Questions tagged [spherical-harmonics]

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Expansion of Green's Function in Spherical Coordinates in Jackson

So I was reading about the expansion of the Green function in Spherical coordinates from Classical Electromagnetism by J.D. Jackson and I'm really confused about a subtle step that he makes to go from ...
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26 views

Understanding vector spherical harmonics

I am studying this classical paper by Regge and Wheeler Stability of a Schwarzschild singularity. In the second page they introduce their formalism with spherical harmonics and generalization thereof. ...
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32 views

Legendre Polynomials Integrals for Laplace Solutions

I know how to normalize Legendre polynomials, but I have a sphere with 0 to pi/3 boundaries where the potential is $V$, otherwise zero. For normalization it is -1 to 1, what changes with different ...
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39 views

Interpreting spherical harmonics angular momentum eigenstates

I'm trying to get some intuition behind the spherical harmonics being the angular momentum eigenstates in quantum mechanics. Firstly, am I correct in saying that we can imagine the angular momentum (|...
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Coordinates for coupled transition dipole moments

I am working on some calculations on coupled transition dipole moments interacting with light pulses with different polarisations. I am currently trying to find a way of writing $\theta_{\beta}$ and $\...
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41 views

Spherical harmonic expansion with radius

I'm having some trouble in computing the spherical harmonic's expansion of a general wave function $\psi(\textbf{x})$ where $\textbf{x}$ is a vector in 3D . The result on the article I'm working on is ...
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227 views

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 ...
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44 views

Nodes in Spherical Harmonics

In Rydberg Atoms book by Thomas F. Gallagher, Page 14, the author provides the general equation of spherical harmonics which is well-known. He states that there exist $l-m$ nodes in the $\theta$ ...
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91 views

In actuality, light can only approximate spherical waves, as it can only approximate plane waves

I am currently studying Optics, fifth edition, by Hecht. In chapter 2.9 Spherical Waves, the author says the following: The outgoing spherical wave emanating from a point source and the incoming wave ...
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23 views

The pulse has the same extent in space at any point along any radius $r$

I am currently studying Optics, fifth edition, by Hecht. In chapter 2.9 Spherical Waves, the author says the following: $$\dfrac{\partial^2}{\partial{r}^2}(r \psi) = \dfrac{1}{v^2} \dfrac{\partial^2}{...
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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{\...
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Why do we care about real form spherical harmonics?

I'm studying atomic orbitals and the shape is usually represented with real form spherical harmonics, taken as an appropriate linear combination of the complex ones. If, however, the physical quantity ...
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44 views

Eigenstate of spherical harmonics

$$ \psi(\theta,\phi) = 1/(2\sqrt{3\pi}) [ \sqrt{5} \cos\theta + \sin (\theta + \phi) + \sin(\theta -\phi) $$ Calculate $\hat{L}^2 \psi$ and $\hat{L}_z \psi$. Is $\psi$ an eigenstate of $\hat{L}^2$ and ...
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20 views

Integration over all directions in spontaneous emission, complex dipole element and polarization vectors

When calculating the rate of spontaneous emission in the Weisskopf-Wigner theory the time derivative of the excited state is related to a sum of couplings to all modes (directions and polarizations) ...
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81 views

How to express a rank-2 tensor as a spherical tensor?

A common example how to write a rank-2 tensor in the spherical basis is an outer product of two vectors, $$ T_{ij} = a_i b_j $$ such that $$ T_{ij} = \frac{\textbf{a}\cdot\textbf{b}}{3}\delta_{ij} + ...
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110 views

Electric dipole field and spherical harmonics

The field of the electric dipole is $\displaystyle\vec{E}=\frac{3(\vec{p}\cdot\hat{r})\hat{r}-\vec{p}}{4\pi\epsilon_{0}r^3}$, show that it can be written as linear combination of $\displaystyle\frac{...
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50 views

Spherical tensor operators relation to spin operators

I am reading a research paper discussing theoretical calculations of electron paramagnetic resonance parameters (EPR)*. In the section about higher-order EPR parameters, it states that according to ...
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53 views

How to decompose Cartesian into spherical tensor with Wigner-Eckart theorem?

I'm solving a question that required me to work with a Cartesian tensor, $A_i$, with Wigner-Eckart theorem. The book was Sakurai Modern Quantum Mechanics with information given in related posts: Ra....
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34 views

Potential between a metal sphere and spherical shell which are not quite concentric

I have been struggling with this problem for some time by now. An insulated metal sphere of radius $a$ with total charge $q$ is placed inside a hollow grounded metal sphere of radius $b$. The centre ...
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61 views

Analytical expressions for acceleration due to zonal harmonics of a gravitational field?

