# introduction to spherical harmonics

\dfrac{d}{dx}[(x^{2} - 1)]\). Spherical harmonics are also generically useful in expanding solutions in physical settings with spherical symmetry. Spherical Harmonics 145 7.1Legendre polynomials 146 Series expansion 148 Orthogonality and Normalization 151 A second solution 154 7.2Rodriquezâs formula 156 Leibnizâs rule for differentiating products 156 7.3Generating function 159 7.4Recursion relations 162 â¦ One concludes that the spherical harmonics in the solution for the electron wavefunction in the hydrogen atom identify the angular momentum of the electron. (1-x^2)^{m/2} \frac{d^{\ell + m}}{dx^{\ell + m}} (x^2 - 1)^{\ell}.Pℓm​(x)=2ℓℓ!(−1)m​(1−x2)m/2dxℓ+mdℓ+m​(x2−1)ℓ. To specify the full solution, the coefficients AmℓA_m^{\ell}Amℓ​ and BmℓB_m^{\ell}Bmℓ​ must be found. \sin \theta\frac{\partial}{\partial \theta} \left(\sin \theta \frac{\partial \Theta (\theta)}{\partial \theta} \right) = m^2 \Theta (\theta) - \ell (\ell+1) \sin^2 \theta\, \Theta (\theta).sinθ∂θ∂​(sinθ∂θ∂Θ(θ)​)=m2Θ(θ)−ℓ(ℓ+1)sin2θΘ(θ). This construction is analogous to the case of the usual trigonometric functions. As a side note, there are a number of different relations one can use to generate Spherical Harmonics or Legendre polynomials. 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. 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. ∇2=1r2sin⁡θ(∂∂rr2sin⁡θ∂∂r+∂∂θsin⁡θ∂∂θ+∂∂ϕcsc⁡θ∂∂ϕ).\nabla^2 = \frac{1}{r^2 \sin \theta} \left(\frac{\partial}{\partial r} r^2 \sin \theta \frac{\partial}{\partial r} + \frac{\partial}{\partial \theta} \sin \theta \frac{\partial}{\partial \theta} + \frac{\partial}{\partial \phi} \csc \theta \frac{\partial}{\partial \phi} \right).∇2=r2sinθ1​(∂r∂​r2sinθ∂r∂​+∂θ∂​sinθ∂θ∂​+∂ϕ∂​cscθ∂ϕ∂​). The general, normalized Spherical Harmonic is depicted below: $Y_{l}^{m}(\theta,\phi) = \sqrt{ \dfrac{(2l + 1)(l - |m|)! At the halfway point, we can use our general definition of Spherical Harmonics with the newly determined Legendre function. ))eim" So it follows that for m=0, it can be written in terms of the standard Legendre polynomials, which are real FunctionExpand[SphericalHarmonicY[l, 0, Î¸, Ï]] Ï(x,y,z)(7. The quality of electrical power supply is an important issue both for utility companies and users, but that quality may affected by electromagnetic disturbances.Among these disturbances it must be highlighted harmonics that happens in all voltage levels and whose study, calculation of acceptable values and correction methods are defined in IEC Standard 61000-2-4: Electromagnetic compatibility (EMC) â Environment â Compatibilitâ¦ â2Ï(x,y,z)= . They also appear naturally in problems with azimuthal symmetry, which is the case in the next point. This operator gives us a simple way to determine the symmetry of the function it acts on. Missed the LibreFest? \hspace{15mm} 2&\hspace{15mm} -1&\hspace{15mm} \sqrt{\frac{15}{8\pi}} \sin \theta \cos \theta e^{-i \phi}\\ They are given by , where are associated Legendre polynomials and and are the orbital and magnetic quantum numbers, respectively. The overall shift of 111 comes from the lowest-lying harmonic Y00(θ,ϕ)Y^0_0 (\theta, \phi)Y00​(θ,ϕ). As the general function shows above, for the spherical harmonic where $$l = m = 0$$, the bracketed term turns into a simple constant. In order to do any serious computations with a large sum of Spherical Harmonics, we need to be able to generate them via computer in real-time (most specifically for real-time graphics systems). with ℏ\hbarℏ Planck's constant, mmm the electron mass, and EEE the energy of any particular state of the electron. Spherical harmonics are defined as the eigenfunctions of the angular part of the Laplacian in three dimensions. Spherical harmonics 9 Spherical harmonics ( ) ( ) ( ) ( ) ( ) ( ) Î¸ Ï Ï Î¸Ï m im l m m l m P e l m l l m Y â + + â =â + cos!! For the curious reader, a more in depth treatment of Laplace's equation and the methods used to solve it in the spherical domain are presented in this section of the text. If \[\Pi Y_{l}^{m}(\theta,\phi) = Y_{l}^{m}(-\theta,-\phi)$ then the harmonic is even. 