Questions tagged [spherical-harmonics]

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Discrepancy in Legendre function for angular momentum

On wikipedia: https://en.wikipedia.org/wiki/Associated_Legendre_polynomials It is clear that the formula to solve for the eigenfunctions of $\hat{L}^2$ are the spherical harmonics where: $$ Y_{l, m} ...
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84 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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What are the eigenfunctions of a spin 1/2 particle in an infinite spherically symmetric potential? [closed]

A spin $1/2$ particle in an infinite spherically symmetric potential is described by a hamiltonian $H= T+V(r)+(\omega _0/3\hbar)$$J^2$ $J=L+S$ What are the eigenfunctions of $H$? and what are the ...
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42 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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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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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 ...
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56 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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33 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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66 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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45 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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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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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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78 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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79 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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136 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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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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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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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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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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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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98 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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84 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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90 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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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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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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55 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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88 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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270 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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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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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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193 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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590 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 ...
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Quantization of electrons' angular momentum in atoms and molecules

It is known that the Schrödinger's equation of the electron's wave function in atoms can be solved analitically only when a single electron is present (the "hydrogenlike atom"). In that case, the ...
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257 views

Systematic expansion of $e^{i\vec{k}\cdot\vec{r}}$ in atomic physics in terms of Legendre polynomials and identifying different $l$ terms

In the context of light-matter interaction one often makes the approximation $e^{i\vec{k}\cdot\vec{r}}\approx 1$. Keeping higher order terms in $e^{i\vec{k}\cdot\vec{r}}$ give magnetic dipole, ...
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Directional Eigenket?

I'm reading Sakurai's Modern Quantum Mechanics. In page 99, there is a concept of Directional Eigenket. It doesn't have a definition, or any properties about it. Because the angular dependence is ...
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183 views

Linear combination of 2 spherical harmonic functions

The task is to form 2 linear combinations out of the 2 given spherical harmonic functions. I dont understand why the resultant wave function has to be multiplied with the constant $1/sqrt(2)$?
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Can spherical harmonics be used in relativity equations?

I have a neutral pion of mass $m_{\pi}$, and it decays into two photons. In it's reference frame the decay is isotropic. One of the photons has a helicity of $+\hbar$ and the other $-\hbar$. In ...
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293 views

What are the boundary conditions for the Hydrogen Atom which cause the polar power series to need to terminate?

I am trying to solve the Hydrogen Atom, and I am stuck in the polar equation. To simplify, I have taken the special case in which $m=0$ to get the Legendre Equation: $$(1-x^2)P''(x)-2xP'(x)+AP(x)$$ $$(...
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107 views

matrix elements of $\hat{z}$ operator under the angular momentum basis

I found a quite challenge quantum mechanics problem in a preparation sample test for a midterm. The sample test does not have a solution, so it is bothering. The question reads as follows: ...
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1answer
81 views

How to evaluate the matrix element of coulomb repulsion term between electrons in an atom suing spherical harmonics multipole expansion?

This is a lecture notes take from the following link on numerical calculation of atomic physics:http://www.phys.ubbcluj.ro/~lnagy/pdf/1curs.pdf I am trying to evaluate the two electron matrix element ...
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95 views

Musical Instrument that Exhibits Spherical Harmonics

A guitar string exhibits standing wave patterns when its struck, some superposition of sines and cosines, a drum head exhibits a superposition of Bessel functions when its struck. Is there any ...
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204 views

Definition of the spherical harmonics do not agree

I Griffiths' Introduction to quantum mechanics, the spherical harmonics are defined as $$Y_l^m(\theta,\phi) = \epsilon\sqrt{\frac{2l+1}{4\pi} \frac{(l-|m|)!}{(l+|m|)!}}e^{im\phi} P_l^m(\cos \theta)$$ ...
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183 views

Probability of finding a particle in the solid angle $d\Omega$ at $\theta$ and $\phi$ [closed]

For a spinless particle with the wavefunction \begin{equation} \psi(x,y,z)= K(x+y+2z)\exp(-\alpha r) \end{equation} with $r=\sqrt{x^2+y^2+z^2}$ and K and $\alpha$ are real constants. I have to ...
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310 views

What does the spherical-harmonic notation $Y^{m}_l(\hat{\textbf{r}})$ mean, and how does it relate to the usual $Y^m_l(\theta, \varphi)$?

By using the plane wave expansion, the decomposition of stationary harmonic plane wave into partial waves can be given by $$ e^{i\textbf{k}\cdot\textbf{r}} = e^{ikz} = e^{ikr\cos\theta} = \sum^{\infty}...
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110 views

Vector Spherical Harmonics and total angular momentum

In their book Akhiezer et al. give a definition of vector spherical harmonics (p.18 of Russian Edition) as $$\pmb{Y}_{j\ell m}(\pmb \Omega) = \sum_{m' \lambda} \langle \ell m' 1\lambda| jm \rangle Y_{...
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58 views

Putting the ket $|l,m_x \rangle$ in terms of the ket $|l, m_z \rangle$

Could someone guide me in my thought process of this problem? I don’t know if I’m thinking about it the right way. The problem is the following: I have a system which possible states are generated ...
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151 views

Orthonormal Basis integration and Kronecker delta

Given that this integral I'm trying to solve is $$\frac{2}{\pi}\sum^{\infty}_{l=0}\sum^{l}_{m=-l}\int_{r=0}^{\infty}\int_{k=0}^{\infty} R_{nl}(r)b_{lm}(k)j_{l}(kr)k^2 r^2 \int_{\theta = 0}^{\pi}\int_{\...