Wikipedia's Geopotential_model; The deviations of Earth's gravitational field from that of a homogeneous sphere discusses the expansion of the potential in spherical harmonics. The first few zonal ...
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36 views

Vector Spherical Harmonics

I've been looking at Mie scattering and I was having some trouble with the idea of Vector Spherical Harmonics on a conceptual level. From what I understand they are a basis set in spherical ...
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1answer
86 views

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 ...
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62 views

Eigenstates for a particle in a spherically symmetric potential

Consider a particle in a spherically symmetric potential. Write down a complete set of commuting observables for the system, the relevant eingenvalue equations, and the corresponding eigenstates in a ...
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22 views

What do the stationary states of a particle subject to a central potential look like?

What do the stationary-state, position wavefunctions of a particle in three dimensions subject to a time-independent central potential look like in spherical coordinates?
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24 views

Scalar hamiltonian and electromagnetic transitions

How can a Hamiltonian be a scalar and allow transitions between states with different angular momentum at the same time? Electromagnetic induced transitions are usually represented as a perturbation ...
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1answer
120 views

Transforming Cartesian Position Operators into Spherical Coordinates

Context: (not asking for solution) I'm attempting to show $\langle n,l',m'|\hat z|n,l,m \rangle = 0$ for $m\neq m'$ using the explicit form of $Y_{l,m}(\theta,\phi)$. Question: I wasn't sure how ...
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269 views

Why is quadruple moment zero for spherically symmetric charge distribution about centre?

How can we show that for a spherically symmetric charge distribution, the dipole, quadrupole and all higher moments about the centre of the distribution are identically zero. As we already know that ...
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1answer
85 views

In General Relativity, can I represent a Tetrad/Frame field in terms of ladder operators?

I've been interested in expressing the metric tensor $g$ in terms of it's harmonic expansions. In particular I'm interested in writing the tetrad/frame-fields in terms of such expansions. For ...
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142 views

Why is the azimuthal part of spherical harmonics in the form of $Ae^{im\phi}$, but not $Ae^{im\phi}+Be^{-im\phi}$?

The general solution to the azimuthal equation for a quantum-mechanical rigid rotor (spherical harmonics) $$\frac{d^2}{d\phi^2}\psi(\phi)=n^2\psi(\phi)$$ ($n^2$ is the separation constant) is given by ...
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40 views

Question on Kleinman-Bylander pseudopotentials

When Kleinman-Bylander pseudopotentials are used the Hamiltonian operator is given by $$\hat{H} = -\frac{1}{2}\nabla^2+V_{\textrm{local}}+\delta \hat{V}_{\textrm{NL}}$$ where $$\hat{V}_{\textrm{NL}} = ...
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76 views

Griffiths Quantum Mechanics - How can magnetic quantum number $m$ possibly be negative?

I apologize if this is already somewhere on this site, I searched with relevant keywords but could find nothing. This is from David Griffiths' Introduction to QM. In section 4.1.2 (Angular Equation),...
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Why in Landau the spherical harmonics had a multiple of $i^l$?

Quote Landau The Classical Theory of Fields Page 100 Equation 41.11 $$\displaystyle Y_{lm}(\theta,\phi)=(-1)^m i^l \sqrt{\frac{2l+1}{4\pi} \frac{(l-m)!}{(l+m)!} } P_l^{|m|}(\cos\theta) e^{im\phi} ...
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36 views

Why are some associated Legendre functions not orthogonal to each other?

For example, $P_1^1 = sin\theta$ and $P_2^2 = 3 sin^2\theta$ seem to be not orthogonal to each other because the integral $$\int_0^\pi (sin\theta)(3 sin^2\theta) sin\theta d\theta$$ is not zero. Am I ...
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13 views

What is the unit of the Angular Power Spectrum $C_\ell$ of Galaxy Clustering and Weak Lensing?

If the matter power spectrum $P_m(k,z)$ is measured in $h/Mpc^3$ and $k$ is measured in $h/Mpc$, what about its Fourier Transform $C_\ell$ and the conjugate vector to $k$, $\ell$? Thank you so much
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51 views

What happened to the factor of $\pi$ in this question?