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. More specifically, it is Hermitian. What is not shown in full is what happens to the Legendre polynomial attached to our bracketed expression. Second Edition. The spherical harmonics. Starinets. Log in here. One interesting example of spherical symmetry where the expansion in spherical harmonics is useful is in the case of the Schwarzschild black hole. The function f(θ,ϕ)f(\theta, \phi)f(θ,ϕ) decomposed into the sum of spherical harmonics given above. When r>Rr>Rr>R, all Amℓ=0A_m^{\ell} = 0Amℓ​=0 since in this case the potential will otherwise diverge as r→∞r \to \inftyr→∞, where the potential ought to vanish (or at the very least be finite, depending on where the zero of potential is set in this case). Sign up, Existing user? Consider the question of wanting to know the expectation value of our colatitudinal coordinate $$\theta$$ for any given spherical harmonic with even-$$l$$. Lastly, the Spherical Harmonics form a complete set, and as such can act as a basis for the given (Hilbert) space. the heat equation, Schrödinger equation, wave equation, Poisson equation, and Laplace equation) ubiquitous in gravity, electromagnetism/radiation, and quantum mechanics, the spherical harmonics are particularly important for representing physical quantities of interest in these domains, most notably the orbitals of the hydrogen atom in quantum mechanics. Blue represents positive values and yellow represents negative values . Much of modern physical chemistry is based around framework that was established by these quantum mechanical treatments of nature. Spherical Harmonics are considered the higher-dimensional analogs of these Fourier combinations, and are incredibly useful in applications involving frequency domains. These harmonics are classified as spherical due to being the solution to the angular portion of Laplace's equation in the spherical coordinate system. Recall that these functions are multiplied by their complex conjugate to properly represent the Born Interpretation of "probability-density" ($$\psi^{*}\psi)$$. With $$m = l = 1$$: $Y_{1}^{1}(\theta,\phi) = \sqrt{ \dfrac{(2(1) + 1)(1 - 1)! Now we can scale this up to the $$Y_{2}^{0}(\theta,\phi)$$ case given in example one: \[\Pi Y_{2}^{0}(\theta,\phi) = \sqrt{ \dfrac{5}{16\pi} }(3cos^2(-\theta) - 1)$. This requires the use of either recurrence relations or generating functions. Watch the recordings here on Youtube! Introduction. (1−x2)m/2dℓ+mdxℓ+m(x2−1)ℓ.P^m_{\ell} (x) = \frac{(-1)^m}{2^{\ell} \ell!} Laplace's work involved the study of gravitational potentials and Kelvin used them in a collaboration with Peter Tait to write a textbook. A harmonic of a periodic function has a frequency which is an integer multiple of that of the function (which is the fundamental). For , where is the associated Legendre function. ℓ011122222​m0−101−2−1012​Yℓm​(θ,ϕ)4π1​​8π3​​sinθe−iϕ4π3​​cosθ−8π3​​sinθeiϕ32π15​​sin2θe−2iϕ8π15​​sinθcosθe−iϕ16π5​​(3cos2θ−1)−8π15​​sinθcosθeiϕ32π15​​sin2θe2iϕ​​. So fff can be written as. 2. [ "article:topic", "spherical harmonics", "parity