# introduction to spherical harmonics

January 12, 2021 4:38 am Leave your thoughtsSpherical harmonics are defined as the eigenfunctions of the angular part of the Laplacian in three dimensions. SphericalHarmonicY can be evaluated to arbitrary numerical precision. In quantum mechanics, the total angular momentum operator is defined as the Laplacian on the sphere up to a constant: L^2=−ℏ2(1sinθ∂∂θ(sinθ∂∂θ)+1sin2θ∂2∂ϕ2),\hat{L}^2 = -\hbar^2 \left(\frac{1}{\sin \theta} \frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial}{\partial \theta} \right) + \frac{1}{\sin^2 \theta} \frac{\partial^2}{\partial \phi^2} \right),L^2=−ℏ2(sinθ1∂θ∂(sinθ∂θ∂)+sin2θ1∂ϕ2∂2), and similarly the operator for the angular momentum about the zzz-axis is. They are a higher-dimensional analogy of Fourier series, which form a complete basis for the set of periodic functions of a single variable (((functions on the circle S1).S^1).S1). for \(I\) equal to the moment of inertia of the represented system. (ℓ+m)!Pℓm(cosθ)eimϕ.Y^m_{\ell} (\theta, \phi) = \sqrt{\frac{2\ell + 1}{4\pi} \frac{(\ell - m)! We have described these functions as a set of solutions to a differential equation but we can also look at Spherical Harmonics from the standpoint of operators and the field of linear algebra. The two major statements required for this example are listed: \( P_{l}(x) = \dfrac{1}{2^{l}l!} These notes provide an introduction to the theory of spherical harmonics in an arbitrary dimension as well as an overview of classical and recent results on some aspects of the approximation of functions by spherical polynomials and numerical integration over the unit sphere. As a final topic, we should take a closer look at the two recursive relations of Legendre polynomials together. More specifically, it is Hermitian. \hspace{15mm} 1&\hspace{15mm} -1&\hspace{15mm} \sqrt{\frac{3}{8\pi}} \sin \theta e^{-i \phi}\\ A conducting sphere of radius RRR with a layer of charge QQQ distributed on its surface has the electric potential on the surface of the sphere given by. These two properties make it possible to deduce the reconstruction formula of the surface to be modeled. units as follows: −ℏ22m∇2ψ−e24πϵ0rψ=Eψ,-\frac{\hbar^2}{2m} \nabla^2 \psi - \frac{e^2}{4\pi \epsilon_0 r}\psi = E\psi,−2mℏ2∇2ψ−4πϵ0re2ψ=Eψ. ∂r∂(r2∂r∂R(r))sinθ1∂θ∂(sinθ∂θ∂Y(θ,ϕ))+sin2θ1dϕ2d2Y(θ,ϕ)=ℓ(ℓ+1)R(r)=−ℓ(ℓ+1)Y(θ,ϕ),. L2=ℏ2ℓ(ℓ+1),Lz=ℏm.L^2 = \hbar^2 \ell (\ell + 1), \quad L_z = \hbar m.L2=ℏ2ℓ(ℓ+1),Lz=ℏm. A photo-set reminder of why an eigenvector (blue) is special. P l m(cos(! New user? Pearson: Upper Saddle River, NJ, 2006. with ℏ\hbarℏ Planck's constant, mmm the electron mass, and EEE the energy of any particular state of the electron. but cosine is an even function, so again, we see: \[ Y_{2}^{0}(-\theta,-\phi) = Y_{2}^{0}(\theta,\phi)\]. \begin{aligned} From https://en.Wikipedia.org/wiki/Eigenvalues_and_eigenvectors. One interesting example of spherical symmetry where the expansion in spherical harmonics is useful is in the case of the Schwarzschild black hole. V=14πϵ0QRsinθcosθcos(ϕ).V = \frac{1}{4\pi \epsilon_0} \frac{Q}{R} \sin \theta \cos \theta \cos (\phi).V=4πϵ01RQsinθcosθcos(ϕ). Spherical harmonics are often used to approximate the shape of the geoid. As the general function shows above, for the spherical harmonic where \(l = m = 0\), the bracketed term turns into a simple constant. If this is the case (verified after the next example), then we now have a simple task ahead of us. In Cartesian coordinates, the three-dimensional Laplacian is typically defined as. Visually, this corresponds to the decomposition below: Any harmonic is a function that satisfies Laplace's differential equation: These harmonics are classified as spherical due to being the solution to the angular portion of Laplace's equation in the spherical coordinate system. For a brief review, partial differential equations are often simplified using a separation of variables technique that turns one PDE into several ordinary differential equations (which is easier, promise). The Yℓm(θ,ϕ)Y^m_{\ell} (\theta, \phi)Yℓm(θ,ϕ) thus correspond to the different possible electron orbitals; they label the unique states of the electron in hydrogen at a single fixed energy. Color represents the phase of the spherical harmonic. This means that when it is used in an eigenvalue problem, all eigenvalues will be real and the eigenfunctions will be orthogonal. This correspondence can be made more precise by considering the angular momentum of the electron. Utilized first by Laplace in 1782, these functions did not receive their name until nearly ninety years later by Lord Kelvin. Missed the LibreFest? The first two cases ~ave, of course~ been handled before~ without resorting to tensors. The \({Y_{1}^{0}}^{*}Y_{1}^{0}\) and \({Y_{1}^{1}}^{*}Y_{1}^{1}\) functions are plotted above. Spherical Harmonics are a group of functions used in math and the physical sciences to solve problems in disciplines including geometry, partial differential equations, and group theory. When we consider the fact that these functions are also often normalized, we can write the classic relationship between eigenfunctions of a quantum mechanical operator using a piecewise function: the Kronecker delta. As \(l = 1\): \( P_{1}(x) = \dfrac{1}{2^{1}1!