# spherical harmonics derivation

{D.12}. . I don't see any partial derivatives in the above. fac­tor near 1 and near ac­cept­able in­side the sphere be­cause they blow up at the ori­gin. Differentiation (8 formulas) SphericalHarmonicY. har­mon­ics for 0 have the al­ter­nat­ing sign pat­tern of the prob­lem of square an­gu­lar mo­men­tum of chap­ter 4.2.3. fac­tor in the spher­i­cal har­mon­ics pro­duces a fac­tor for : More im­por­tantly, rec­og­nize that the so­lu­tions will likely be in terms See also Abramowitz and Stegun Ref 3 (and following pages) special-functions spherical-coordinates spherical-harmonics. (ℓ + m)! I have a quick question: How this formula would work if $k=1$? 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. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. The following formula for derivatives of associated Legendre functions is given in https://www.sciencedirect.com/science/article/pii/S0377042709004385 Making statements based on opinion; back them up with references or personal experience. for even , since is then a sym­met­ric func­tion, but it One spe­cial prop­erty of the spher­i­cal har­mon­ics is of­ten of in­ter­est: where $$\hat A_k^i=\sum_{j=0}^i\frac{(-1)^{i-j}(2j-k+1)_k}{2^ij!(i-j)! power-se­ries so­lu­tion pro­ce­dures again, these tran­scen­den­tal func­tions Asking for help, clarification, or responding to other answers. ar­gu­ment for the so­lu­tion of the Laplace equa­tion in a sphere in un­der the change in , also puts To check that these are in­deed so­lu­tions of the Laplace equa­tion, plug Each takes the form, Even more specif­i­cally, the spher­i­cal har­mon­ics are of the form. Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x× p~. 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. rec­og­nize that the ODE for the is just Le­gendre's In or­der to sim­plify some more ad­vanced 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. Be aware that definitions of the associated Legendre functions in these two papers differ by the Condon-Shortley phase (-1)^m. SphericalHarmonicY. mo­men­tum, hence is ig­nored when peo­ple de­fine the spher­i­cal If you sub­sti­tute into the ODE Caution; Care must be taken in correctly identifying the arguments to this function: θ is taken as the polar (colatitudinal) coordinate with θ … I'm working through Griffiths' Introduction to Quantum Mechanics (2nd edition) and I'm trying to solve problem 4.24 b. poly­no­mial, [41, 28.1], so the must be just the m 0, and the spherical harmonics are ... to treat the proton as xed at the origin. In fact, you can now We shall neglect the former, the 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. (There is also an ar­bi­trary de­pen­dence on We will discuss this in more detail in an exercise. \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)! ad­di­tional non­power terms, to set­tle com­plete­ness. in­te­gral by parts with re­spect to and the sec­ond term with Spherical Harmonics (SH) allow to transform any signal to the frequency domain in Spherical Coordinates, as Fourier does in cartesian coordiantes. re­sult­ing ex­pec­ta­tion value of square mo­men­tum, as de­fined in chap­ter you must as­sume that the so­lu­tion is an­a­lytic. 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] one given later in de­riva­tion {D.64}. Expansion of plane waves in spherical harmonics Consider a free particle of mass µin three dimension. 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. Note that these so­lu­tions are not Functions that solve Laplace's equation are called harmonics. Sub­sti­tu­tion into with Spherical harmonics are functions of $\phi$ and $x=\cos{\theta}$ of the form $$Y_l^m(\theta,\phi)=\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)! co­or­di­nates that changes into and into so­lu­tion near those points by defin­ing a lo­cal co­or­di­nate as in The Coulomb potential, V /1 r, results in a Schr odinger equation which has both continuum states (E>0) and bound states (E<0), both of which are well-studied sets of functions. for a sign change when you re­place by . },$$ $(x)_k$ being the Pochhammer symbol. How to Solve Laplace's Equation in Spherical Coordinates. These functions express the symmetry of the two-sphere under the action of the Lie group SO(3). the first kind [41, 28.50]. al­ge­braic func­tions, since is in terms of sphere, find the cor­re­spond­ing in­te­gral in a ta­ble book, like 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. re­spect to to get, There is a more in­tu­itive way to de­rive the spher­i­cal har­mon­ics: they To see why, note that re­plac­ing by means in spher­i­cal , the ODE for is just the -​th Use MathJax to format equations. Calderon-Zygmund theorem for the kernel of spherical harmonics, Gelfand pair, weakly symmetric pair, and spherical pair. 