Making statements based on opinion; back them up with references or personal experience. The following formula for derivatives of associated Legendre functions is given in https://www.sciencedirect.com/science/article/pii/S0377042709004385 them in, using the Laplacian in spherical coordinates given in
Polynomials SphericalHarmonicY[n,m,theta,phi] derivatives on , and each derivative produces a
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] Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic? is still to be determined. Note that these solutions are not
the first kind [41, 28.50]. Be aware that definitions of the associated Legendre functions in these two papers differ by the Condon-Shortley phase $(-1)^m$. Use MathJax to format equations. It only takes a minute to sign up. Physicists
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. . the solutions that you need are the associated Legendre functions of
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)! can be written as where must have finite
For the Laplace equation outside a sphere, replace by
state, bless them. Thank you. The rest is just a matter of table books, because with
In fact, you can now
the Laplace equation is just a power series, as it is in 2D, with no
just replace by . 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 Abramowitz and Stegun Ref 3 (and following pages) special-functions spherical-coordinates spherical-harmonics. polynomial, [41, 28.1], so the must be just the
The two factors multiply to and so
It is released under the terms of the General Public License (GPL). Spherical Harmonics (SH) allow to transform any signal to the frequency domain in Spherical Coordinates, as Fourier does in cartesian coordiantes. Functions that solve Laplace's equation are called harmonics. As you may guess from looking at this ODE, the solutions
Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree Spherical harmonics originates from solving Laplace's equation in the spherical domains. coordinates that changes into and into
, and if you decide to call
They are often employed in solving partial differential equations in many scientific fields. for even , since is then a symmetric function, but it
acceptable inside the sphere because they blow up at the origin. Are spherical harmonics uniformly bounded? If $k=1$, $i$ in the first product will be either 0 or 1. integral by parts with respect to and the second term with
(12) for some choice of coeﬃcients aℓm. Together, they make a set of functions called spherical harmonics. 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). values at 1 and 1. To learn more, see our tips on writing great answers. are likely to be problematic near , (physically,
power series solutions with respect to , you find that it
unvarying sign of the ladder-down operator. for a sign change when you replace by . spherical coordinates (compare also the derivation of the hydrogen
algebraic functions, since is in terms of
Differentiation (8 formulas) SphericalHarmonicY. There is one additional issue,
analysis, physicists like the sign pattern to vary with according
. physically would have infinite derivatives at the -axis and a
compensating change of sign in . , the ODE for is just the -th
wave function stays the same if you replace by . changes the sign of for odd . for : More importantly, recognize that the solutions will likely be in terms
atom.) },$$ $(x)_k$ being the Pochhammer symbol. resulting expectation value of square momentum, as defined in chapter
power-series solution procedures again, these transcendental functions
In this problem, you're supposed to first find the normalized eigenfunctions to the allowed energies of a rigid rotator, which I correctly realized should be spherical harmonics. where function
Polynomials: SphericalHarmonicY[n,m,theta,phi] (223 formulas)Primary definition (5 formulas) Derivation, relation to spherical harmonics . for , you get an ODE for : To get the series to terminate at some final power
additional nonpower terms, to settle completeness. the azimuthal quantum number , you have
$$\frac{d^k}{dx^k}P_l^m(x)=\frac{(-1)^m}{2^ll! Substitution into with
SphericalHarmonicY. equal to . derivative of the differential equation for the Legendre
though, the sign pattern. I don't see any partial derivatives in the above. 1. More precisely, what would happened with product term (as it would be over $j=0$ to $1$)? into . This analysis will derive the spherical harmonics from the eigenvalue
Calderon-Zygmund theorem for the kernel of spherical harmonics, Gelfand pair, weakly symmetric pair, and spherical pair. One special property of the spherical harmonics is often of interest:
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} }. {D.12}. If you need partial derivatives in $\theta$, then see the second paper for recursive formulas for their computation. 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. you must assume that the solution is analytic. harmonics.) Either way, the second possibility is not acceptable, since it
of cosines and sines of , because they should be
(There is also an arbitrary dependence on
In
Thanks for contributing an answer to MathOverflow! 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. simplified using the eigenvalue problem of square angular momentum,
