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Questions tagged [spherical-harmonics]

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1answer
289 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
390 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
978 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 ...
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1answer
1k views

Position operator in spherical basis

In a set of notes it states that we can define spherical basis vectors in terms of the Cartesian basis vectors $\hat{x}, \hat{y}$ and $\hat{z}$ by $$\hat{e}_{\pm 1} := \mp \frac{1}{\sqrt{2}}(\hat{x} \...
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0answers
79 views

Spherical Formulation of Quantum Mechanics

I always wondered, during my QM courses, if we don't explore enough of the freedom that the Lagrangian and Hamiltonian Classical Dynamics give us. Classically, we can always make canonical ...
4
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1answer
607 views

The dipole radiation pattern and spherical harmonics $Y_{10}$

I am studying the multipole expansion of electromagnetic wave radiation pattern, and it is said that any fields can be decomposed into the spherical harmonics $Y_{lm}$. However, for $l=1$, which ...
1
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1answer
157 views

Is it a coincidence or a convention that the spherical harmonics $Y_{\ell m}(\theta, \phi)$ are eigenstates of $L_z$?

The $z$-component of the angular momentum $L_z$ is an eigenstate of the spherical harmonics $Y_{\ell m}(\theta, \phi)$. The spherical harmonics $Y_{\ell m}(\theta, \phi)$ are also the eigenstates of ...
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1answer
397 views

Monopole Spherical Harmonics

I was following Yakov Shnir his book about magnetic monopoles, there they derive the monopole spherical harmonics. I will sketch the derivation briefly, The starting equation is the eigenvalue ...
1
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2answers
444 views

Spherical harmonics and angular momentum

I'm studying the relation between rotation matrices and spherical harmonics $\langle\hat{r}|l,m\rangle = Y_{lm}(\theta,\phi)$. In one demonstration it is stated that if $|\hat{r}\rangle=|\hat{e}_z\...
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2answers
160 views

Why are circular unit vectors often defined as $\hat{\mathbf e}_\pm = \mp (\hat{\mathbf e}_x \pm i \hat{\mathbf e}_y)/\sqrt{2}$

When dealing with circular polarizations, spherical harmonics, and generally with any vector-valued rotationally-invariant quantity, it is often a requirement to define complex-valued unit vectors of ...
2
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0answers
398 views

Meaning of different Stokes' coefficients in harmonic expansion of gravitational potential

I know that geophysical quantities, such as gravitational potential, can be written in terms of spherical harmonics: $$W(r,\phi,\lambda;t) = \frac{G m_e}{r} + \frac{G m_e}{r} \sum_{l=2}^{N_{max}}\left(...
1
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1answer
416 views

Can a standing wave exist on a spherical surface?

I've often seen the DeBroglie wave illustrated by a two dimensional surface as a standing wave, but then the 'electron cloud' surrounding an atom is hardly two dimensional and furthermore held to the ...
2
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0answers
166 views

Is there an intuitive interpretation of the shape of the angular momentum eigenstate?

I was watching a MIT lecture video on angular momentum eigenstate. Toward the end of the lecture, the professor had shown some plots of the first few spherical harmonics, in an attempt to explain ...
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0answers
47 views

Angular power spectrum: Calculating bias from N weighted events

I'm interested in calculating the angular power spectrum $C_{l,N,\omega}$ of $N$ weighted (weight $\omega_i$ for event $i$) events from a full sky map with distribution $C_l$? Interesting quantities ...
6
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2answers
600 views

How to calculate the angular momentum states of isotropic quantum harmonic oscillator?

While trying to calculate the angular momentum states for the first non trivial even and odd states ($N=2$ and $N=3$). When $N=n_x + n_y + n_z$ By solving the radial problem one can see that there 6 ...
3
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1answer
1k views

Klein Gordon equation in Schwarzschild spacetime (spherical harmonic mode expansion)

My Question: In his GR text, Robert Wald claims that solutions, $\phi$, to the Klein-Gordon equation, $\nabla_a\nabla^a \phi = m^2 \phi$, in Schwarzschild spacetime can be expanded in modes of the ...
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176 views

EM Field Quantization in Spherical polars

Is it possible to quantize the electromagnetic field in spherical polar coordinates instead of cartesian ones? Such that creation and annihilation operators correspond to harmonic oscillator modes ...
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2answers
862 views

Which direction do electrons orbit around/near the nucleus differ in aligned magnetic atoms?

Atoms can be aligned in magnets to create magnetic fields. Does that alignment give an atom a north and south pole or certain atoms have a unique electron orbitals giving an atom a north and south ...
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1answer
1k views

Why do we need the Condon-Shortley phase in spherical harmonics?