$\\ $ I was going through the answer to this problem, when I noticed that a factor of $\pi$ in the denominator disappeared and a factor of 4 appeared in the numerator when the author started ...
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133 views

Eigenfunctions in Spherically Symmetric Well

I am looking at a problem that has a potential $$ V(r) = \begin{cases} 0 & a<r<b\\ \infty & \text{elsewhere} \end{cases} $$ This is a modification of the infinite spherical well ...
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106 views

Neumann boundary condition in spherical coordinates

I'm trying to solve heat equation $$\nabla^2 u = \frac{1}{k}\frac{\partial u}{\partial t}$$ in the region $$ a \leq r \leq b, \ \ \ \ 0 \leq \varphi \leq 2\pi, \ \ \ \ 0 \leq \theta \leq \theta_0 $$ ...
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196 views

Expanding the Green's function in spherical harmonics

Hello and thanks for reading. I'm currently working through electrodynamics from Purcell supplemented by Jackson and online notes. I've read up the basic cases demonstrating the method of image ...
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109 views

How do you apply the antisymmetrization operator?

I have an expression like, $Y^{L M_L}_{l_1 l_2}(\Omega_1, \Omega_2) = \sum_{m_1 m_2} \langle l_1 m_1l_2m_2|L m_L\rangle Y_{l_1m_1}(\Omega_1) Y_{l_2m_2}(\Omega_2)$ , as the angular part of a two ...
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36 views

Plasmon modes of cylinder metalic particle

Solving Laplace equation gives plasmon modes of spherical metalic particle radius $R$, plasma frequency $\omega_p$. Famous result is $l = 0, 1, 2, ...$ $$\omega_l = \sqrt{\frac{l}{2l+1}} \omega_p$$ ...
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70 views

If there are eigenstates of $L_z$ in a degenerate subspace, are there also eigenstates of $L^2$?

The question arises from an exercise but tackles deeper understanding of angular momentum operators. Suppose we have a 2D harmonic oscillator and an infinite square well in the third dimension: \...
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45 views

Matrix representation in angular momentum basis

I'm trying to find a way to verify that the following expansion is valid for any potential, including noncentral ones, $$ \langle \textbf{k}' |V|\textbf{k}\rangle = \frac2\pi\sum_{lm} V_l (k', k) Y_{...
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65 views

What is this angular momentum coupling notation? $\langle \ell 2 m_\ell 0|\ell 2\ell' m'_\ell\rangle \langle \ell 2 0 0|\ell 2\ell' 0\rangle$

I'm reading this unsigned powerpoint presentation of the Nilsson model in nuclear structure physics. On p. 15, they have this: $$\langle \ell'm'_\ell|Y_{20}|\ell m_\ell\rangle = i^{\ell-\ell'}\sqrt{\...
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93 views

Quantum mechanics angular momentum spherical tensor components

In Sakurai Quantum Mechanics, problem 3.25b we imagine $J_z^2$ as the component of a tensor with components $T_{ij} = J_iJ_j$. $J_z^2 = \frac{1}{3}\pmb{J}^2 + (J_z^2 - \frac{1}{3}\pmb{J}^2) $ The ...
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319 views

Expectation value of $x,y,z$ for general $nlm$ state of hydrogen atom

How to calculate expectation value of $\langle x\rangle, \langle y\rangle,\langle z\rangle$ for the general $\psi_{nlm}$ state? $x$ has $\sin(\theta)\cos(\phi)$ angular part which can be expressed as $...
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51 views

How to determine the best-fit magnetic dipole Gauss coefficient in a shifted coordinate system?

Introduction A planetary magnetic field $\vec{B}$ can be described outside of the planet using Gauss coefficients $g_n^m$ and $h_n^m$ and a spherical harmonic expansion: $$\vec{B} \;=\; -\vec{\...
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27 views

Depolarization factors of a drude metal plasmonic spheroid

A practice question asks for the depolarization factors $ L_{i=x,y,z} $ of a plasmonic spheroid made of a drude metal having the same resonance as the SPP resonance frequency. The answer turns out to ...
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223 views

Action Of Time-Reversal Operator On Spherical Harmonics

Given some spherical harmonic of the form $ \textbf{Y}_l^m = (i)^lY_l^m$ Where $Y_l^m$ is a standard spherical harmonic, I would like to find the action of the time-reversal operator $T$. My attempt ...
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How does the hydrogen atom actually “look like”? [duplicate]

When deriving the solutions for the "simple" quantum mechanical hydrogen problem, one normally uses spherical coordinates $(r,\theta,\phi)$, since the problem has rotational symmetry. The solution has ...
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642 views

In a spherically symmetric central potential why do we look for eigenfunctions of the angular momentum operator?

In finding the solutions to the wave equation for a spherically symmetric potential $V(r)$, we look for the eigenfunctions of $\hat L^2$ and $\hat L_z$ operators. However, what is the reasoning behind ...