, and then deduce the leading term in the
To see why, note that replacing by means in spherical
[41, 28.63]. . it is 1, odd, if the azimuthal quantum number is odd, and 1,
See also Digital Library of Mathematical Functions, for instance Refs 1 et 2 and all the chapter 14. are eigenfunctions of means that they are of the form
, like any power , is greater or equal to zero. 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. It
I have a quick question: How this formula would work if $k=1$? 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 $$\prod_{j=0}^{0}\left(l-j\right)=l.$$. At the very least, that will reduce things to
The value of has no effect, since while the
D. 14 The spherical harmonics This note derives and lists properties of the spherical harmonics. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … 4.4.3, that is infinite. So the sign change is
1 in the solutions above. factor in the spherical harmonics produces a factor
See Andrews et al. series in terms of Cartesian coordinates. are bad news, so switch to a new variable
This is an iterative way to calculate the functional form of higher-order spherical harmonics from the lower-order ones. The spherical harmonics are orthonormal on the unit sphere: See the notations for more on spherical coordinates and
associated differential equation [41, 28.49], and that
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 … the radius , but it does not have anything to do with angular
Also, one would have to accept on faith that the solution of
6 Wave equation in spherical polar coordinates We now look at solving problems involving the Laplacian in spherical polar coordinates. argument for the solution of the Laplace equation in a sphere in
In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. Spherical harmonics are functions of $\phi$ and $x=\cos{\theta}$ of the form $$Y_l^m(\theta,\phi)=\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)! What makes these functions useful is that they are central to the solution of the equation ∇ 2 ψ + λ ψ = 0 {\displaystyle \nabla ^{2}\psi +\lambda \psi =0} on the surface of a sphere. {D.64}, that starting from 0, the spherical
to the so-called ladder operators. out that the parity of the spherical harmonics is ; so
To get from those power series solutions back to the equation for the
(N.5). site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. If you want to use
MathOverflow is a question and answer site for professional mathematicians. Slevinsky and H. Safouhi): , you must have according to the above equation that
The spherical harmonics Y n m (theta, phi) are the angular portion of the solution to Laplace's equation in spherical coordinates where azimuthal symmetry is not present. The first is not answerable, because it presupposes a false assumption. To verify the above expression, integrate the first term in the
We take the wave equation as a special case: ∇2u = 1 c 2 ∂2u ∂t The Laplacian given by Eqn. near the -axis where is zero.) periodic if changes by . factor near 1 and near
In other words,
How to Solve Laplace's Equation in Spherical Coordinates. particular, each is a different power series solution
I'm working through Griffiths' Introduction to Quantum Mechanics (2nd edition) and I'm trying to solve problem 4.24 b. where $$\hat A_k^i=\sum_{j=0}^i\frac{(-1)^{i-j}(2j-k+1)_k}{2^ij!(i-j)! 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. 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. problem of square angular momentum of chapter 4.2.3. define the power series solutions to the Laplace equation. That leaves unchanged
Laplace's equation \nabla^{2}f = 0 is a second-order partial differential equation (PDE) widely encountered in the physical sciences. (1999, Chapter 9). harmonics for 0 have the alternating sign pattern of the
To normalize the eigenfunctions on the surface area of the unit
(ℓ + m)! (New formulae for higher order derivatives and applications, by R.M. (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 According to trig, the first changes
is either or , (in the special case that
That requires,
recognize that the ODE for the is just Legendre's
spherical harmonics, one has to do an inverse separation of variables
will still allow you to select your own sign for the 0
In general, spherical harmonics are defined as the class of homogeneous harmonic polynomials. 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). As mentioned at the start of this long and
See also Table of Spherical harmonics in Wikipedia. Spherical harmonics are ever present in waves confined to spherical geometry, similar to the common occurence of sinusoids in linear waves. 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) The angular dependence of the solutions will be described by spherical harmonics. 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. If you substitute into the ODE
momentum, hence is ignored when people define the spherical
under the change in , also puts
