What is Spherical harmonics: Definition and 126 Discussions

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.
Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).
Spherical harmonics originates from solving Laplace's equation in the spherical domains. Functions that solve Laplace's equation are called harmonics. Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree

{\displaystyle \ell }


{\displaystyle (x,y,z)}
that obey Laplace's equation. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence


{\displaystyle r^{\ell }}
from the above-mentioned polynomial of degree

{\displaystyle \ell }
; the remaining factor can be regarded as a function of the spherical angular coordinates


{\displaystyle \theta }


{\displaystyle \varphi }
only, or equivalently of the orientational unit vector


{\displaystyle {\mathbf {r} }}
specified by these angles. In this setting, they may be viewed as the angular portion of a set of solutions to Laplace's equation in three dimensions, and this viewpoint is often taken as an alternative definition.
A specific set of spherical harmonics, denoted




{\displaystyle Y_{\ell }^{m}(\theta ,\varphi )}






{\displaystyle Y_{\ell }^{m}({\mathbf {r} })}
, are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above.
Spherical harmonics are important in many theoretical and practical applications, including the representation of multipole electrostatic and electromagnetic fields, electron configurations, gravitational fields, geoids, the magnetic fields of planetary bodies and stars, and the cosmic microwave background radiation. In 3D computer graphics, spherical harmonics play a role in a wide variety of topics including indirect lighting (ambient occlusion, global illumination, precomputed radiance transfer, etc.) and modelling of 3D shapes.

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  1. F

    A Poisson noise on ##a_{\ell m}## complex number: real or complex?

    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...
  2. Junaidjami

    A Real vs complex spherical harmonics for hexagonal symmetry

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

    Rotation of spherical harmonics

    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...
  4. F

    A Coefficient correlation between 2 cosmological probes

    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...
  5. F

    A Relation between a_{\ell m} noise and Poisson noise with C_{\ell}

    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...
  6. The Bill

    Analysis Resource(s) for introduction to spherical harmonics with exercises?

    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...
  7. P

    Commutation relations between Ladder operators and Spherical Harmonics

    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?
  8. Salmone

    I Inner products with spherical harmonics in quantum mechanics

    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...
  9. F

    A Need help about a demo with inverse weighted variance average

    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...
  10. F

    A Calculating the variance of integrated Poisson noise on a defined quantity

    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}...
  11. R

    Expanding potential in Legendre polynomials (or spherical harmonics)

    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...
  12. F

    A Cl's : sum into a chi^2 when we have a sum of chi^2

    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)...
  13. F

    A Introduction of a factor Δℓ when summing equal distants 𝐶ℓ

    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...
  14. docnet

    Solve Spherical Harmonics: Y_{1,1} Eigenfunction of L^2 & L_z

    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}...
  15. P

    I Second Matrices from Spherical Harmonics with Eigenvalue l+1

    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...
  16. Athenian

    I Transforming Cartesian Coordinates in terms of Spherical Harmonics

    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...
  17. Athenian

    Finding the Monopole and Multipole Moments of the Electric Potential

    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...
  18. P

    A Matrices from Spherical Harmonics with Eigenvalue l+1

    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\\...
  19. Taz

    A Normalization of the radial part of the spherical harmonics

    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...
  20. Coltrane8

    I Spherical Harmonics Expansion convergence

    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...
  21. F

    Does operator L^2 commute with spherical harmonics?

    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...
  22. M

    A What are the uses of spherical harmonics?

    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!
  23. M

    I Spherical Harmonics Axisymmetry

    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?
  24. beefbrisket

    I Computing inner products of spherical harmonics

    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...
  25. Faizan Samad

    Sph Harmonics Homework: Find Potential for r>a, Contribution Using Superposition

    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...
  26. D

    Integration of Spherical Harmonics with a Gaussian (QM)

    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...
  27. renec112

    QM: Issues with parity of spherical harmonics and Heisenberg

    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}##...
  28. S

    Solving Spherical Harmonics Homework

    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)...
  29. B

    I Can the Schrodinger equation satisfy Laplace's equation?

    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...
  30. C

    Coefficient Matching for different series

    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...
  31. amjad-sh

    A Spherical Harmonics: A Primer on Barton's Relations & Addition Theorem

    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...
  32. davidge

    I Spherical Harmonics from operator analysis

    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...
  33. davidge

    I Spherical Harmonics: Arriving at Equation

    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...
  34. C

    A Angular power spectrum, bias from N weighted events

    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...
  35. davidge

    I Deriving Spherical Harmonics from Sakurai's Book

    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...
  36. M

    Orthonormality of Spherical Harmonics Y_1,1 and Y_2,1

    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 =...
  37. M

    Solving Radial Schrodinger Equation

    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...
  38. M

    Where am I going wrong in my radial equation substitution derivation?

    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) =...
  39. T

    I Obtaining Spherical Harmonics: A Guide to Normalized Angular Wave Functions

    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...
  40. M

    Time-Dependent Wave Function of Spherical Harmonics

    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...
  41. Y

    Spherical Harmonics Change of Coordinate System

    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...
  42. L

    Eigenstates of Orbital Angular Momentum

    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...
  43. K

    Fortran [Fortran 77] Subroutine for computing a global spherical har

    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.
  44. Safinaz

    Spherical Harmonics (QM)

    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}-...
  45. KostasV

    What is the role of spherical harmonics in quantum mechanics?

    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...
  46. C

    CMB , Spherical Harmonics and Rotational Invariance

    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...
  47. A

    Why are spherical instruments not more common?

    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...
  48. S

    Why Gravitational Waves are Decomposed in Spin Weighted Spherical Harmonics

    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...
  49. Raman Choudhary

    Superposition of spherical harmonics

    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...
  50. H

    Spherical Harmonics: Degree l & Order m Structure & Variation

    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...