Spherical harmonics Definition and 126 Threads

1) 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}+\bar{a}_{\ell m}^{p}\right)\left(a_{\ell m}+\bar{a}_{\ell m}^{p}\right)^* ## Is Poisson...

Are real spherical harmonics better than complex spherical harmonics for hexagonal symmetry, which are directly associated to a finite Lz?

I tried using the Wigner matrices: $$\sum_{m'=-2}^{2} {d^{(2)}}_{1m'} Y_{2; m'}={d^{(2)}}_{1 -2} Y_{2; -2} + {d^{(2)}}_{1 -1} Y_{2; -1} + ...= -\frac{1-\cos(\beta)}{2} \sin(\beta) \sqrt{\frac{15}{32 \pi}} \sin^2(\theta) e^{-i \phi} + ...$$ where $$\beta=\frac{\pi}{4}$$. But I don't know if...

Hello, I have the demonstration below. A population represents the spectroscopic proble and B the photometric probe. I would like to know if, from the equation (13), the correlation coeffcient is closed to 0 or to 1 since I don't know if ##\mathcal{N}_{\ell}^{A}## Poisson noise of spectroscopic...

We assume two galaxy population, ##\mathrm{A}## and ##\mathrm{B}##; the corresponding maps have the following ##a_{\ell m}## : ## \begin{aligned} &a_{\ell m}^{A}=b_{A} a_{\ell m}^{M}+a_{\ell m}^{p A} \\ &a_{\ell m}^{B}=b_{B} a_{\ell m}^{M}+a_{\ell m}^{p B} \end{aligned} ## Here, ##b_{A}## and...

What combination of resources can you recommend for introducing people to spherical harmonics? Assume that the audience has the mathematical maturity of first-year grad students in mathematics, and will want a decent introduction to the theory and constructions. But also assume that this is part...

I've tried figuring out commutation relations between ##L_+## and various other operators and ##L^2## could've been A, but ##L_z, L^2## commute. Can someone help me out in figuring how to actually proceed from here?

Let ##|l,m\rangle## be a simultaneous eigenstate of operators ##L^2## and ##L_z## and we want to calculate ##\langle l,m|cos(\theta)|l,m'\rangle## where ##\theta## is the angle ##[0,\pi]##. It is true that in general ##\langle l,m|cos(\theta)|l,m'\rangle=0## ##(1)## for the same ##l## even if...

I have a problem of understanding in the following demo : In a cosmology context with 2 probes (spectroscopic and photometric), let notice ##a_{\ell m, s p}## the spectroscopic and ##a_{\ell m, p h}## the photometric coefficients of the decomposition in spherical harmonics of the distributions...

It is in cosmology context but actually, but it is also a mathematics/statistical issue. From spherical harmonics with Legendre deccomposition, I have the following definition of the standard deviation of a ##C_\ell## noised with a Poisson Noise ##N_p## : ## \begin{equation}...

Using the generating function for the legendre polynomial: $$ \sum_{n=0}^{\infty} P_{n}(x) t^{n}=\frac{1}{\sqrt{1-2 x t+t^{2}}} $$ It's possible to expand the coulomb potential in a basis of legendre polynomials (or even spherical harmonic ) like this: $$ \begin{aligned} &\frac{1}{\left.\mid...

1) If I take as definition of ##a_{lm}## following a normal distribution with mean equal to zero and ##C_\ell=\langle a_{lm}^2 \rangle=\text{Var}(a_{lm})##, and if I have a sum of ##\chi^2##, can I write the 2 lines below (We use ##\stackrel{d}{=}## to denote equality in distribution)...

Hello, In the context of Legendre expansion with ##C_\ell## quantities, below the following formula which is the error on a ##C_\ell## : ##\sigma_(C_{\ell})=\sqrt{\frac{2}{(2 \ell+1)\Delta\ell}}\,C_{\ell}\quad(1)## where ##\Delta\ell## is the width of the multipoles bins used when computing...

To show ##Y_{1,1}(\theta,\phi)## is an eigenfunction of ##\hat{L}^2## we operate on ##Y_{1,1}(\theta,\phi)## with ##\hat{L}^2## \begin{equation} \hat{L}^2Y_{1,1}(\theta,\phi)=\hat{L}^2\Big(-\sqrt{{\frac{3}{8\pi}}}sin\theta e^{i\phi}\Big) \end{equation} \begin{equation}...

See the first post in the previous thread ‘Matrices from Spherical Harmonics with Eigenvalue l+1’ first. Originally when I came across the Lxyz operator and the Rlm matrices I had a different question. If this had to do with something like the quantum Hydrogen atom then why did it appear that...

As the subject title states, I am wondering how would one go about transforming Cartesian coordinates in terms of spherical harmonics. To my understanding, cartesian coordinates can be transformed into spherical coordinates as shown below. $$x=\rho \sin \phi \cos \theta$$ $$y= \rho \sin \phi...