} For , where is the associated Legendre function. Laplace's work involved the study of gravitational potentials and Kelvin used them in a collaboration with Peter Tait to write a textbook. An even function multiplied by an odd function is an odd function (like even and odd numbers when multiplying them together). Sign up to read all wikis and quizzes in math, science, and engineering topics. V=14πϵ0QRsinθcosθcos(ϕ)=14πϵ0QR2π15(Y2−1(θ,ϕ)−Y21(θ,ϕ)).V = \frac{1}{4\pi \epsilon_0} \frac{Q}{R} \sin \theta \cos \theta \cos (\phi) = \frac{1}{4\pi \epsilon_0} \frac{Q}{R} \sqrt{\frac{2\pi}{15}} \big(Y^{-1}_2 (\theta, \phi) - Y^1_2 (\theta, \phi) \big).V=4πϵ01RQsinθcosθcos(ϕ)=4πϵ01RQ152π(Y2−1(θ,ϕ)−Y21(θ,ϕ)). The full solution may only include a combination of Y2−1Y^{-1}_2Y2−1 and Y21Y^1_2Y21 in the angular part because the angular dependence is completely independent of the radial dependence. Log in here. Is an electron in the hydrogen atom in the orbital defined by the superposition Y1−1(θ,ϕ)+Y2−1(θ,ϕ)Y^{-1}_1 (\theta, \phi) + Y^{-1}_2 (\theta, \phi)Y1−1(θ,ϕ)+Y2−1(θ,ϕ) an eigenfunction of the (total angular momentum operator, angular momentum about zzz axis)? Due to the spherical symmetry of the black hole and the presence of the Laplacian on the sphere, the general solution for perturbations can be written as a Fourier transform: Φ(t,r,θ,ϕ)=∫dωe−iωt∑ℓ,mΨ(r)rYℓm(θ,ϕ).\Phi(t,r, \theta, \phi) = \int d\omega e^{-i\omega t} \sum_{\ell ,m} \frac{\Psi (r)}{r} Y_{\ell m} (\theta, \phi).Φ(t,r,θ,ϕ)=∫dωe−iωtℓ,m∑rΨ(r)Yℓm(θ,ϕ). Combining this with \(\Pi\) gives the conditions: Using the parity operator and properties of integration, determine \(\langle Y_{l}^{m}| Y_{k}^{n} \rangle\) for any \( l\) an even number and \(k\) an odd number. Starinets. â¢ In quantum mechanics, they (really the spherical harmonics; Section 11.5) represent angular momentum eigenfunctions. The ability to expand in the basis of spherical harmonics is essential in permitting the separation of the radial dependence which ultimately constrains the modes ω\omegaω. The spherical harmonics are constructed to be the eigenfunctions of the angular part of the Laplacian in three dimensions, also called the Laplacian on the sphere. Introduction Spherical harmonic analysis is a process of decom-posing a function on a sphere into components of various wavelengths using surface spherical harmonics as base functions. where ℓ(ℓ+1)\ell(\ell+1)ℓ(ℓ+1) is some constant called the separation constant, written in what will ultimately be the most convenient form. The first is determining our \(P_{l}(x)\) function. [2] Griffiths, David J. One of the most well-known applications of spherical harmonics is to the solution of the Schrödinger equation for the wavefunction of the electron in a hydrogen atom in quantum mechanics. This allows us to say \(\psi(r,\theta,\phi) = R_{nl}(r)Y_{l}^{m}(\theta,\phi)\), and to form a linear operator that can act on the Spherical Harmonics in an eigenvalue problem. These can be found by demanding continuity of the potential at r=Rr=Rr=R. Now, another ninety years later, the exact solutions to the hydrogen atom are still used to analyze multi-electron atoms and even entire molecules. Plots of the real parts of the first few spherical harmonics, where distance from origin gives the value of the spherical harmonic as a function of the spherical angles, https://brilliant.org/wiki/spherical-harmonics/. \end{aligned} Note that the normalization factor of (−1)m(-1)^m(−1)m here included in the definition of the Legendre polynomials is sometimes included in the definition of the spherical harmonics instead or entirely omitted. As Spherical Harmonics are unearthed by working with Laplace's equation in spherical coordinates, these functions are often products of trigonometric functions. When we plug this into our second relation, we now have to deal with \(|m|\) derivatives of our \(P_{l}\) function. \hspace{15mm} 2&\hspace{15mm} 0&\hspace{15mm} \sqrt{\frac{5}{16\pi}} (3\cos^2 \theta -1 )\\ $\begingroup$ This paper by Volker Schönefeld shows a good introduction to SH with excellent visualizations $\endgroup$ â bobobobo Sep 3 '13 at 1 ... factors in front of the defining expression for spherical harmonics were set so that the integral of the square of a spherical harmonic over the sphere's surface is 1. A similar analysis obtains the solution for r

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