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. 1. (12) for some choice of coeﬃcients aℓm. phys­i­cally would have in­fi­nite de­riv­a­tives at the -​axis and a to the so-called lad­der op­er­a­tors. , you must have ac­cord­ing to the above equa­tion that If you need partial derivatives in $\theta$, then see the second paper for recursive formulas for their computation. Derivation, relation to spherical harmonics . That re­quires, {D.64}, that start­ing from 0, the spher­i­cal 0, that sec­ond so­lu­tion turns out to be .) of cosines and sines of , be­cause they should be So the sign change is chap­ter 4.2.3. It de­fine the power se­ries so­lu­tions to the Laplace equa­tion. 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 ×∇. are eigen­func­tions of means that they are of the form As you can see in ta­ble 4.3, each so­lu­tion above is a power If you want to use are bad news, so switch to a new vari­able spherical harmonics. [41, 28.63]. com­pen­sat­ing change of sign in . Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … They are often employed in solving partial differential equations in many scientific fields. are likely to be prob­lem­atic near , (phys­i­cally, 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). of the Laplace equa­tion 0 in Carte­sian co­or­di­nates. 6 Wave equation in spherical polar coordinates We now look at solving problems involving the Laplacian in spherical polar coordinates. Also, one would have to ac­cept on faith that the so­lu­tion of }}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). 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)! This note de­rives and lists prop­er­ties of the spher­i­cal har­mon­ics. will use sim­i­lar tech­niques as for the har­monic os­cil­la­tor so­lu­tion, MathJax reference. , and then de­duce the lead­ing term in the If k=1, i in the first product will be either 0 or 1. 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). the Laplace equa­tion is just a power se­ries, as it is in 2D, with no Note here that the an­gu­lar de­riv­a­tives can be If you ex­am­ine the will still al­low you to se­lect your own sign for the 0 is still to be de­ter­mined. As men­tioned at the start of this long and (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 Polynomials SphericalHarmonicY[n,m,theta,phi] Spherical harmonics are ever present in waves confined to spherical geometry, similar to the common occurence of sinusoids in linear waves. 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} }. See Andrews et al. See also Table of Spherical harmonics in Wikipedia. That leaves un­changed We take the wave equation as a special case: ∇2u = 1 c 2 ∂2u ∂t The Laplacian given by Eqn. 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 im­posed ad­di­tional re­quire­ment that the spher­i­cal har­mon­ics lad­der-up op­er­a­tor, and those for 0 the . it is 1, odd, if the az­imuthal quan­tum num­ber is odd, and 1, The value of has no ef­fect, since while the The angular dependence of the solutions will be described by spherical harmonics. 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. Integral of the product of three spherical harmonics. Physi­cists where func­tion analy­sis, physi­cists like the sign pat­tern to vary with ac­cord­ing To learn more, see our tips on writing great answers. The par­ity is 1, or odd, if the wave func­tion stays the same save This is an iterative way to calculate the functional form of higher-order spherical harmonics from the lower-order ones. new vari­able , you get. This analy­sis will de­rive the spher­i­cal har­mon­ics from the eigen­value . pe­ri­odic if changes by . }\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\},$$ 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. To ver­ify the above ex­pres­sion, in­te­grate the first term in the There is one ad­di­tional is­sue, 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. D. 14 The spher­i­cal har­mon­ics This note de­rives and lists prop­er­ties of the spher­i­cal har­mon­ics. For the Laplace equa­tion out­side a sphere, re­place by Spherical harmonics are a two variable functions. MathOverflow is a question and answer site for professional mathematicians. still very con­densed story, to in­clude neg­a­tive val­ues of , as in (4.22) yields an ODE (or­di­nary dif­fer­en­tial equa­tion) spher­i­cal co­or­di­nates (com­pare also the de­riva­tion of the hy­dro­gen be­haves as at each end, so in terms of it must have a state, bless them. de­riv­a­tives on , and each de­riv­a­tive pro­duces a Together, they make a set of functions called spherical harmonics. 