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Questions tagged [spherical-harmonics]

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42 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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1answer
38 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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34 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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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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33 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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40 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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1answer
46 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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3answers
148 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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1answer
21 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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125 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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2answers
455 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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116 views

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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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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2answers
102 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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4answers
257 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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1answer
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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
62 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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75 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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2answers
158 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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1answer
139 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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219 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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1answer
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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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2answers
55 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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1answer
119 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_{\...
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Do relativistically-contracted electron states have the same energy and angular momentum values?

I've been reading that electron bound states are defined by four quantum numbers, $n$, $l$, $m_l$, and $m_s$, respectively the principal quantum number, the azimuthal quantum number, the magnetic ...
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3answers
145 views

Why are there three $p$-orbitals?

This question is specifically about Schrödinger quantum mechanics, but if an answer in some other mode would illuminate it could be acceptable, as demonstrating a physical or mathematical reason for ...
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1answer
181 views

What are spin-2 spherical harmonics and why are they needed?

A function $f(\theta,\phi)$ (with $\theta,\phi\in \mathbb{S}^2$) can be expanded in terms of spherical harmonics $Y_{l.m}(\theta,\phi)$. Recently, in this Particle Data Group review titled Cosmic ...
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1answer
125 views

Simplifying CMB correlation function with spherical harmonics

I originally asked this on the physics Stack Exchange site, but perhaps it could be more easily answered here. Given the definition of the correlation function for CMB temperature fluctuations as $$ ...
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2answers
344 views

Expectation Value $\langle \frac{1}{r^2} \rangle$ using Hellmann–Feynman theorem

Suppose we have the hydrogen atom$$ H ~=~ \frac{p_r ^2}{2m} + \frac{L^2}{2mr^2} -\frac{e^2}{r} \,.$$And have solved the Schrodinger equation finding $$E_n = - \frac{me^4}{2 \hbar ^2 n^2} $$ and $$ Ψ_{...
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44 views

Hydrogen Atom, polar equation eigenfunctions

In my textbook, Quantum Mechanics by David McIntyre on page 235, the solutions to the polar equation resulting from the separation of variables to the hydrogen atom are the eigenstates: The book ...
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0answers
43 views

What is a force multipole?

In a recent talk about physics and mechanics inside the cell, I heard such terms as 'force monopole' and 'force dipole'. What do such terms mean? Are they talking about the angular distributions, ...
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1answer
122 views

Increase in energy with increasing orbital quantum number

If we consider the problem in a spherically symmetric potential, we can see that with an increase in the orbital quantum number, the energy state in the spectrum increases. This is observed both in ...
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5answers
463 views

What is the geometry of DeBroglie standing waves?

I asked a similar question here . But never received a complete answer. So I've made the question more specific to DeBroglie waves. So from what I've read DeBroglie waves are indeed standing waves ...
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0answers
120 views

Integral of spherical harmonics and their derivatives

What is the Integral of the product of spherical harmonics and derivatives of spherical harmonics? More precisely, I am looking for $$\int_\Omega d\Omega\, Y_{l}^m Y_{l^\prime}^{m^\prime} \partial_\...
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2answers
441 views

Compact Formula for Wigner D-matrix

I have done an elementary calculation that appears to be giving me a simple (intuitive) formula for an arbitrary Wigner D-matrix. I can't seem to find this formula anywhere. In the following, $\...
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1answer
426 views

Why spherical harmonics are related to certain rotations (and not others)?

Let's take a direction eigenket $|{\bf\hat{n}}\rangle$ in 3-dimensional space oriented with angles $\theta\in\left[0,\pi\right]$ and $\phi\in\left[0,2\pi\right]$ in spherical coordinates. Next take ...
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1answer
447 views

Laplace equation azimuthal and non azimuthal symmetry

In general we can say that if the potential is specified in the surface of a sphere with azimuthal symmetry $(m=0)$ has a solution: $$ \Phi(r,\theta)= \sum_{l=0}^\infty \left[ A_lr^l + B_lr^{-(l+1)}\...
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2answers
334 views

Is there a relation between the Legendre generator function and Spherical Harmonics for a Potential?

Recently I had to solve a simple problem in which I had a sphere of radius $R$ with a constant potential (but with different sign), on both of the hemispheres, and I was asked to get the electrostatic ...
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1answer
428 views

How do multipole moments relate to a Taylor expansion, with regards to Newtonian potential?

Given the Newtonian gravitational potential, $$ \phi(\mathbf{x}) = - \int \frac{G \rho(\mathbf{x'})}{|\mathbf{x} - \mathbf{x'}|}$$ One can construct a 'multipole expansion' by using the Taylor ...
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0answers
180 views

Can you write the Einstein equation as an eigenvalue equation for analytical solutions (to local problems) on non-flat spacetimes

In reading about various local solutions to Einstein's field equation it is easy to forget that they almost all assume a flat background spacetime (at least asymptotically). Considering this made me ...
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1answer
194 views

Expanding Schwarzschild solution using spherical harmonics?

The well known Schwarzschild solution is given by \begin{equation} (ds)^2 = \left( 1 - \frac{2GM}{c^2r} \right) dt^2 - \left( 1 - \frac{2GM}{c^2r} \right)^{-1} dr^2 - r^2 d\Omega^2. \end{equation} ...
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1answer
286 views

Queries of Proof of Wigner-Eckart Theorem

With regard to the Wigner-Eckart Theorem the following is stated: The following is an outline of the proof in a text I am using: "Consider the action of a tensor-operator component on an angular-...
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1answer
352 views

What does it mean to normalize a combination of Spherical Harmonics?

Using the following as an example: Show that the combinations $$-\frac{1}{\sqrt{2}}\left(Y_{11}-Y_{1-1}\right)\quad\text{&}\quad\frac{i}{\sqrt{2}}\left(Y_{11}+Y_{1-1}\right)$$ are real and ...
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2answers
922 views

Irreducible form of Spherical tensor operators

In the section on spherical tensors in Sakurai, he introduces the idea of going from Cartesian tensors to irreducible spherical tensors. He states the following: A spherical harmonic can be written ...