My first attempt revolved mostly around the solution method shown in this "site" or PowerPoint: http://physics.gmu.edu/~joe/PHYS685/Topic4.pdf . However, after studying the content and writing down my answer for the monopole moment as equal to ##\sqrt{\frac{1}{4 \pi}} \rho##, I found out the...

I’m New to the forum. I’m Interested if a certain set of matrices have any significance. To start out the unit vectors ##\vec i , \vec j, and ~\vec k ## are replaced with two dimensional matrices. ##\sigma r = \begin{pmatrix}1&0\\0&1\\\end{pmatrix}, ~\sigma z = \begin{pmatrix}1&0\\...

Im trying to solve the equation 62.7 of this numerical on mathematica. Whenever i try to normalized the function it shows function diverges. As the Bessel function contains trigonometry term so it diverges. I don't know how to solve the integral. Can i use the hydrogen atom wavefunction in exp...

In the contex of ##L^2## space, it is usually stated that any square-integrable function can be expanded as a linear combination of Spherical Harmonics: $$ f(\theta,\varphi)=\sum_{\ell=0}^\infty \sum_{m=-\ell}^\ell f_\ell^m \, Y_\ell^m(\theta,\varphi)\tag 2 $$ where ##Y_\ell^m( \theta , \varphi...

My teacher said me this commutator is zero because the spherical harmonics are eigenfunctions of L^2. Actually, he said that any operator must commute with its eigenfunctions. I tried an example: [L^2,Y_20] expressing L^2 on spherical coordinates and I determined this commutator is not zero...

Hi PF! When solving the Laplace equation in spherical coordinates, the spherical harmonics are functions of ##\phi,\theta## but not ##r##. Why don't they include the ##r## component? Thanks!

I'm expanding a function in spherical harmonics. I want to conserve axisymmetry of the function. what harmonics would respect that? Should I only include m=0 terms?

In this video, at around 37:10 he is explaining the orthogonality of spherical harmonics. I don't understand his explanation of the \sin \theta in the integrand when taking the inner product. As I interpret this integral, we are integrating these two spherical harmonics over the surface of a...

Homework Statement A sphere of radius a has V = 0 everywhere except between 0 < θ < π/2 and 0 < φ < π. Write an expression in spherical harmonics for the potential for r > a. For which values of m are there contributions? Determine the contributions through l= 2. How could you determine the...

Homework Statement I wish to solve this integral $$b_{lm}(k) = \frac{1}{2(\hbar)^{9/4}(2\pi)^{5/2}\sqrt{\sigma_{px} \sigma_{py} \sigma_{pz}}} \int_{\theta_k = 0}^{\pi}\int_{\varphi_k = 0}^{2\pi} i^l \text{exp}\left[ - \frac{1}{(2\hbar)^2}\left(\frac{(k_z - k_{z0})^2}{\sigma_{pz}^2} + \frac{(k_y...

Hi physics forms! I'm practicing to for an Quantum mechanics exam, and i have a problem. 1. Homework Statement I have two problems, but it's all related to the same main task. I have a state for the Hydrogen: ## \Psi = \frac{1}{\sqrt{2}}(\psi_{100} + i \psi_{211})## where ## \psi_{nlm}##...

Homework Statement The spherical harmonic, Ym,l(θ,φ) is given by: Y2,3(θ,φ) = √((105/32π))*sin2θcosθe2iφ 1) Use the ladder operator, L+ = +ħeiφ(∂/∂θ+icotθ∂/∂φ) to evaluate L+Y2,3(θ,φ) 2) Use the result in 1) to calculate Y3,3(θ,φ) Homework Equations L+Ym,l(θ,φ)=Am,lYm+1,l(θ,φ)...

The time-dependent Schrodinger equation is given by: ##-\frac{\hslash^{2}}{2m}\triangledown^{2}\psi+V\psi=i\hslash\frac{\partial }{\partial t}\psi## Obviously, there is a laplacian in the kinetic energy operator. So, I was wondering if the equation was rearranged as...

Homework Statement Hello, I have a general question regarding to coefficient matching when spanning some function, say , f(x) as a linear combination of some other basis functions belonging to real Hilbert space. Homework Equations - Knowledge of power series, polynomials, Legenedre...

Hello. I was recently reading Barton's book. I reached the part where he proved that in spherical polar coordinates ##δ(\vec r - \vec r')=1/r^2δ(r-r')δ(cosθ-cosθ')δ(φ-φ')## ##=1/r^2δ(r-r')δ(\Omega -\Omega')## Then he said that the most fruitful presentation of ##δ(\Omega-\Omega')## stems from...

I found an interesting thing when trying to derive the spherical harmonics of QM by doing what I describe below. I would like to know whether this can be considered a valid derivation or it was just a coincidence getting the correct result at the end. Starting making a Fundamental Assumption...

How does one arrive at the equation $$\bigg( (1-z^2) \frac{d^2}{dz^2} - 2z \frac{d}{dz} + l(l+1) - \frac{m^2}{1-z^2} \bigg) P(z) = 0$$ Solving this equation for ##P(z)## is one step in deriving the spherical harmonics "##Y^{m}{}_{l}(\theta, \phi)##". The problem is that the book I'm following...