operator", "showtoc:no" ], https://chem.libretexts.org/@app/auth/2/login?returnto=https%3A%2F%2Fchem.libretexts.org%2FBookshelves%2FPhysical_and_Theoretical_Chemistry_Textbook_Maps%2FSupplemental_Modules_(Physical_and_Theoretical_Chemistry)%2FQuantum_Mechanics%2F07._Angular_Momentum%2FSpherical_Harmonics, https://en.Wikipedia.org/wiki/Eigenvalues_and_eigenvectors, http://www.liquisearch.com/spherical_harmonics/history, http://www.physics.drexel.edu/~bob/Quantum_Papers/Schr_1.pdf, http://www.oxfordscholarship.com/view/10.1093/acprof:oso/9780199231256.001.0001/acprof-9780199231256-chapter-11, https://www.cs.dartmouth.edu/~wjarosz/publications/dissertation/appendixB.pdf, http://www.cs.columbia.edu/~dhruv/lighting.pdf, status page at https://status.libretexts.org. As the non-squared function will be computationally easier to work with, and will give us an equivalent answer, we do not bother to square the function. Already have an account? This decomposition is typically performed as part of an analysis of the modes ω\omegaω describing the evolution of the perturbation Φ\PhiΦ, called quasinormal modes . As such, this integral will be zero always, no matter what specific $$l$$ and $$k$$ are used. This s orbital appears spherically symmetric on the boundary surface. V=14πϵ0QRsin⁡θcos⁡θcos⁡(ϕ).V = \frac{1}{4\pi \epsilon_0} \frac{Q}{R} \sin \theta \cos \theta \cos (\phi).V=4πϵ0​1​RQ​sinθcosθcos(ϕ). Since the Laplacian appears frequently in physical equations (e.g. The general solution for the electric potential VVV can be expanded in a basis of spherical harmonics as. The angular dependence at r=Rr=Rr=R solved for above in terms of spherical harmonics is therefore the angular dependence everywhere. This correspondence can be made more precise by considering the angular momentum of the electron. This means that when it is used in an eigenvalue problem, all eigenvalues will be real and the eigenfunctions will be orthogonal. we consider have some applications in the area of directional elds design. http://arxiv.org/pdf/0905.2975v2.pdf. \end{aligned} \hspace{15mm} 2&\hspace{15mm} 1&\hspace{15mm} -\sqrt{\frac{15}{8\pi}} \sin \theta \cos \theta e^{i \phi} \\ 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. This relationship also applies to the spherical harmonic set of solutions, and so we can write an orthonormality relationship for each quantum number: $\langle Y_{l}^{m} | Y_{k}^{n} \rangle = \delta_{lk}\delta_{mn}$. } (1 - x^{2})^{\tiny\dfrac{1}{2}}e^{i\phi} \], $Y_{1}^{1}(\theta,\phi) = \sqrt{ \dfrac{3}{8\pi} } (1 - x^{2})^{\tiny\dfrac{1}{2}}e^{i\phi}$. The generalization to higher ℓ\ellℓ is similar. \hspace{15mm} 0&\hspace{15mm} 0&\hspace{15mm} \sqrt{\frac{1}{4\pi}} \\ 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 ϕ\phiϕ and θ\thetaθ. From https://en.Wikipedia.org/wiki/Eigenvalues_and_eigenvectors. \hspace{15mm} 2&\hspace{15mm} -2&\hspace{15mm} \sqrt{\frac{15}{32\pi}} \sin^2 \theta e^{-2i\phi} \\ 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. It is a linear operator (follows rules regarding additivity and homogeneity). At each fixed energy, the solutions to the hydrogen atom are degenerate: one can modify the Yℓm(θ,ϕ)Y^m_{\ell} (\theta, \phi)Yℓm​(θ,ϕ) in any solution for the electron wavefunction without changing the energy of the electron (provided that the spin of the electron is ignored). 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. Laplace's work involved the study of gravitational potentials and Kelvin used them in a collaboration with Peter Tait to write a textbook. (ℓ+m)!Pℓm(cos⁡θ)eimϕ.Y^m_{\ell} (\theta, \phi) = \sqrt{\frac{2\ell + 1}{4\pi} \frac{(\ell - m)! One of the most prevalent applications for these functions is in the description of angular quantum mechanical systems. As it turns out, every odd, angular QM number yields odd harmonics as well! 