h X_{lm})$?

So I want to solve the problem of an electromagnetic plane wave scattering by a sphere myself, and one of the crucial steps is to solve the scattering of a converging TE- or TM-polarized vector ...
Denis Baranov's user avatar
3 votes
1 answer
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Can someone identify this vector representation of $\rm SO(3)$ in terms of multi-variable polynomials?

I am trying to get some deeper intuition for the representations of $\rm SO(3)$ and how they combine with each other, and I ran into an odd object that I'm hoping that folks here might help me ...
Emilio Pisanty's user avatar
1 vote
1 answer
106 views

Multipole expansion and spectral decomposition

We can always write a Hermitian operator in the form of its spectral decomposition: $\hat{A}=\sum_i \lambda_i | \chi_i(\boldsymbol{r})\rangle\langle\chi_i(\boldsymbol{r}')|$ where $\lambda_i$ are the ...
Guiste's user avatar
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Spherical coordinates using ladder operators [closed]

In an exercise I am dealing with, where I consider a system resembling a 3D quantum harmonic oscillator, it says that the transition to spherical coordinates can now be made in the operator language. ...
imbAF's user avatar
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Why is the square of the magnitude of a spherical harmonic function related to the zenith angle $θ$?

My intuition tells me that the wave function corresponding to a spherically symmetric potential should only depend on r (of course, intuition can be misleading), otherwise changing the orientation of ...
John Title's user avatar
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Calculating minimum $l$ energy in central potential problems using this generalization of the variational theorem?

The variational theorem talks about giving an upper bound on the lowest eigenvalue of a given Hermitian operator, and there is a simple generalization if we allow ourselves to constrain the space of ...
EE18's user avatar
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How to calculate angular velocity from forces on spherical pendulum head?

I have a bunch of forces that add up the force vector $(a_x, a_y, a_z)$ which is applied to the head of a spherical pendulum with given angles and angular velocities $ (\phi, \theta, \phi', \theta') $ ...
2080's user avatar
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1 vote
3 answers
625 views

Some intuitions for irreducible representations $SO(3)$ in classical physics

With a very naive and intuitive understanding of representation theory, I make my point below. Feel free to correct my intuition. In order to find out the representations of $SO(3)$ on a vector space $...
Solidification's user avatar
3 votes
2 answers
2k views

What are spherical tensors?

Following Sakurai, I know how the Cartesian components of a tensor transform under rotation, in classical physics and also in quantum physics. For example, the Cartesian components of a vector change ...
Solidification's user avatar
2 votes
0 answers
48 views

Orbital phase and gravitational wave phase

I am trying to understand the relation between the orbital phase of binary and the phase of the gravitational wave when expressed as spin-weighted spherical harmonics. The metric perturbation can be ...
Khushal's user avatar
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Using spherical harmonics for the charged interaction of particles

I am currently inversitgating a system consists of positive and negative point particles satisfying charge neutrality condition. They have a unit charge +1 and -1. In most case, we deal with coulomb ...
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1 vote
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How to write a hamiltonian in minimal basis as a spherical tensor

I am reading this paper where the authors write the atom-blocks of the hamiltonian in a minimal basis set and use some regression technique to fit the hamiltonian to data. This question is about how ...
Mikkel Rev's user avatar
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2 votes
1 answer
216 views

Probability and the Magnetic Quantum Number

I am currently self-studying quantum mechanics, and I'm working problems on angular momentum. The problem I'm currently working on asks one to consider a particle subjected to a spherically symmetric ...
kandb's user avatar
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2 votes
1 answer
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How are multipolar expansion of earth magnetic field computed?

In the study of the geomagnetic field, an expansion in spherical harmonics is used to represent the scalar magnetic potential: the first terms give the dipole approximation, then the quadrupole, etc. ...
Weier's user avatar
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4 votes
1 answer
139 views

Angular Momentum Operators and Spherical Harmonics in Higher Dimensions

Suppose we have a $d$-dimensional quantum system with a rotationally symmetric Hamiltonian $\hat{H}$. Extrapolating from the two and three dimensional cases, one might expect that the eigenstates of $\...
Andrew's user avatar
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0 answers
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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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