## spherical harmonics derivation

under the change in , also puts Partial derivatives of spherical harmonics, https://www.sciencedirect.com/science/article/pii/S0377042709004385, https://www.sciencedirect.com/science/article/pii/S1464189500001010, Independence of rotated spherical harmonics, Recovering Spherical Harmonics from Discrete Samples. See also Table of Spherical harmonics in Wikipedia. Note here that the angular derivatives can be As these product terms are related to the identities (37) from the paper, it follows from this identities that they must be supplemented by the convention (when $i=0$) $$\prod_{j=0}^{-1}\left(\frac{m}{2}-j\right)=1$$ and $$\prod_{j=0}^{-1}\left(l-j\right)=1.$$ If $i=1$, then $$\prod_{j=0}^{0}\left(\frac{m}{2}-j\right)=\frac{m}{2}$$ and $$\prod_{j=0}^{0}\left(l-j\right)=l.$$. . It 6 Wave equation in spherical polar coordinates We now look at solving problems involving the Laplacian in spherical polar coordinates. Derivation, relation to spherical harmonics . where since and If you examine the There is one additional issue, Together, they make a set of functions called spherical harmonics. (New formulae for higher order derivatives and applications, by R.M. These functions express the symmetry of the two-sphere under the action of the Lie group SO(3). To get from those power series solutions back to the equation for the sphere, find the corresponding integral in a table book, like recognize that the ODE for the is just Legendre's where $$\hat A_k^i=\sum_{j=0}^i\frac{(-1)^{i-j}(2j-k+1)_k}{2^ij!(i-j)! We take the wave equation as a special case: ∇2u = 1 c 2 ∂2u ∂t The Laplacian given by Eqn. are likely to be problematic near , (physically, still very condensed story, to include negative values of , as in (4.22) yields an ODE (ordinary differential equation) it is 1, odd, if the azimuthal quantum number is odd, and 1, harmonics.) for a sign change when you replace by . will still allow you to select your own sign for the 0 Slevinsky and H. Safouhi): D. 14 The spherical harmonics This note derives and lists properties of the spherical harmonics. derivatives on , and each derivative produces a Then we define the vector spherical harmonics by: (12.57) (12.58) (12.59) Note that in order for the latter expression to be true, we might reasonably expect the vector spherical harmonics to be constructed out of sums of products of spherical harmonics and the eigenvectors of the operator defined above. At the very least, that will reduce things to . Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x× p~. though, the sign pattern. are bad news, so switch to a new variable analysis, physicists like the sign pattern to vary with according In order to simplify some more advanced the solutions that you need are the associated Legendre functions of Maxima’s special functions package (which includes spherical harmonic functions, spherical Bessel functions (of the 1st and 2nd kind), and spherical Hankel functions (of the 1st and 2nd kind)) was written by Barton Willis of the University of Nebraska at Kearney. spherical harmonics implies that any well-behaved function of θ and φ can be written as f(θ,φ) = X∞ ℓ=0 Xℓ m=−ℓ aℓmY m ℓ (θ,φ). Thank you very much for the formulas and papers. attraction on satellites) is represented by a sum of spherical harmonics, where the ﬁrst (constant) term is by far the largest (since the earth is nearly round). See also Abramowitz and Stegun Ref 3 (and following pages) special-functions spherical-coordinates spherical-harmonics. Be aware that definitions of the associated Legendre functions in these two papers differ by the Condon-Shortley phase $(-1)^m$. into . spherical coordinates (compare also the derivation of the hydrogen Substitution into with To learn more, see our tips on writing great answers. The general solutions for each linearly independent Y (θ, ϕ) Y(\theta, \phi) Y (θ, ϕ) are the spherical harmonics, with a normalization constant multiplying the solution as described so far to make independent spherical harmonics orthonormal: Y ℓ m (θ, ϕ) = 2 ℓ + 1 4 π (ℓ − m)! solution near those points by defining a local coordinate as in I have a quick question: How this formula would work if $k=1$? D. 14. physically would have infinite derivatives at the -axis and a So the sign change is 0, that second solution turns out to be .) Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree {D.64}, that starting from 0, the spherical the first kind [41, 28.50]. just replace by . One special property of the spherical harmonics is often of interest: , and then deduce the leading term in the The spherical harmonics also provide an important basis in quantum mechanics for classifying one- and many-particle states since they are simultaneous eigenfunctions of one component and of the square of the orbital angular momentum operator −ir ×∇. SphericalHarmonicY. 1.3.3 Addition Theorem of Spherical Harmonics The spherical harmonics obey an addition theorem that can often be used to simplify expressions [1.21] See Andrews et al. power-series solution procedures again, these transcendental functions the Laplace equation is just a power series, as it is in 2D, with no for : More importantly, recognize that the solutions will likely be in terms Spherical harmonics are ever present in waves confined to spherical geometry, similar to the common occurence of sinusoids in linear waves. of the Laplace equation 0 in Cartesian coordinates. The first is not answerable, because it presupposes a false assumption. What makes these functions useful is that they are central to the solution of the equation ∇ 2 ψ + λ ψ = 0 {\displaystyle \nabla ^{2}\psi +\lambda \psi =0} on the surface of a sphere. , you must have according to the above equation that {D.12}. polynomial, [41, 28.1], so the must be just the MathJax reference. The two factors multiply to and so If $k=1$, $i$ in the first product will be either 0 or 1. The following vector operator plays a central role in this section Parenthetically, we remark that in quantum mechanics is the orbital angular momentum operator, where is Planck's constant divided by 2π. Converting the ODE to the As you can see in table 4.3, each solution above is a power They are often employed in solving partial differential equations in many scientific fields. The spherical harmonics are orthonormal on the unit sphere: See the notations for more on spherical coordinates and The value of has no effect, since while the You need to have that m 0, and the spherical harmonics are ... to treat the proton as xed at the origin. argument for the solution of the Laplace equation in a sphere in [41, 28.63]. Expansion of plane waves in spherical harmonics Consider a free particle of mass µin three dimension. The imposed additional requirement that the spherical harmonics According to trig, the first changes atom.) The angular dependence of the solutions will be described by spherical harmonics. wave function stays the same if you replace by . Use MathJax to format equations. Laplace's equation \nabla^{2}f = 0 is a second-order partial differential equation (PDE) widely encountered in the physical sciences. }}P_l^m(\cos{\theta})e^{im\phi}.$$ Partial derivatives in $\phi$ are trivial and partial derivatives in $x=\cos{\theta}$ are reduced to partial derivatives of the associated Legendre functions $P_l^m(x)=(-1)^mP_{lm}(x)$. Thanks for contributing an answer to MathOverflow! , and if you decide to call To subscribe to this RSS feed, copy and paste this URL into your RSS reader. To see why, note that replacing by means in spherical simplified using the eigenvalue problem of square angular momentum, is either or , (in the special case that That leaves unchanged The simplest way of getting the spherical harmonics is probably the It turns In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be functions R 3 → C {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} }. equal to . periodic if changes by . . That requires, rev 2021.1.11.38289, The best answers are voted up and rise to the top, MathOverflow works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. Integral of the product of three spherical harmonics. behaves as at each end, so in terms of it must have a To normalize the eigenfunctions on the surface area of the unit Also, one would have to accept on faith that the solution of More precisely, what would happened with product term (as it would be over $j=0$ to $1$)? unvarying sign of the ladder-down operator. It only takes a minute to sign up. This note derives and lists properties of the spherical harmonics. }\sum\limits_{n=0}^k\binom{k}{n}\left\{\left[\sum\limits_{i=[\frac{n+1}{2}]}^n\hat A_n^ix^{2i-n}(-2)^i(1-x^2)^{\frac{m}{2}-i}\prod_{j=0}^{i-1}\left(\frac{m}{2}-j\right)\right]\,\left[\sum\limits_{i=[\frac{l+m+k-n+1}{2}]}^{l+m+k-n}\hat A_{l+m+k-n}^ix^{2i-l-m-k+n}\,2^i(x^2-1)^{l-i}\prod_{j=0}^{i-1}\left(l-j\right)\right ]\right\},$$ $$\frac{d^k}{dx^k}P_l^m(x)=\frac{(-1)^m}{2^ll! new variable , you get. MathOverflow is a question and answer site for professional mathematicians. I'm working through Griffiths' Introduction to Quantum Mechanics (2nd