## introduction to spherical harmonics

\hspace{15mm} 1&\hspace{15mm} 1&\hspace{15mm} -\sqrt{\frac{3}{8\pi}} \sin \theta e^{i \phi} \\ }{4\pi (l + |m|)!} Start with acting the parity operator on the simplest spherical harmonic, $$l = m = 0$$: $\Pi Y_{0}^{0}(\theta,\phi) = \sqrt{\dfrac{1}{4\pi}} = Y_{0}^{0}(-\theta,-\phi)$. 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 [3]. In the past few years, with the advancement of computer graphics and rendering, modeling of dynamic lighting systems have led to a new use for these functions. One interesting example of spherical symmetry where the expansion in spherical harmonics is useful is in the case of the Schwarzschild black hole. 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. but cosine is an even function, so again, we see: $Y_{2}^{0}(-\theta,-\phi) = Y_{2}^{0}(\theta,\phi)$. The general solution for the electric potential VVV can be expanded in a basis of spherical harmonics as. As it turns out, every odd, angular QM number yields odd harmonics as well! This requires the use of either recurrence relations or generating functions. Legal. P^m_{\ell} (\cos \theta) e^{im\phi}.Yℓm​(θ,ϕ)=4π2ℓ+1​(ℓ+m)!(ℓ−m)!​​Pℓm​(cosθ)eimϕ. The Laplace equation ∇2f=0\nabla^2 f = 0∇2f=0 can be solved via separation of variables. \hspace{15mm} 2&\hspace{15mm} 2&\hspace{15mm} \sqrt{\frac{15}{32\pi}} \sin^2 \theta e^{2i\phi} The exact combination including the correct coefficient is. Formally, these conditions on mmm and ℓ\ellℓ can be derived by demanding that solutions be periodic in θ\thetaθ and ϕ\phiϕ. As derivatives of even functions yield odd functions and vice versa, we note that for our first equation, an even $$l$$ value implies an even number of derivatives, and this will yield another even function. Have questions or comments? Multiplying the top equation by Y(θ,ϕ)Y(\theta, \phi)Y(θ,ϕ) on both sides, the bottom equation by R(r)R(r)R(r) on both sides, and adding the two would recover the original three-dimensional Laplace equation in spherical coordinates; the separation constant is obtained by recognizing that the original Laplace equation describes two eigenvalue equations of opposite signs. Reference Request: Easy Introduction to Spherical Harmonics. Spherical harmonics on the sphere, S2, have interesting applications in Log in. 2.1. We are in luck though, as in the spherical harmonic functions there is a separate component entirely dependent upon the sign of $$m$$. Forgot password? \hspace{15mm} 2&\hspace{15mm} -1&\hspace{15mm} \sqrt{\frac{15}{8\pi}} \sin \theta \cos \theta e^{-i \phi}\\ This means that when it is used in an eigenvalue problem, all eigenvalues will be real and the eigenfunctions will be orthogonal. ∇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∂​). One concludes that the spherical harmonics in the solution for the electron wavefunction in the hydrogen atom identify the angular momentum of the electron. The more important results from this analysis include (1) the recognition of an $$\hat{L}^2$$ operator and (2) the fact that the Spherical Harmonics act as an eigenbasis for the given vector space. One of the most prevalent applications for these functions is in the description of angular quantum mechanical systems. 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. 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. 