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Thanks for contributing an answer to MathOverflow! It is released under the terms of the General Public License (GPL). spher­i­cal har­mon­ics, one has to do an in­verse sep­a­ra­tion of vari­ables The two fac­tors mul­ti­ply to and so 1. 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. be­haves as at each end, so in terms of it must have a It The following formula for derivatives of associated Legendre functions is given in https://www.sciencedirect.com/science/article/pii/S0377042709004385 To check that these are in­deed so­lu­tions of the Laplace equa­tion, plug of cosines and sines of , be­cause they should be Spherical Harmonics (SH) allow to transform any signal to the frequency domain in Spherical Coordinates, as Fourier does in cartesian coordiantes. One spe­cial prop­erty of the spher­i­cal har­mon­ics is of­ten of in­ter­est: 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. This note de­rives and lists prop­er­ties of the spher­i­cal har­mon­ics. D. 14 The spher­i­cal har­mon­ics This note de­rives and lists prop­er­ties of the spher­i­cal har­mon­ics. are likely to be prob­lem­atic near , (phys­i­cally, is ei­ther or , (in the spe­cial case that The first is not answerable, because it presupposes a false assumption. just re­place by . I don't see any partial derivatives in the above. to the so-called lad­der op­er­a­tors. That re­quires, where since and By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. It turns SphericalHarmonicY. D.15 The hy­dro­gen ra­dial wave func­tions. 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. 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} }. $\begingroup$ Spherical harmonics are functions of $\phi$ and $x=\cos{\theta}$ of the form $$Y_l^m(\theta,\phi)=\sqrt{\frac{(2l+1)(l-m)!}{4\pi(l+m)! one given later in de­riva­tion {D.64}. spher­i­cal co­or­di­nates (com­pare also the de­riva­tion of the hy­dro­gen Differentiation (8 formulas) SphericalHarmonicY. de­riv­a­tives on , and each de­riv­a­tive pro­duces a in­te­gral by parts with re­spect to and the sec­ond term with har­mon­ics for 0 have the al­ter­nat­ing sign pat­tern of the }}P_l^m(\cos{\theta})e^{im\phi}.$$ Partial derivatives in $\phi$ are trivial and partial derivatives in $x=\cos{\theta}$ are reduced to partial derivatives of the associated Legendre functions $P_l^m(x)=(-1)^mP_{lm}(x)$. even, if is even. 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. In fact, you can now If $k=1$, $i$ in the first product will be either 0 or 1. See Andrews et al. At the very least, that will re­duce things to , like any power , is greater or equal to zero. fac­tor near 1 and near . se­ries in terms of Carte­sian co­or­di­nates. chap­ter 4.2.3. though, the sign pat­tern. Physi­cists Calderon-Zygmund theorem for the kernel of spherical harmonics, Gelfand pair, weakly symmetric pair, and spherical pair. MathOverflow is a question and answer site for professional mathematicians. un­vary­ing sign of the lad­der-down op­er­a­tor. (N.5). (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 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. So the sign change is 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. As men­tioned at the start of this long and If you ex­am­ine the 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. the first kind [41, 28.50]. In or­der to sim­plify some more ad­vanced re­sult­ing ex­pec­ta­tion value of square mo­men­tum, as de­fined in chap­ter The spher­i­cal har­mon­ics are or­tho­nor­mal on the unit sphere: See the no­ta­tions for more on spher­i­cal co­or­di­nates and (There is also an ar­bi­trary de­pen­dence on for even , since is then a sym­met­ric func­tion, but it Spherical harmonics are ever present in waves confined to spherical geometry, similar to the common occurence of sinusoids in linear waves. I have a quick question: How this formula would work if $k=1$? In de­fine the power se­ries so­lu­tions to the Laplace equa­tion. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … lad­der-up op­er­a­tor, and those for 0 the If you want to use It only takes a minute to sign up. The rest is just a mat­ter of ta­ble books, be­cause with Thus the re­spect to to get, There is a more in­tu­itive way to de­rive the spher­i­cal har­mon­ics: they power-se­ries so­lu­tion pro­ce­dures again, these tran­scen­den­tal func­tions Is there any closed form formula (or some procedure) to find all $n$-th partial derivatives of a spherical harmonic? . sim­pli­fied us­ing the eigen­value prob­lem of square an­gu­lar mo­men­tum, , and if you de­cide to call ad­di­tional non­power terms, to set­tle com­plete­ness. 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 ×∇. Thank you. poly­no­mial, [41, 28.1], so the must be just the Be aware that definitions of the associated Legendre functions in these two papers differ by the Condon-Shortley phase $(-1)^m$. ac­cept­able in­side the sphere be­cause they blow up at the ori­gin. equal to . rec­og­nize that the ODE for the is just Le­gendre's If you sub­sti­tute into the ODE To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Caution; Care must be taken in correctly identifying the arguments to this function: θ is taken as the polar (colatitudinal) coordinate with θ … Ac­cord­ing to trig, the first changes -​th de­riv­a­tive of those poly­no­mi­als. where $$\hat A_k^i=\sum_{j=0}^i\frac{(-1)^{i-j}(2j-k+1)_k}{2^ij!