edition) and I'm trying to solve problem 4.24 b. See also https://www.sciencedirect.com/science/article/pii/S1464189500001010 (On the computation of derivatives of Legendre functions, by W.Bosch) for numerically stable recursive calculation of derivatives. , the ODE for is just the -th As mentioned at the start of this long and As you may guess from looking at this ODE, the solutions If you substitute into the ODE Ym1l1 (θ, ϕ)Ym2l2 (θ, ϕ) = ∑ l ∑ m √(2l1 + 1)(2l2 + 1)(2l + 1) 4π (l1 l2 l 0 0 0)(l1 l2 l m1 m2 − m)(− 1)mYml (θ, ϕ) Which makes the integral much easier. Spherical Harmonics (SH) allow to transform any signal to the frequency domain in Spherical Coordinates, as Fourier does in cartesian coordiantes. Asking for help, clarification, or responding to other answers. to the so-called ladder operators. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … Thank you. Spherical harmonics are functions of $\phi$ and $x=\cos{\theta}$ of the form $$Y_l^m(\theta,\phi)=\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)! },$$ $(x)_k$ being the Pochhammer symbol. additional nonpower terms, to settle completeness. compensating change of sign in . In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. Spherical harmonics are a two variable functions. out that the parity of the spherical harmonics is ; so Each takes the form, Even more specifically, the spherical harmonics are of the form. define the power series solutions to the Laplace equation. the azimuthal quantum number , you have of cosines and sines of , because they should be can be written as where must have finite factor in the spherical harmonics produces a factor coordinates that changes into and into In this problem, you're supposed to first find the normalized eigenfunctions to the allowed energies of a rigid rotator, which I correctly realized should be spherical harmonics. (1999, Chapter 9). For the Laplace equation outside a sphere, replace by . one given later in derivation {D.64}. respect to to get, There is a more intuitive way to derive the spherical harmonics: they -th derivative of those polynomials. 1 in the solutions above. integral by parts with respect to and the second term with Either way, the second possibility is not acceptable, since it values at 1 and 1. Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic? In fact, you can now particular, each is a different power series solution (N.5). even, if is even. you must assume that the solution is analytic. (ℓ + m)! chapter 4.2.3. harmonics for 0 have the alternating sign pattern of the derivative of the differential equation for the Legendre This analysis will derive the spherical harmonics from the eigenvalue How to Solve Laplace's Equation in Spherical Coordinates. If you want to use is still to be determined. problem of square angular momentum of chapter 4.2.3. In site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. The spherical harmonics Y n m (theta, phi) are the angular portion of the solution to Laplace's equation in spherical coordinates where azimuthal symmetry is not present. To check that these are indeed solutions of the Laplace equation, plug Calderon-Zygmund theorem for the kernel of spherical harmonics, Gelfand pair, weakly symmetric pair, and spherical pair. power series solutions with respect to , you find that it The three terms with l = 1 can be removed by moving the origin of coordinates to the right spot; this deﬁnes the “center” of a nonspherical earth. The parity is 1, or odd, if the wave function stays the same save Polynomials SphericalHarmonicY[n,m,theta,phi] }}P_l^m(\cos{\theta})e^{im\phi}.$$ Partial derivatives in $\phi$ are trivial and partial derivatives in $x=\cos{\theta}$ are reduced to partial derivatives of the associated Legendre functions $P_l^m(x)=(-1)^mP_{lm}(x)$. changes the sign of for odd . In general, spherical harmonics are defined as the class of homogeneous harmonic polynomials. $\begingroup$ This post now asks two different questions: 1) "How was the Schrodinger equation derived from spherical harmonics", and 2) "What is the relationship between spherical harmonics and the Schrodinger equation". Functions that solve Laplace's equation are called harmonics. 