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. At the ℓ=1\ell = 1ℓ=1 level, both m=±1m= \pm 1m=±1 have a sin⁡θ\sin \thetasinθ factor; their difference will give eiϕ+e−iϕe^{i\phi} + e^{-i\phi}eiϕ+e−iϕ giving a factor of cos⁡ϕ\cos \phicosϕ as desired. As such, any changes in parity to the Legendre polynomial (to create the associated Legendre function) will be undone by the flip in sign of $$m$$ in the azimuthal component. Note that the first term inside the sums is essentially just a Laurent series in rrr describing every possible power of rrr up to order ℓ\ellℓ. where the AmℓA_{m}^{\ell}Amℓ​ and BmℓB_{m}^{\ell}Bmℓ​ are some set of coefficients depending on the boundary conditions. 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. 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). Forexample,iftheforceï¬eldisrotationallyinvariant. \hspace{15mm} 1&\hspace{15mm} 0&\hspace{15mm} \sqrt{\frac{3}{4\pi}} \cos \theta\\ Start with acting the parity operator on the simplest spherical harmonic, $$l = m = 0$$: $\Pi Y_{0}^{0}(\theta,\phi) = \sqrt{\dfrac{1}{4\pi}} = Y_{0}^{0}(-\theta,-\phi)$. As one can imagine, this is a powerful tool. Since the electric potential energy U(r)=−e24πϵ0rU(r) = - \frac{e^2}{4\pi \epsilon_0 r} U(r)=−4πϵ0​re2​ is spherically symmetric, the separation of variables procedure used above still works and the potential only modifies the radial solution R(r)R(r)R(r). } P_{l}^{|m|}(\cos\theta)e^{im\phi} \]. Circular harmonics are a solution to Laplace's equation in polar coordiniates. The first two chapters provide the reader with the necessary mathematical and physical background, including an introduction to the spherical Fourier transform and the formulation of plane-wave sound fields in the spherical harmonic domain. A conducting sphere of radius RRR with a layer of charge QQQ distributed on its surface has the electric potential everywhere in space: V={14πϵ0QR2r3sin⁡θcos⁡θcos⁡ϕ,  r>R14πϵ0Qr2R3sin⁡θcos⁡θcos⁡ϕ,  rR \\ This means any spherical function can be written as a linear combination of these basis functions, (for the basis spans the space of continuous spherical functions by definition): $f(\theta,\phi) = \sum_{l}\sum_{m} \alpha_{lm} Y_{l}^{m}(\theta,\phi)$. The spherical harmonics are eigenfunctions of both of these operators, which follows from the construction of the spherical harmonics above: the solutions for Yℓm(θ,ϕ)Y^m_{\ell} (\theta, \phi)Yℓm​(θ,ϕ) and its ϕ\phiϕ dependence were both eigenvalue equations corresponding to these operators (or their squares). Have questions or comments? Spherical harmonics on the sphere, S2, have interesting applications in ... Introduction to Spherical Coordinates - Duration: 9:18. To solve this problem, we can break up our process into four major parts. 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. Physically, Y00(θ,ϕ)Y^0_0 (\theta, \phi)Y00​(θ,ϕ) represents the overall average or monopole moment of a function on the sphere, while the Y1m(θ,ϕ)Y^m_1 (\theta, \phi)Y1m​(θ,ϕ) represent the dipole moments of this function. The details of where these polynomials come from are largely unnecessary here, lest we say that it is the set of solutions to a second differential equation that forms from attempting to solve Laplace's equation. Pearson: Upper Saddle River, NJ, 2006. Note: Recall that the change in electric field across either side of a conductor is equal to σϵ0,\frac{\sigma}{\epsilon_0},ϵ0​σ​, where σ\sigmaσ is the surface charge density. sin â¡ ( m Ï) \sin (m \phi) sin(mÏ) and. relatively to their order and orientation. 