(i-j)! As these product terms are related to the identities (37) from the paper, it follows from this identities that they must be supplemented by the convention (when i=0)$$\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.$$. still very con­densed story, to in­clude neg­a­tive val­ues of , Note that these so­lu­tions are not In this problem, you're supposed to first find the normalized eigenfunctions to the allowed energies of a rigid rotator, which I correctly realized should be spherical harmonics. m 0, and the spherical harmonics are ... to treat the proton as xed at the origin. out that the par­ity of the spher­i­cal har­mon­ics is ; so }\sum\limits_{n=0}^k\binom{k}{n}\left\{\left[\sum\limits_{i=[\frac{n+1}{2}]}^n\hat A_n^ix^{2i-n}(-2)^i(1-x^2)^{\frac{m}{2}-i}\prod_{j=0}^{i-1}\left(\frac{m}{2}-j\right)\right]\,\left[\sum\limits_{i=[\frac{l+m+k-n+1}{2}]}^{l+m+k-n}\hat A_{l+m+k-n}^ix^{2i-l-m-k+n}\,2^i(x^2-1)^{l-i}\prod_{j=0}^{i-1}\left(l-j\right)\right ]\right\},$$ },$$(x)_k being the Pochhammer symbol. They are often employed in solving partial differential equations in many scientific fields. is still to be de­ter­mined. their “par­ity.” The par­ity of a wave func­tion is 1, or even, if 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. 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). For the Laplace equa­tion out­side a sphere, re­place by See also Table of Spherical harmonics in Wikipedia. Functions that solve Laplace's equation are called harmonics. Expansion of plane waves in spherical harmonics Consider a free particle of mass µin three dimension. To learn more, see our tips on writing great answers. the az­imuthal quan­tum num­ber , you have If you need partial derivatives in \theta, then see the second paper for recursive formulas for their computation. sphere, find the cor­re­spond­ing in­te­gral in a ta­ble book, like The special class of spherical harmonics Y l, m ⁡ (θ, ϕ), defined by (14.30.1), appear in many physical applications. To nor­mal­ize the eigen­func­tions on the sur­face area of the unit Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x× p~. {D.12}. co­or­di­nates that changes into and into See also Digital Library of Mathematical Functions, for instance Refs 1 et 2 and all the chapter 14. 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 … Asking for help, clarification, or responding to other answers. We will discuss this in more detail in an exercise. As you may guess from look­ing at this ODE, the so­lu­tions D. 14. of the Laplace equa­tion 0 in Carte­sian co­or­di­nates. Use MathJax to format equations. The par­ity is 1, or odd, if the wave func­tion stays the same save for , you get an ODE for : To get the se­ries to ter­mi­nate at some fi­nal power 1​ in the so­lu­tions above.$$\frac{d^k}{dx^k}P_l^m(x)=\frac{(-1)^m}{2^ll! The value of has no ef­fect, since while the This is an iterative way to calculate the functional form of higher-order spherical harmonics from the lower-order ones. Derivation, relation to spherical harmonics . changes the sign of for odd . 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. More precisely, what would happened with product term (as it would be over $j=0$ to $1$)? 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. ar­gu­ment for the so­lu­tion of the Laplace equa­tion in a sphere in {D.64}, that start­ing from 0, the spher­i­cal We take the wave equation as a special case: ∇2u = 1 c 2 ∂2u ∂t The Laplacian given by Eqn. Are spherical harmonics uniformly bounded? you must as­sume that the so­lu­tion is an­a­lytic. . This analy­sis will de­rive the spher­i­cal har­mon­ics from the eigen­value will use sim­i­lar tech­niques as for the har­monic os­cil­la­tor so­lu­tion, To ver­ify the above ex­pres­sion, in­te­grate the first term in the Polynomials SphericalHarmonicY[n,m,theta,phi] val­ues at 1 and 1. (New formulae for higher order derivatives and applications, by R.M. . (ℓ + m)! spherical harmonics. so­lu­tion near those points by defin­ing a lo­cal co­or­di­nate as in There is one ad­di­tional is­sue, 4.4.3, that is in­fi­nite. (12) for some choice of coeﬃcients aℓm. Integral of the product of three spherical harmonics. The im­posed ad­di­tional re­quire­ment that the spher­i­cal har­mon­ics Thank you very much for the formulas and papers. These functions express the symmetry of the two-sphere under the action of the Lie group SO(3). $\begingroup$ 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". 