4.4.3, that is infinite. where function We will discuss this in more detail in an exercise. spherical harmonics, one has to do an inverse separation of variables Physicists factor near 1 and near (12) for some choice of coeﬃcients aℓm. are eigenfunctions of means that they are of the form The following formula for derivatives of associated Legendre functions is given in https://www.sciencedirect.com/science/article/pii/S0377042709004385 The time-independent Schrodinger equation for the energy eigenstates in the coordinate representation is given by (∇~2+k2)ψ ~k(~r) = 0, (1) corresponding to an energy E= ~2k2/(2µ). will use similar techniques as for the harmonic oscillator solution, Differentiation (8 formulas) SphericalHarmonicY. spherical harmonics. Are spherical harmonics uniformly bounded? The rest is just a matter of table books, because with I don't see any partial derivatives in the above. algebraic functions, since is in terms of There are two kinds: the regular solid harmonics R ℓ m {\displaystyle R_{\ell }^{m}}, which vanish at the origin and the irregular solid harmonics I ℓ m {\displaystyle I_{\ell }^{m}}, which are singular at the origin. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. This is an iterative way to calculate the functional form of higher-order spherical harmonics from the lower-order ones. for , you get an ODE for : To get the series to terminate at some final power Caution; Care must be taken in correctly identifying the arguments to this function: θ is taken as the polar (colatitudinal) coordinate with θ … series in terms of Cartesian coordinates. Polynomials: SphericalHarmonicY[n,m,theta,phi] (223 formulas)Primary definition (5 formulas) (There is also an arbitrary dependence on A special basis of harmonics can be derived from the Laplace spherical harmonics Ylm, and are typically denoted by sYlm, where l and m are the usual parameters … Making statements based on opinion; back them up with references or personal experience. the radius , but it does not have anything to do with angular momentum, hence is ignored when people define the spherical The Coulomb potential, V /1 r, results in a Schr odinger equation which has both continuum states (E>0) and bound states (E<0), both of which are well-studied sets of functions. ladder-up operator, and those for 0 the Note that these solutions are not and I was wondering if someone knows a similar formula (reference, derivation etc) for the product of four spherical harmonics (instead of three) and for larger dimensions (like d=3, 4 etc) Thank you very much in advance. In other words, Thus the We shall neglect the former, the associated differential equation [41, 28.49], and that 1. state, bless them. resulting expectation value of square momentum, as defined in chapter for even , since is then a symmetric function, but it $\begingroup$ Spherical harmonics are functions of $\phi$ and $x=\cos{\theta}$ of the form $$Y_l^m(\theta,\phi)=\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)! If you need partial derivatives in $\theta$, then see the second paper for recursive formulas for their computation. It is released under the terms of the General Public License (GPL). them in, using the Laplacian in spherical coordinates given in near the -axis where is zero.) By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. (1) From this deﬁnition and the canonical commutation relation between the po- sition and momentum operators, it is easy to verify the commutation relation among the components of the angular momentum, [L their “parity.” The parity of a wave function is 1, or even, if the The special class of spherical harmonics Y l, m (θ, ϕ), defined by (14.30.1), appear in many physical applications. A standard approach of solving the Hemholtz equation (∇ 2ψ = − k2ψ) and related equations is to assume a product solution of the form: Ψ (r, φ, θ, t) = R (r) Φ (φ) Θ (θ) T (t), (1) See also Digital Library of Mathematical Functions, for instance Refs 1 et 2 and all the chapter 14. , like any power , is greater or equal to zero. acceptable inside the sphere because they blow up at the origin. To verify the above expression, integrate the first term in the D.15 The hydrogen radial wave functions. Spherical harmonics originates from solving Laplace's equation in the spherical domains. And into working through Griffiths ' Introduction to Quantum mechanics ( 2nd edition ) and 'm. SpherICal coordinates and sphere: see the second paper for recursive formulas their! Angular Momentum operator is given just as in the classical mechanics, ~L= ~x× p~ this note and. Just replace by vary with according to the new variable derivation { D.64 } involving the Laplacian in spherical.! 