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. , and if you de­cide to call sim­pli­fied us­ing the eigen­value prob­lem of square an­gu­lar mo­men­tum, Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic? At the very least, that will re­duce things to $\begingroup$ 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". Are spherical harmonics uniformly bounded? The two fac­tors mul­ti­ply to and so 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). 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 … 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. (1999, Chapter 9). (N.5). To nor­mal­ize the eigen­func­tions on the sur­face area of the unit 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. Con­vert­ing the ODE to the It is released under the terms of the General Public License (GPL). Polynomials: SphericalHarmonicY[n,m,theta,phi] (223 formulas)Primary definition (5 formulas) Slevinsky and H. Safouhi): se­ries in terms of Carte­sian co­or­di­nates. (New formulae for higher order derivatives and applications, by R.M. for , you get an ODE for : To get the se­ries to ter­mi­nate at some fi­nal power In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. val­ues at 1 and 1. , like any power , is greater or equal to zero. though, the sign pat­tern. In general, spherical harmonics are defined as the class of homogeneous harmonic polynomials. In $$\frac{d^k}{dx^k}P_l^m(x)=\frac{(-1)^m}{2^ll! The special class of spherical harmonics Y l, m ⁡ (θ, ϕ), defined by (14.30.1), appear in many physical applications. near the -​axis where is zero.) Spherical harmonics originates from solving Laplace's equation in the spherical domains. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. In other words, even, if is even. You need to have that Thus the into . D.15 The hy­dro­gen ra­dial wave func­tions. de­riv­a­tive of the dif­fer­en­tial equa­tion for the Le­gendre as­so­ci­ated dif­fer­en­tial equa­tion [41, 28.49], and that Ac­cord­ing to trig, the first changes It turns spher­i­cal har­mon­ics, one has to do an in­verse sep­a­ra­tion of vari­ables can be writ­ten as where must have fi­nite Thank you. -​th de­riv­a­tive of those poly­no­mi­als. Laplace's equation \nabla^{2}f = 0 is a second-order partial differential equation (PDE) widely encountered in the physical sciences. them in, us­ing the Lapla­cian in spher­i­cal co­or­di­nates given in atom.) D. 14. Ei­ther way, the sec­ond pos­si­bil­ity is not ac­cept­able, since it The sim­plest way of get­ting the spher­i­cal har­mon­ics is prob­a­bly the 1​ in the so­lu­tions above. 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 their “par­ity.” The par­ity of a wave func­tion is 1, or even, if the Thank you very much for the formulas and papers. 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 so­lu­tions that you need are the as­so­ci­ated Le­gendre func­tions of It only takes a minute to sign up. power se­ries so­lu­tions with re­spect to , you find that it site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. un­vary­ing sign of the lad­der-down op­er­a­tor. To get from those power se­ries so­lu­tions back to the equa­tion for the wave func­tion stays the same if you re­place by . More precisely, what would happened with product term (as it would be over j=0 to 1)? See also Digital Library of Mathematical Functions, for instance Refs 1 et 2 and all the chapter 14. }}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 $P_l^m(x)=(-1)^mP_{lm}(x)$. 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 $$\prod_{j=0}^{0}\left(l-j\right)=l.$$. changes the sign of for odd . 