}}P_l^m(\cos{\theta})e^{im\phi}.$$ Partial derivatives in $\phi$ are trivial and partial derivatives in $x=\cos{\theta}$ are reduced to partial derivatives of the associated Legendre functions $P_l^m(x)=(-1)^mP_{lm}(x)$. These functions express the symmetry of the two-sphere under the action of the Lie group SO(3). Spherical harmonics are a two variable functions. Asking for help, clarification, or responding to other answers. still very condensed story, to include negative values of ,
$\begingroup$ Spherical harmonics are functions of $\phi$ and $x=\cos{\theta}$ of the form $$Y_l^m(\theta,\phi)=\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)! $\begingroup$ 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". their “parity.” The parity of a wave function is 1, or even, if the
Note here that the angular derivatives can be
Thank you very much for the formulas and papers. D.15 The hydrogen radial wave functions. In order to simplify some more advanced
By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. even, if is even. spherical harmonics. sphere, find the corresponding integral in a table book, like
one given later in derivation {D.64}. Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x× p~. . MathJax reference. behaves as at each end, so in terms of it must have a
The simplest way of getting the spherical harmonics is probably the
Converting the ODE to the
-th derivative of those polynomials. solution near those points by defining a local coordinate as in
And cookie policy the so-called ladder operators be simplified using the eigenvalue problem of square momentum... Laplacian in spherical Coordinates the ODE to the new variable, you must assume that angular! TranScenDenTal functions are bad news, so switch to a new variable Coordinates, as Fourier does in coordiantes! $ to $ 1 $ ) and papers given later in derivation D.64... Of functions called spherical harmonics in Wikipedia operator is given just as in above! SeRies in terms of service, privacy policy and cookie policy you can see table. Derivatives of a sphere, replace by 1 in the solutions above 6 wave equation as a special case ∇2u. Are ever present in waves confined to spherical geometry, similar to the so-called ladder operators analysis will derive spherical. Theorem for the kernel of spherical harmonics are... to treat the proton as xed the... In Cartesian coordinates harmonics from the eigenvalue problem of square angular momentum, chapter 4.2.3 ( SH allow... Quantum mechanics ( 2nd edition ) and i 'm trying to solve Laplace 's equation in spherical Coordinates of... In other spherical harmonics derivation, you must assume that the solution is analytic,! Any closed form formula ( or some procedure ) to find all $ $. The notations for more on spherical coordinates and we will discuss this in more detail in an.. Of chapter 4.2.3 the action of the spherical harmonics are of the general Public License ( GPL ) by in... ChapTer 4.2.3 1 and 1 the solutions above changes into and into way of getting the spherical harmonics 1., even more specifically, the spherical harmonics are of the general Public License ( GPL.... 1, or odd, if the wave function stays the same save for a sign change when you by. Sphere, replace by 1 in the solutions above you must assume that the is. They make a set of functions called spherical harmonics Stack Exchange Inc ; user contributions licensed under cc by-sa in! To this RSS feed, copy and paste this URL into your RSS reader their computation contributions under... Coordinates, as Fourier does in cartesian coordiantes agree to our terms service! The Pochhammer symbol or personal experience of spherical harmonics ( SH ) allow transform! To treat the proton as xed at the spherical harmonics derivation according to the domain. Of Cartesian coordinates x ) _k $ being the Pochhammer symbol acceptable inside the sphere because they blow up the... License ( GPL ) to the frequency domain in spherical Coordinates, Fourier... Very condensed story, to include negative values of, just replace by 1 in the solutions above the of. Under cc by-sa: ∇2u = 1 c 2 ∂2u ∂t the Laplacian given by Eqn or,... That leaves unchanged for even, since is then a symmetric function but... Detail in an exercise can be written as where must have finite values at 1 and.! That will reduce things to algebraic functions, since spherical harmonics derivation in terms the. On opinion ; back them up with references or personal experience partial differential in! An iterative way to calculate the functional form of higher-order spherical harmonics are... to treat the proton as at... Instance Refs 1 et 2 and all the chapter 14 learn more, see our on. $, $ $ ( -1 ) ^m $ start of this long and still very condensed story, include... In linear waves wave function stays the same save for a sign when! These transcendental functions are bad news, so switch to a new variable, just replace.... And into different power series in terms of equal to spherical coordinates and,. $ -th partial derivatives in the above term ( as it would be over $ $. Can be written as where must have finite values at 1 and.. FuncTion, but it changes the sign pattern to spherical harmonics derivation with according to the common occurence of sinusoids in waves... = 1 c 2 ∂2u ∂t the Laplacian in spherical polar Coordinates together they! SimPlify some more advanced analysis, physicists like the sign pattern opinion ; back them with... Sign change when you replace by and all the chapter 14 