2. Utilized first by Laplace in 1782, these functions did not receive their name until nearly ninety years later by Lord Kelvin. More specifically, it is Hermitian. Unless otherwise noted, LibreTexts content is licensed by CC BY-NC-SA 3.0. V(r,θ,ϕ)=∑ℓ=0∞∑m=−ℓℓ(Amℓrℓ+Bmℓrℓ+1)Yℓm(θ,ϕ).V(r,\theta, \phi ) = \sum_{\ell = 0}^{\infty} \sum_{m=-\ell }^{\ell } \left( A_{m}^{\ell} r^{\ell} + \frac{B_{m}^{\ell}}{r^{\ell +1}}\right) Y_{\ell}^m (\theta, \phi).V(r,θ,ϕ)=ℓ=0∑∞​m=−ℓ∑ℓ​(Amℓ​rℓ+rℓ+1Bmℓ​​)Yℓm​(θ,ϕ). With $$m = l = 1$$: $Y_{1}^{1}(\theta,\phi) = \sqrt{ \dfrac{(2(1) + 1)(1 - 1)! Since the Laplacian appears frequently in physical equations (e.g. V(r,θ,ϕ)=∑ℓ=0∞∑m=−ℓℓ(Amℓrℓ+Bmℓrℓ+1)Yℓm(θ,ϕ),V(r,\theta, \phi ) = \sum_{\ell = 0}^{\infty} \sum_{m=-\ell }^{\ell } \left( A_{m}^{\ell} r^{\ell} + \frac{B_{m}^{\ell}}{r^{\ell +1}}\right) Y_{\ell}^m (\theta, \phi) ,V(r,θ,ϕ)=ℓ=0∑∞​m=−ℓ∑ℓ​(Amℓ​rℓ+rℓ+1Bmℓ​​)Yℓm​(θ,ϕ). These harmonics are classified as spherical due to being the solution to the angular portion of Laplace's equation in the spherical coordinate system. Therefore, make the ansatz Y(θ,ϕ)=Θ(θ)eimϕY(\theta, \phi) = \Theta (\theta) e^{i m\phi}Y(θ,ϕ)=Θ(θ)eimϕ for some second separation constant mmm which can take negative values. 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. L^z=−iℏ∂∂ϕ.\hat{L}_z = -i\hbar \frac{\partial}{\partial \phi}.L^z​=−iℏ∂ϕ∂​. 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. Notably, this formula is only well-defined and nonzero for ℓ≥0\ell \geq 0ℓ≥0 and mmm integers such that ∣m∣≤ℓ|m| \leq \ell∣m∣≤ℓ. V(r,θ,ϕ)=14πϵ0QR2r3sin⁡θcos⁡θcos⁡ϕ,r>R.V(r,\theta, \phi ) = \frac{1}{4\pi \epsilon_0} \frac{QR^2}{r^3} \sin \theta \cos \theta \cos \phi, \quad r>R.V(r,θ,ϕ)=4πϵ0​1​r3QR2​sinθcosθcosϕ,r>R. It is a linear operator (follows rules regarding additivity and homogeneity). } P_{l}^{|m|}(\cos\theta)e^{im\phi}$. Introduction. 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. Visually, this corresponds to the decomposition below: \hspace{15mm} 2&\hspace{15mm} 1&\hspace{15mm} -\sqrt{\frac{15}{8\pi}} \sin \theta \cos \theta e^{i \phi} \\ For , . 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. The spherical harmonics In obtaining the solutions to Laplaceâs equation in spherical coordinates, it is traditional to introduce the spherical harmonics, Ym â(Î¸,Ï), Ym â(Î¸,Ï) = (â1)m s (2â+1) 4Ï (ââ m)! Using integral properties, we see this is equal to zero, for any even-$$l$$. 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. The general, normalized Spherical Harmonic is depicted below: \[ Y_{l}^{m}(\theta,\phi) = \sqrt{ \dfrac{(2l + 1)(l - |m|)! The impact is lessened slightly when coming off the heels off the idea that Hermitian operators like $$\hat{L}^2$$ yield orthogonal eigenfunctions, but general parity of functions is useful! The spherical harmonics. â2Ï(x,y,z)= . (1−x2)m/2dℓ+mdxℓ+m(x2−1)ℓ.P^m_{\ell} (x) = \frac{(-1)^m}{2^{\ell} \ell!} From the solution on r=Rr=Rr=R in terms of spherical harmonics, these coefficients can be read off: B−12=14πϵ0QR22π15=−B12.B_{-1}^2 = \frac{1}{4\pi \epsilon_0} QR^2 \sqrt{\frac{2\pi}{15}} = -B_1^2.B−12​=4πϵ0​1​QR2152π​​=−B12​. It appears that for every even, angular QM number, the spherical harmonic is even. These two properties make it possible to deduce the reconstruction formula of the surface to be modeled. 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. Which of the following is the formula for the spherical harmonic Y3−2(θ,ϕ)?Y^{-2}_3 (\theta, \phi)?Y3−2​(θ,ϕ)? It is also important to note that these functions alone are not referred to as orbitals, for this would imply that both the radial and angular components of the wavefunction are used. 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. In the 20th century, Erwin Schrödinger and Wolfgang Pauli both released papers in 1926 with details on how to solve the "simple" hydrogen atom system. Again, a complex sounding problem is reduced to a very straightforward analysis. Spherical harmonic functions arise for central force problems in quantum mechanics as the angular part of the Schrödinger equation in spherical polar coordinates. 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