I'm confused with different definitions of spherical harmonics: $$Y_{lm}(\theta,\phi) = (-1)^m \left( \frac{(2l+1)(l-m)!}{4\pi(1+m)!} \right)^{1/2} P_{lm} (\cos\theta) e^{im\phi}$$ For example here ...
3
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1answer
179 views

Do translation formulae for generalised solid spherical harmonics exist?

I'm aware of the solid spherical harmonics functions, which are basically the surface spherical harmonics $Y^m_{\ell}(\theta,\varphi)$ with an additional monomial term along the radial direction: $R^...
2
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1answer
35 views

Choosing $A_l=0$ to guarantee bounded potential in infinity

I'm taking a course in Electrodynamics and quite often, when using the spherical approach $$\Phi=\sum\limits_{l~=~0}^{\infty}\left(A_lr^l+B_lr^{-(l+1)}\right)P_l(\cos\gamma),$$ there's the argument ...
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1answer
955 views

Express how to express cartesian electric field vectors in terms of spherical electric field components

I'm working on Mie scattering from a dielectric sphere. I have expressions for electric fields in terms of $E_{r}, E_{\theta}, E_{\phi}$ however now I would like to visualize these on cartesian planes ...
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1answer
49 views

Why am I not getting the correct spherical harmonic? [closed]

I defined $L_x$ and $iL_y$ in Mathematica. Then I used them to operate on $Y_2^1(\theta, \phi)$. Finally, I used $A_l^m = \hbar \sqrt{l(l+1)-m(m+1)}$. The code: But for some reason, my code returns $...
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1answer
888 views

Why is $\langle x \rangle =0$ for the ground state hydrogen atom?

From Griffiths, Introduction to Quantum Mechanics, 2nd ed: I found $\langle r \rangle =\frac{3a}{2}$ and $\langle r^2 \rangle =3a^2$. Now I need to find the expectation value of x. However, I don't ...
2
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1answer
482 views

Why do closed shells have a spherical symmetric charge density?

When repeating multi-electron atoms, I often read that closed shells have a symmetric charge density. It was justified by the following addition theorem for spherical harmonics everytime: $\sum\...
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0answers
50 views

Validity of analysing spherical harmonics in real-space using the probability amplitude [closed]

While looking up spherical harmonics (on the validity of analysing them in real-space in a transition metal crystal structure), I came across this: http://shpenkov.janmax.com/hybridizationshpenkov.pdf ...
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3answers
1k views

How do you take the inner product of a spherical Bessel function?

The equation \begin{equation} (r^2)R''+2rR'+[(kr)^{2}-l(l+1)]R=0, \end{equation} where $R=R(r)$, $R'=\frac{\mathrm{d}R}{\mathrm{d}r}$, etc. and $k, l$ are constants, is the Spherical Bessel equation ...
3
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1answer
135 views

Are there higher genus equivalent of spherical harmonics?

So the equation of a surface with topology S2 can be expanded out in terms of spherical harmonic functions. (I believe). A torus T2 which is just S1xS1 can be expanded out in terms of ordinary ...
0
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2answers
353 views

Normalization of time-independent Schroedinger equation in Spherical Coordinates

I have a short question about the time-independent Schroedinger equation in Spherical coordinates: $$\psi(r, \theta, \phi) = R(r)Y(\theta, \phi)$$ then the normalization condition becomes $$\int |\...
0
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2answers
508 views

Multipole expansion of the electromagnetic field

In Jackson's Classical Electrodynamics, section 9.7, he develops the multipole expansion of the electromagnetic fields in terms of the vector spherical harmonics and the spherical Bessel and Hankel ...
1
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1answer
2k views

Selection rules for electric quadrupole radiation

The selection rules for electric quadrupole radiation in a Hydrogen-like atom are: $$ \begin{aligned} \Delta l &= 0,\pm2 \hspace{1cm}(l=0\leftrightarrow l'=0 \textrm{ is forbidden}) \\ \Delta m &...
8
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1answer
556 views

Eigenvalues of spherical harmonics in $d$ dimensions

I'm working on the Schrodinger equation for a hydrogen atom in a $d$-dimensional space, so I'm interested in the possible eigenvalues of the angular momentum part of the $d$-dimensional Laplace ...
0
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2answers
312 views

Energy of central potential in QM

A hydrogen atom (Coulomb potential) has energy that only depends on $n$ (if we ignore other effects like spin-orbit coupling). In general (not necessarily Coulomb, can be any V), does $E$ depend on ...
1
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1answer
631 views

Conservation of energy in a sound wave

I have two ultrasonic transducers, an emitter and a receiver, and I'd like to know how the energy of the spherical wave is conserved. I guess the energy is proportional to its amplitude and it ...
5
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1answer
770 views