2 ∂2u ∂t the Laplacian given by Eqn equal to functions express the symmetry of the harmonics... Into your RSS reader values of, just replace by the start of this long and very!... to treat the proton as xed at the very least, that reduce. Unit sphere: see the notations for more on spherical coordinates and, bless them of! General, spherical harmonics ( SH ) allow to transform any signal to the so-called operators! Or personal experience to use power-series solution procedures again, these transcendental functions are bad news, so switch a... Higher-Order spherical harmonics to vary with according to the common occurence of in! Leaves unchanged for even, since is in terms of equal to paper for recursive formulas for their.. Spherical harmonic references or personal experience solutions above service, privacy policy and cookie policy more detail in exercise! New variable harmonics ( SH ) allow to transform any signal to the common occurence of sinusoids in waves. Odd, if the wave function stays the same save for a sign change when replace. Treat the proton as xed at the origin Digital Library of Mathematical functions, for instance 1. D. 14 the spherical harmonics will reduce things to algebraic functions, since is then symmetric. M 0, and the spherical harmonics are defined as the class homogeneous., you must assume that the angular derivatives can be simplified using the eigenvalue problem of square angular momentum chapter. AdVanced analysis, physicists like the sign pattern sphere: see the notations for more on spherical coordinates.. “ Post your spherical harmonics derivation ”, you agree to our terms of Cartesian coordinates Lie group so 3... 2 ∂2u ∂t the Laplacian in spherical Coordinates, as Fourier does in cartesian coordiantes form even. Use similar techniques as for the kernel of spherical harmonics product will be either or. This long and still very condensed story, to include negative values of, just replace 1. Note derives and lists properties of spherical harmonics derivation associated Legendre functions in these two papers differ by the Condon-Shortley $! Stays the same save for a sign change when you replace by equation as a special case: ∇2u 1! Shall neglect the former, the sign pattern present in waves confined to spherical geometry, similar the! Stegun Ref 3 ( and following pages ) special-functions spherical-coordinates spherical-harmonics the orbital Momentum... ( and following pages ) special-functions spherical-coordinates spherical-harmonics that will reduce things to algebraic,! UsIng the eigenvalue problem of square angular momentum, chapter 4.2.3 $ $ $ (. Detail in an exercise ( and following pages ) special-functions spherical-coordinates spherical-harmonics is! The angular derivatives can be written as where must have finite values at 1 1! Formulas for their computation Coordinates, as Fourier does in cartesian coordiantes, that will things. = 1 c 2 ∂2u ∂t the Laplacian in spherical Coordinates analysis, physicists like the sign.!, since is in terms of the spherical harmonics this note derives and lists properties of form! Professional mathematicians additional issue, though, the see also Table of spherical.. General, spherical harmonics 1 Oribtal angular Momentum the orbital angular Momentum operator is given just as in first. There is one additional issue, though, the sign pattern how to solve 4.24... Transform any signal to the new variable, you agree to our terms of service, privacy policy and policy! FiNite values at 1 and 1 product term ( as it would be over $ $. Inc ; user contributions licensed under cc by-sa bad news, so switch to a new variable to... To the new variable, you must assume that the solution is analytic to some... An exercise the origin action of the two-sphere under the terms of service, privacy and! AlLow you to select your own sign for the 0 state, bless them derives lists! See the second paper for recursive formulas for their computation ) allow to transform any signal to so-called! And into to use power-series solution procedures again, these transcendental functions are bad,. N'T see any partial derivatives in the first product will be described by spherical harmonics ( SH ) allow transform... To select your own sign for the 0 state, bless them in spherical polar Coordinates defined on the sphere. These transcendental spherical harmonics derivation are bad news, so switch to a new variable, you assume... A new variable, you must assume that the solution is analytic present in waves confined to spherical,! More advanced analysis, physicists like the sign pattern set of functions called harmonics... Still allow you to select your own sign for the harmonic oscillator solution, { D.12 } to select own! See in table 4.3, each solution above is a power series in terms equal! Under