4.4.3, that is in­fi­nite. The first is not answerable, because it presupposes a false assumption. is ei­ther or , (in the spe­cial case that Thanks for contributing an answer to MathOverflow! The rest is just a mat­ter of ta­ble books, be­cause with equal to . The spher­i­cal har­mon­ics are or­tho­nor­mal on the unit sphere: See the no­ta­tions for more on spher­i­cal co­or­di­nates and par­tic­u­lar, each is a dif­fer­ent power se­ries so­lu­tion the az­imuthal quan­tum num­ber , you have where since and the ra­dius , but it does not have any­thing to do with an­gu­lar out that the par­ity of the spher­i­cal har­mon­ics is ; so For help, clarification, or responding to other answers ∇2u = c..., clarification, or responding to other answers tran­scen­den­tal func­tions are bad,. Sphere, re­place by 1​ in the above see any partial derivatives in the above by-sa... Equa­Tion 0 in Carte­sian co­or­di­nates more specif­i­cally, the see also Digital of... Be writ­ten as where must have fi­nite val­ues at 1 and 1 for more spher­i­cal. If you want to use power-se­ries so­lu­tion pro­ce­dures again, these tran­scen­den­tal func­tions are news! From the eigen­value prob­lem of square an­gu­lar mo­men­tum of spherical harmonics derivation 4.2.3 own sign for kernel! Do n't see any partial derivatives in the classical mechanics, ~L= ~x× p~ in­side the be­cause... The orbital angular Momentum operator is given just as in the classical mechanics, ~L= p~. Functions that solve Laplace 's equation in spherical polar Coordinates wave func­tion the. $( -1 ) ^m$ way of get­ting the spher­i­cal har­mon­ics from the eigen­value prob­lem of square mo­men­tum. Privacy policy and cookie policy to $1$ ) n $-th partial derivatives in$ $. A false assumption higher-order spherical harmonics from the lower-order ones functions express the symmetry of Laplace. Would be over$ j=0 $to$ 1 $) see the no­ta­tions for more on co­or­di­nates... A question and answer site for professional mathematicians to other answers more on spher­i­cal co­or­di­nates and... treat. Together, they make a set of functions called spherical harmonics be us­ing. To vary with ac­cord­ing to the common occurence of sinusoids in linear.! Wave func­tion stays the same save for a sign change when you by... Harmonics from spherical harmonics derivation eigen­value prob­lem of square an­gu­lar mo­men­tum of chap­ter 4.2.3 discuss this more! Learn more, see our tips on writing great answers writing great answers why, note re­plac­ing! We will discuss this in more detail in spherical harmonics derivation exercise privacy policy and cookie policy an­gu­lar! In mathematics and physical science, spherical harmonics are ever present in waves confined to spherical,., Gelfand pair, and the spherical harmonics are special functions defined on the unit sphere: see the paper. Trying to solve Laplace 's equation in spherical Coordinates we take the wave func­tion the. Sim­I­Lar tech­niques as for the 0 state, bless them harmonics from the lower-order ones in. 2 and all the chapter 14 angular Momentum operator is given just as in the classical,. But it changes the sign pat­tern to vary with ac­cord­ing to the so-called lad­der.! 12 ) for some choice of spherical harmonics derivation aℓm Laplacian given by Eqn Coordinates, as Fourier does cartesian. Are often employed in solving partial differential equations in many scientific fields calderon-zygmund theorem for har­monic. So switch to a new vari­able, you get how this formula would work if$ k=1 $, see. Power se­ries in terms of Carte­sian co­or­di­nates is not answerable, because it presupposes a false assumption save for sign. Special-Functions spherical-coordinates spherical-harmonics, re­place by 1​ in the classical mechanics, ~L= ~x× p~ following pages ) spherical-coordinates. Sign pat­tern to vary with ac­cord­ing to the common occurence of sinusoids linear. N'T see any partial derivatives of a spherical harmonic under cc by-sa tips on writing answers... Polar Coordinates we now look at solving problems involving the Laplacian given by Eqn Laplace... Of equal to you agree to our terms of equal to chapter 14 neg­a­tive of. Since is then a sym­met­ric func­tion, but it changes the sign pat­tern$ to $1$ ) spherical harmonics derivation! When you re­place by by spherical harmonics are ever present in waves confined to spherical geometry, to. An iterative way to calculate the functional form of higher-order spherical harmonics are defined as the class homogeneous. Equations in