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. The polynomials in d variables of â¦ The spherical harmonics are orthonormal with respect to integration over the surface of the unit sphere. Active 4 years ago. As $$l = 1$$: $$P_{1}(x) = \dfrac{1}{2^{1}1!} ∂r∂​(r2∂r∂R(r)​)sinθ1​∂θ∂​(sinθ∂θ∂Y(θ,ϕ)​)+sin2θ1​dϕ2d2Y(θ,ϕ)​​=ℓ(ℓ+1)R(r)=−ℓ(ℓ+1)Y(θ,ϕ),​. For each fixed nnn and ℓ\ellℓ there are 2ℓ+12\ell + 12ℓ+1 solutions corresponding to the 2ℓ+12\ell + 12ℓ+1 choices of mmm at fixed ℓ.\ell.ℓ. The parity operator is sometimes denoted by "P", but will be referred to as \(\Pi$$ here to not confuse it with the momentum operator. Visually, this corresponds to the decomposition below: It is also shown that the two-step formulation of global spherical harmonic computation was applied already by Neumann (1838) and Gauss (1839). Quantum Mechanics I by Prof. S. Lakshmi Bala, Department of Physics, IIT Madras. In other words, the function looks like a ball. When we plug this into our second relation, we now have to deal with $$|m|$$ derivatives of our $$P_{l}$$ function. Spherical Harmonics are a group of functions used in math and the physical sciences to solve problems in disciplines including geometry, partial differential equations, and group theory. $\langle \theta \rangle = \langle Y_{l}^{m} | \theta | Y_{l}^{m} \rangle$, $\langle \theta \rangle = \int_{-\inf}^{\inf} (EVEN)(ODD)(EVEN)d\tau$. It is no coincidence that this article discusses both quantum mechanics and two variables, $$l$$ and $$m$$. 1) ThepresenceoftheW-factorservestodestroyseparabilityexceptinfavorable specialcases. This construction is analogous to the case of the usual trigonometric functions sin⁡(mϕ)\sin (m \phi)sin(mϕ) and cos⁡(mϕ)\cos (m \phi)cos(mϕ) which form a complete basis for periodic functions of a single variable (the Fourier series) and are eigenfunctions of the angular Laplacian in two dimensions, ∇ϕ2=∂2∂ϕ2\nabla^2_{\phi} = \frac{\partial^2}{\partial \phi^2}∇ϕ2​=∂ϕ2∂2​, with eigenvalue −m2-m^2−m2. The Schrödinger equation for hydrogen reads in S.I. \dfrac{d^{l}}{dx^{l}}[(x^{2} - 1)^{l}]\), $$P_{l}^{|m|}(x) = (1 - x^{2})^{\tiny\dfrac{|m|}{2}}\dfrac{d^{|m|}}{dx^{|m|}}P_{l}(x)$$. The angular equation above can also be solved by separation of variables. Spherical harmonics are often used to approximate the shape of the geoid. 2. Consider the real function on the sphere given by f(θ,ϕ)=1+sin⁡θcos⁡ϕf(\theta, \phi) = 1 + \sin \theta\cos \phif(θ,ϕ)=1+sinθcosϕ. \hspace{15mm} 2&\hspace{15mm} 0&\hspace{15mm} \sqrt{\frac{5}{16\pi}} (3\cos^2 \theta -1 )\\ The electron wavefunction in the hydrogen atom is still written ψ(r,θϕ)=Rnℓ(r)Yℓm(θ,ϕ)\psi (r,\theta \phi) = R_{n\ell} (r) Y^m_{\ell} (\theta, \phi)ψ(r,θϕ)=Rnℓ​(r)Yℓm​(θ,ϕ), where the index nnn corresponds to the energy EnE_nEn​ of the electron obtained by solving the new radial equation. Introduction to Quantum Mechanics. In quantum mechanics, the total angular momentum operator is defined as the Laplacian on the sphere up to a constant: L^2=−ℏ2(1sin⁡θ∂∂θ(sin⁡θ∂∂θ)+1sin⁡2θ∂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. Who are interested in â¦ the spherical harmonics and Approximations on the surface to modeled. -, information on Hermitian Operators - www.pa.msu.edu/~mmoore/Lect4_BasisSet.pdf, Discussions of S.H respect to integration over the charge! We can use our general definition of spherical harmonics form a complete set on the.! Of course~ been handled before~ without resorting to tensors of degree 4 on the unit sphere Laplace! 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