My general question is: What is the angular power spectrum C_{l,N,ω} of N weighted (weight ω_i for event i) events from a full sky map with distribution C_l? I'm interested in: Mean of C_{l,N,ω}: <C_{l,N,ω}> Variance of C_{l,N,ω}: Var(C_{l,N,ω}) The question is important, since we observe in...

Hi everyone. I'm looking for a derivation of the Spherical Harmonics that result in the form below given in Sakurai's book. I looked up on web and I found just that these are related with Legendre Polynomials. Has anyone a source, pdf, or similar to indicate me? (I would appreciate a derivation...

Homework Statement Here is a copy of the pdf problem set {https://drive.google.com/open?id=0BwiADXXgAYUHOTNrZm16NHlibUU} the problem in question is problem number 1 which asks you to prove the orthonormality of the spherical Harmonics Y_1,1 and Y_2,1. Homework Equations Y_1,1 =...

Homework Statement This is a (long) multi-part question working through the various stages of solving the radial Schrodinger equation and as such it would be impractical to type it all out here but I will upload the pdf (https://drive.google.com/open?id=0BwiADXXgAYUHOTNrZm16NHlibUU) of the...

Homework Statement Essentially we are describing the ODE for the radial function in quantum mechanics and in the derivation a substitution of u(r) = rR(r) is made, the problem then asks you to show that {(1/r^2)(d/dr(r^2(dR/dr))) = 1/r(d^(2)u/dr^2) Homework Equations The substitution: u(r) =...

The normalized angular wave functions are called spherical harmonics: $$Y^m_l(\theta,\phi)=\epsilon\sqrt{\frac{(2l+1)}{4\pi}\frac{(l-|m|)!}{(l+|m|)!}}e^{im\phi}*P^m_l(cos\theta)$$ How do I obtain this from this(http://www.physics.udel.edu/~msafrono/424-2011/Lecture 17.pdf) (Page 8)? The...

Homework Statement In t=0, wave function of the particle that moves freely on the surface of the sphere has the wave function: Ψ(Φ,θ) = (4+√5 +3√5cos2θ)/(8√2π) what is time-dependent wave function?Homework Equations Spherical harmonics The Attempt at a Solution I tried normalizing this wave...

Homework Statement Let $$\vec H = ih_4^{(1)}(kr)\vec X_{40}(\theta,\phi)\cos(\omega t)$$ where ##h## is Hankel function of the first kind and ##\vec X## the vector spherical harmonic. a) Find the electric field in the area without charges; b) Find both fields in a spherical coordinate system...

Recently I've been studying Angular Momentum in Quantum Mechanics and I have a doubt about the eigenstates of orbital angular momentum in the position representation and the relation to the spherical harmonics. First of all, we consider the angular momentum operators L^2 and L_z. We know that...

Greetings, I want to ask if there is any subroutine for computing a global spherical harmonic reference field. I read journal and they say it exists, I hope we can share information regarding this subject. Thank you in advance.

1. Homework Statement Homework Equations Here we have to express ##\psi(\theta,\phi)## in terms of spherical harmonics ##Y_{lm}## to find the angular momentum. If ##\psi(\theta,\phi) = i \sqrt{\frac{3}{4\pi}} \sin{\theta} \sin{\phi} ##, it can be written as: $$ \frac{i}{\sqrt{2}} (Y_{1,1}-...

Hello people ! I have been studying Zettili's book of quantum mechanics and found that spherical harmonics are written <θφ|L,M>. Does this mean that |θφ> is a basis? What is more, is it complete and orthonormal basis in Hilbert? More evidence that it is a basis, in the photo i uploaded , in...

In Dodelson's "Modern Cosmology" on p.241 he states that the ##a_{lm}##-s -- for a given ##l##-- corresponding to a spherical harmonic expansion of the photon-temperature fluctuations, are drawn from the same probability distribution regardless of the value of ##m##. Dodelson does not explain...

When it comes to waves, spherical harmonics are, like, da bomb. I'm no expert - probably obvious from the question - but it seem to me that an instrument which maximises the utilisation of harmonics/resonances would be spherical. And yet, I can think of no spherical instruments - the most...

Hi All, Can someone tell me why gravitational waves are always decomposed in spin weighted spherical harmonics with spin weight -2 ? I'm assuming you can hand wave the answer with something to do with the 'graviton' being a spin 2 particle but this isn't very satisfying to me. Are there any...

i am a beginner and was going through (Donald Mcquarie's "quantum chemistry" ) some discussion regarding orbitals of H-atom but i didn't get the logic behind writing px and py orbitals as linear combinations of spherical harmonics? according to what i understood, a given spherical harmonic in...

I am studying the Earths main magnetic field (internal, specifically the stuff at the Core-Mantle boundary) which has led me to spherical harmonics. I am curious... how is the structure of a spherical harmonic determined by its degree l and order m? What role do the first three coefficients...