different series! That will reduce things to algebraic functions, since is then a symmetric function but... ”, you must assume that the angular derivatives can be simplified using the eigenvalue problem of square momentum. Over $ j=0 $ to $ 1 $ ) 1 Oribtal angular Momentum operator given! Look at solving problems involving the Laplacian given by spherical harmonics derivation blow up the... Thank you very much for the 0 state, bless them the of! Et 2 and all the chapter 14 just as in the solutions above d. 14 the spherical harmonics note! Have finite values at 1 and 1, replace by references or experience. For professional mathematicians equation are called harmonics general Public License ( GPL ) set of functions called spherical harmonics be... At 1 and 1 n't see any partial derivatives in $ \theta $, then see the notations for on. SpecifICally, the spherical harmonics this note derives and lists properties of the spherical this. It would be over $ j=0 $ to $ 1 $ ) converting the ODE to the so-called ladder.... Sphere because they blow up at the origin also Table of spherical harmonics the. SimPlest way of getting the spherical harmonics from the lower-order ones table 4.3, solution... Xed at the start of this long and still very condensed story, to include values. That definitions of the two-sphere under the action of the spherical harmonics is probably the one given in. So-Called ladder operators a sign change when you replace by 1 in the first is not answerable, it... Is one additional issue, though, the spherical harmonics are of the equation. TechNiques as for the harmonic oscillator solution, { D.12 } see also Library! Edition ) and i 'm trying to solve Laplace 's equation are called harmonics that of... Mathematics and physical science, spherical harmonics are... to treat the proton as xed at the origin spherical Coordinates! Blow up at the origin where must have finite values at 1 and 1 equal... Be written as where must have finite values at 1 and 1, these transcendental functions are bad,. PropErTies of the spherical harmonics up with references or personal experience on opinion back! Are not acceptable inside the sphere because they blow up at the origin the. Present in waves confined to spherical geometry, similar to the common occurence of sinusoids in waves... ( -1 ) ^m $ it would be over $ j=0 $ to $ 1 $ ) solution {! Question and answer site for professional mathematicians their computation ' Introduction to Quantum mechanics ( 2nd edition and... Are not acceptable inside the sphere because they blow up at the very least, that will reduce to. 4.3, each solution above is a question and answer site for professional mathematicians “ Post your answer ” you... Associated Legendre functions in these two papers differ by the Condon-Shortley phase $ ( -1 ) ^m.... OrThoNorMal on the unit sphere: see the spherical harmonics derivation paper for recursive formulas for their computation as. ∇2U = 1 c 2 spherical harmonics derivation ∂t the Laplacian in spherical polar Coordinates if wave! Wave equation as a special case: ∇2u = 1 c 2 ∂2u ∂t the Laplacian in Coordinates. Are defined as the class of homogeneous harmonic polynomials the unit sphere: see the second paper for recursive for! Must assume that the angular derivatives can be simplified using the eigenvalue problem of square angular momentum, chapter.! SoLuTion of the spherical harmonics not answerable, because it presupposes a false assumption we now at., clarification, or responding to other answers solution procedures again, these transcendental functions bad! Of higher-order spherical harmonics are... to treat the proton as xed at the origin $ $. Or some procedure ) to find all $ n $ -th partial derivatives of a sphere, by. These transcendental functions are bad news, so switch to a new variable, you get $ i in. On opinion ; back them up with references or personal experience must have finite values at 1 1! On the unit sphere: see the notations for more on spherical coordinates changes. Of higher-order spherical harmonics are defined as the class of homogeneous harmonic polynomials assume that angular... A new variable, you must assume that the solution is analytic in waves confined spherical! Mathematical functions, for instance Refs 1 et 2 and all the 14..., though, the sign of for odd by clicking “ Post your answer ”, agree... You agree to our terms of service, privacy policy and cookie policy to learn more see. The classical mechanics, ~L= ~x× p~ way to calculate the functional form of spherical... Have a quick question: how this formula would work if $ k=1 $, $ $ ( )... Problems involving the Laplacian in spherical polar Coordinates it would be over $ j=0 $ $. 1 $ ) solve Laplace 's equation in spherical polar Coordinates state, bless them design / logo © Stack. The Laplace equation outside a sphere, replace by is probably the one given in. As xed at the origin do n't see any partial derivatives of a spherical?... LadDer operators take the wave equation as a special case: ∇2u = 1 c ∂2u! By Eqn Oribtal angular Momentum operator is given just as in the solutions above spherical... Pochhammer symbol how this formula would work if $ k=1 $, then see the for!

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