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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How to linearly combine the components of $\ell=2$ Cartesian tensor to ensure that the transformation matrix is orthogonal? [migrated]
A Cartesian tensor of rank 2 should can be decomposed into 3 irreducible part:
$$X=\frac{1}{3}tr(X)I + \frac{1}{2}(X-X^T) + \frac{1}{2}(X+X^T-\frac{2}{3}tr(X)I)$$
and $S=\frac{1}{2}(X+X^T-\frac{2}{3}...
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Difference between Mutlipole moments calculated with normal integration and with Wigner-$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 ...
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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
$$
\sum_{l=0}^{\infty} \sum_{m=-l}^{l} |l,...
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Proving spherical harmonics' spherical symmetry and invariance under rotations?
I was told in a lecture the following relationship, where $Y_{\ell}^m$ are the spherical harmonics and the eigenfunctions of the two following operators:
$$ 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 $\...
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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 ...
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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,
$$
f(r,\theta,\phi) = \sum_{\lambda\mu} f_{\lambda\mu}(r) Y_{\lambda\mu}...
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Does the Wigner D-matrix suffer from gimbal lock?
Are there specific Euler angles and initial spherical vectors for which the D-matrix loses a degree of freedom, akin to gimbal lock in the conventional Euler rotation matrices?
(Of course the ...
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Commutator of raising operator in angular momentum with partial derivative wrt z
While fiddling around with certain commutation relations, i noticed the following relation while using spherical coordinates.
What could this relation mean intuitively? Let me know if any information ...
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Function composition and multipole expansion
Assume that I have some function $g(r,\theta,\phi)$ which I have expanded in a multipole series:
$$
g(r,\theta,\phi) = \sum_{\ell=0}^\infty \sum_{m=-\ell}^{+\ell} g_{\ell m}(r)\, Y_{\ell m}(\theta,\...
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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 ...
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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:
$$
Y_{L}^{-M}(\theta, \phi)=(-1)^{M}Y^{*M}_{L}(\theta, \phi)
$$
...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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}+\...
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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 ...
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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 ...
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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 ...
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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.
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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 ...
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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 ...
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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') $ ...
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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 $...
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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 ...
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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 ...
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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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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 ...
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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 ...
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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.
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Angular distribution of two-body decay with orbital angular momentum
I'm currently studying the two-body decay where a particle (Be) with spin-1 decays into a pion (spin-0) and spin-1 particle $X$. The system is prepared so that the Be is polarized with a spin ...
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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 $\...
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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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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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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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