Regular solution vs irregular solution

My Quantum Mechanics textbook (John S. Townsend's A Modern Approach to Quantum Mechanics) mentions regular solutions and irregular solutions. It claims that regular solutions (at the origin) to the ...
0
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1answer
219 views

Orbital angular momentum eigenstates in the $|\mathbf{r}\rangle$ representation

Consider the orbital angular momentum operators $L^2$ and $L_z$. In the $|\mathbf{r}\rangle$ representation using spherical coordinates those operators actions are given by $$L^2\varphi(\mathbf{r})=-\...
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2answers
153 views

Please help me with this doubt from spherical waves

How to calculate phase difference for spherical waves? How to say whether they are in phase or out of phase? In sinusoidal we can easily say whether they are in phase or out of phase just by looking ...
3
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0answers
178 views

Conventions in defining spherical harmonics and associated Legendre polynomials

Relevant Background Spherical harmonics are defined with several different conventions: the definition used in quantum mechanics according to Wikipedia is $Y_l^{\,m}(\theta,\phi) = (-1)^m\sqrt{\frac{...
2
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1answer
325 views

Analogy to Fourier transform in spherical coordinates with boundary at a certain radius

Suppose, we have a wavefuction $\phi(\vec{x})$ which is restricted in a sphere, with the spherical boundary condtion $$\phi(\vec{x}=R)=\phi_0.$$ How can I do the 'Fourier transformation' as the case ...
1
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1answer
53 views

Clarification about first spherical harmonic

If the quantum numer $l$ is equal to zero, each components of L has eigenvalues $m=0$. So the states that have $l=0$ are simulaneous eigenvectors of each components of L. But, my textbook says, it is ...
1
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1answer
322 views

Operators associated to spherical harmonic functions

$$ \newcommand{\op}[1]{\hat{#1}} \renewcommand{\vec}[1]{\mathbf{#1}} \newcommand{\ket}[1]{|{#1}\rangle} \newcommand{\bra}[1]{\langle {#1}|} $$ Let's say we have a spherically symmetric system, like a ...
2
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1answer
316 views

How are spherical harmonics useful outside class? [closed]

I've learned about spherical harmonics (Legendre polynomials $\longrightarrow$ Associated Legendre polynomials $\longrightarrow$ orthogonality relations $\longrightarrow$ normalization coefficient(s) $...
0
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3answers
165 views

Quick way to compute $\langle n^{'}l^{'}m^{'}|r^k|nlm \rangle$, $k \in I$; $|nlm\rangle$ is $H$ atom eigenfunction [closed]

I want to compute quickly (using maybe some scaling arguments) $\langle n^{'}l^{'}m^{'}|r^k|nlm\rangle$, where $k \in I$. $|nlm \rangle$ is the eigenfunction of the Hydrogen atom ($H$). Example: ...
0
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1answer
428 views

Confusing concepts in proof of spherical addition theorem

In http://scipp.ucsc.edu/~haber/ph116C/SphericalHarmonics_12.pdf, section 4, pages 6..9 is a proof of the spherical harmonics addition theorem. Page 8 has eq.(25), an application of Laplace series: ...
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0answers
191 views

Units of angular power spectrum

I am not sure whether this question might be better suited for crossvalidated or stackoverflow, but I will give it a try here: I have a map of the full sky in in the healpix format, units are $\rm ...
1
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0answers
131 views

conducting hollow sphere in magnetic monopole

if a hollow copper sphere(or any conducting hollow sphere) is connected to dc at points diametrical and a magnetic monopole is right at the center of the sphere then will there be any movement of the ...
2
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0answers
188 views

The product of three spherical harmonics in higher dimension

As I see, e.g., in this question, a closed formula for the following integral \begin{equation} \int_{\mathbb{S}^N} Y_{\ell_1\ldots \ell_N}\,Y_{\ell'_1\ldots\ell'_N}\,Y_{\ell''_1\ldots \ell''_N}\, d^N ...
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2answers
346 views

How can mean value of a quantity $be$ an operator?

In Laundau & Lifshitz Quantum Mechanics. Non-relativistic theory in $\S29$ a problem is given: PROBLEM Average the tensor $n_in_k-\frac13\delta_{ik}$ (where $\mathbf{n}$ is a unit vector along ...
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2answers
1k views

Degeneracy of spherical harmonics eigenfunctions

I'm working through Griffiths' Introduction to Quantum Mechanics (2nd edition) and I'm trying to solve problem 4.24 b. In this problem, you're supposed to first find the normalized eigenfunctions to ...
4
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1answer
491 views

Calculating the angular power spectrum of a section of sky

To calculate the angular power spectrum $C_l$ of the whole sky, one uses the variance of the coefficients of the spherical harmonics in the temperature fluctuation field. I.e. $$C_l = \frac{1}{2l+1}\...