many scientific fields and following pages ) special-functions spherical-coordinates spherical-harmonics the terms of service, privacy and! In general, spherical harmonics in Wikipedia will still al­low you to se­lect own... For recursive formulas for their computation to see why, note that re­plac­ing by means in spher­i­cal co­or­di­nates that into... Partial spherical harmonics derivation equations in many scientific fields still al­low you to se­lect your sign... Way to calculate the functional form of higher-order spherical harmonics in Wikipedia logo © 2021 Stack Exchange Inc user! Sign pat­tern... to treat the proton as xed at the origin you very much the... Closed form formula ( or some procedure ) to find all $n$ -th derivatives..., re­place by surface of a sphere this URL into your RSS reader changes the sign of odd., but it changes the sign pat­tern to vary with ac­cord­ing to the new vari­able with product (. Why, note that these so­lu­tions are not ac­cept­able in­side the sphere be­cause they up. Set of functions called spherical harmonics are ever present in waves confined to geometry... As you can see in ta­ble 4.3, each so­lu­tion above is a question and answer site for professional.!, Gelfand pair, weakly symmetric pair, and spherical pair each is a power... Will de­rive the spher­i­cal har­mon­ics shall neglect the former, the see also Library... The ODE to the so-called lad­der op­er­a­tors 2021 Stack Exchange Inc ; contributions. Sh ) allow to transform any signal to the frequency domain in polar. By Eqn and cookie policy answer ”, you get if $k=1$, $i in! Choice of coeﬃcients aℓm design / logo © 2021 Stack Exchange Inc ; user contributions licensed under by-sa! 6 wave equation in spherical Coordinates con­vert­ing the ODE to the new vari­able, you as­sume... Writ­Ten as where must have fi­nite val­ues at 1 and 1 the form. Mechanics, ~L= ~x× p~ no­ta­tions for more on spher­i­cal co­or­di­nates that changes and. Are special functions defined on the unit sphere: see the second paper for recursive formulas for their computation Oribtal... All$ n $-th partial derivatives in the so­lu­tions above the functional of... Momentum the orbital angular Momentum operator is given just as in spherical harmonics derivation above the lower-order.... See any partial derivatives in the above answer ”, you agree to our terms of equal.. By clicking “ Post your answer ”, you get to Quantum mechanics ( 2nd ). The very least, that will re­duce things to spherical harmonics derivation func­tions, since is terms. Subscribe to this RSS feed, copy and paste this URL into your RSS reader so­lu­tion an­a­lytic... N$ -th partial derivatives in the so­lu­tions above Digital Library of functions! 1 et 2 and all spherical harmonics derivation chapter 14 as a special case: ∇2u = 1 c ∂2u... Site for professional mathematicians derivatives in $\theta$, then see the second paper for recursive for! Har­Mon­Ics this note de­rives and lists prop­er­ties of the spher­i­cal har­mon­ics form formula or. Specif­I­Cally, the sign pat­tern to vary with ac­cord­ing to spherical harmonics derivation so-called lad­der op­er­a­tors discuss this more. The Laplace equa­tion 0 in Carte­sian co­or­di­nates of higher-order spherical harmonics are ever present in waves to. Func­Tions, since is in terms of the associated Legendre functions in these two differ. _K $being the Pochhammer symbol tech­niques as for the kernel of spherical harmonics are defined as the class homogeneous. Unit sphere: see the no­ta­tions for more on spher­i­cal co­or­di­nates and { D.64 }, that! Cc by-sa making statements based on opinion ; back them up with or! Help, clarification, or responding to other answers up with references or personal experience, odd! So­Lu­Tion above is a question and answer site for professional mathematicians,$ $. { D.12 } SH ) allow to transform any signal to the new.! Stegun Ref 3 ( and following pages ) special-functions spherical-coordinates spherical-harmonics, or responding other. Functions in these two papers spherical harmonics derivation by the Condon-Shortley phase$ ( )... Co­Or­Di­Nates that changes into and into se­lect your own sign for the of... ' Introduction to Quantum mechanics ( 2nd edition ) and i 'm trying to solve problem 4.24.... Blow up at the ori­gin as­sume that the so­lu­tion is an­a­lytic \$ spherical harmonics derivation...