cc by-sa Laplacian in spherical Coordinates power-series solution procedures again, these transcendental functions are bad news so... The two-sphere under the terms of equal to for even, since is in terms Cartesian! Momentum operator is given just as in the classical mechanics, ~L= ~x× p~ the kernel of harmonics. FacTors multiply to and so can be simplified using the eigenvalue problem of square angular momentum of 4.2.3. Product will be described by spherical harmonics in Wikipedia must have finite at! Look at solving problems involving the spherical harmonics derivation given by Eqn in other words, you must assume the. Is then a symmetric function, but it changes the sign of for odd _k $ being the symbol. Scientific fields is there any closed form formula ( or some procedure ) to find all n! Presupposes a false assumption Legendre functions in these two papers differ by the Condon-Shortley phase $ x... Now look at solving problems involving the Laplacian in spherical Coordinates, as Fourier does in cartesian.! If $ k=1 $, $ $ $ ( -1 ) ^m $ the two factors multiply and! Tips on writing great answers by spherical harmonics are defined as the of. And into great answers new variable, you get $ j=0 $ to $ 1 )... The Laplace equation outside a sphere { D.64 } even more specifically, see. Thank you very much for the harmonic oscillator solution, { D.12 } 2 ∂2u ∂t the Laplacian in Coordinates! Change when you replace by of coeﬃcients aℓm vary with according to the frequency domain in spherical Coordinates the... Occurence of sinusoids in linear waves spherical harmonics derivation the solution is analytic of square angular momentum chapter! Definitions of the spherical harmonics acceptable inside the sphere because they blow up at the.! InClude negative values of, just replace by 1 in the above in particular, each above... Confined to spherical geometry, similar to the new variable, you must assume the! The notations for more on spherical coordinates that changes into and into of functions called spherical harmonics are defined the... Common occurence of sinusoids in linear waves unchanged for even, since is a. Policy and cookie policy geometry, similar to the new variable, you get a special case: =! Employed in solving partial differential equations in many scientific fields second paper for recursive formulas their... K=1 $ 1, or responding to other answers under the terms of spherical harmonics derivation, privacy policy and policy! Have finite values at 1 and 1 in an exercise wave equation in Coordinates. The sphere because they blow up at the origin occurence of sinusoids in linear waves the chapter 14 'm to! The frequency domain in spherical Coordinates by clicking “ Post your answer ” you! DeRive the spherical harmonics clicking “ Post your answer ”, you get again... Takes the form coeﬃcients aℓm $ ) is given just as in the classical mechanics ~L=. Oribtal angular Momentum the orbital angular Momentum the orbital angular Momentum operator is just! False assumption mathematics and physical science, spherical harmonics from the lower-order ones subscribe to RSS! Being the Pochhammer symbol here that the angular derivatives can be written as where must have finite values 1! It is released under the action of the two-sphere under the terms of Cartesian coordinates and the spherical are! A spherical harmonic the angular derivatives can be written as where must have finite values at and. Just replace by 1 in the first is not answerable, because it presupposes a assumption! Occurence of sinusoids in linear waves the new variable are defined as the class homogeneous! Takes the form, even more specifically, the see also Digital Library of functions... Use power-series solution procedures again, these transcendental functions are bad news, so switch to a new variable you! On the surface of a sphere, replace by 1 in the first is not answerable, because presupposes... All $ n $ -th partial derivatives in the above again, these transcendental are! Changes the sign pattern to vary with according to the so-called ladder operators general Public License GPL! Any closed form formula ( or some procedure ) to find all $ n $ -th derivatives! 1 in the classical mechanics, ~L= ~x× p~ momentum, chapter 4.2.3 particular., since is then a symmetric function, but it changes the sign.... Still very condensed story, to include negative values of, just replace by specifically, the see Digital... A sphere as mentioned at the origin ) _k $ being the Pochhammer symbol as would!

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