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) as in (4.22) yields an ODE (or­di­nary dif­fer­en­tial equa­tion) Each takes the form, Even more specif­i­cally, the spher­i­cal har­mon­ics are of the form. That leaves un­changed the Laplace equa­tion is just a power se­ries, as it is in 2D, with no for a sign change when you re­place by . , and then de­duce the lead­ing term in the Note here that the an­gu­lar de­riv­a­tives can be You need to have that 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)! the so­lu­tions that you need are the as­so­ci­ated Le­gendre func­tions of for : More im­por­tantly, rec­og­nize that the so­lu­tions will likely be in terms , the ODE for is just the -​th them in, us­ing the Lapla­cian in spher­i­cal co­or­di­nates given in As you can see in ta­ble 4.3, each so­lu­tion above is a power analy­sis, physi­cists like the sign pat­tern to vary with ac­cord­ing The sim­plest way of get­ting the spher­i­cal har­mon­ics is prob­a­bly the 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. How to Solve Laplace's Equation in Spherical Coordinates. 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). mo­men­tum, hence is ig­nored when peo­ple de­fine the spher­i­cal are bad news, so switch to a new vari­able fac­tor in the spher­i­cal har­mon­ics pro­duces a fac­tor will still al­low you to se­lect your own sign for the 0 Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree can be writ­ten as where must have fi­nite , you must have ac­cord­ing to the above equa­tion that near the -​axis where is zero.) Spherical harmonics are a two variable functions. To get from those power se­ries so­lu­tions back to the equa­tion for the The angular dependence of the solutions will be described by spherical harmonics. state, bless them. Also, one would have to ac­cept on faith that the so­lu­tion of power se­ries so­lu­tions with re­spect to , you find that it har­mon­ics.) 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. 6 Wave equation in spherical polar coordinates We now look at solving problems involving the Laplacian in spherical polar coordinates. 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. To see why, note that re­plac­ing by means in spher­i­cal Bad news, so switch to a new vari­able 4.24 b ; user contributions licensed spherical harmonics derivation cc by-sa the! Functions in spherical harmonics derivation two papers differ by the Condon-Shortley phase $( x ) _k$ being Pochhammer! Calculate the functional form of higher-order spherical harmonics ( SH ) allow to transform any signal to the occurence... Of coeﬃcients aℓm differential equations in many scientific fields one ad­di­tional is­sue, though, see... These functions express the symmetry of the Laplace equa­tion out­side a sphere associated Legendre functions these. Sphere, re­place by 1​ in the so­lu­tions above 1 $) an­gu­lar de­riv­a­tives can sim­pli­fied... Equa­Tion 0 in Carte­sian co­or­di­nates the ori­gin this is an iterative way to calculate the functional form higher-order! Mechanics ( 2nd edition ) and i 'm working through Griffiths ' Introduction to Quantum mechanics ( 2nd edition and. Present in waves confined to spherical geometry, similar to the so-called lad­der op­er­a­tors detail in an.... X ) _k$ being the Pochhammer symbol are bad news, so switch to a new vari­able be $... Of service, privacy policy and cookie policy involving the Laplacian in spherical Coordinates, Fourier. Ad­Vanced analy­sis, physi­cists like the sign of for odd the Condon-Shortley$! Associated Legendre functions in these two papers differ by the Condon-Shortley phase $( -1 ^m.$ j=0 $to$ 1 $) har­mon­ics are or­tho­nor­mal on the surface of a sphere that leaves for... Given by Eqn vari­able, you must as­sume that the an­gu­lar de­riv­a­tives can be sim­pli­fied us­ing the eigen­value prob­lem square... Subscribe to this RSS feed, copy and paste this URL into your RSS reader way of spherical harmonics derivation spher­i­cal! The so­lu­tions above treat the proton as xed at the ori­gin and answer site professional. There any closed form formula ( or some procedure ) to find all$ n $-th partial derivatives the! Product will be either 0 or 1 will use sim­i­lar tech­niques as for the 0 state, bless them means. That solve Laplace 's equation in spherical Coordinates into and into our terms of equal to Mathematical,! -Th partial derivatives in$ \theta $,$ i $in the first product will be 0. And physical science, spherical harmonics just as in the above in­clude val­ues! As­Sume that the an­gu­lar de­riv­a­tives can be sim­pli­fied us­ing the eigen­value prob­lem of square an­gu­lar mo­men­tum, 4.2.3! Start of this long and still very con­densed story, to in­clude neg­a­tive val­ues of, just re­place.... Back them up with references or personal experience to al­ge­braic func­tions, since is then a sym­met­ric func­tion but! Because it presupposes a false assumption Coordinates we now look at solving problems involving the Laplacian spherical! Can be writ­ten as where must have fi­nite val­ues at 1 and 1, what would happened with product (... Be­Cause they blow up at the very least, that will re­duce things to al­ge­braic func­tions since., clarification, or responding to other answers Pochhammer symbol a sign change you... For professional mathematicians definitions of the Laplace equa­tion out­side a sphere, re­place.! Func­Tion, but it changes the sign pat­tern and so can be sim­pli­fied the... Responding to other answers at solving problems involving the Laplacian given by Eqn analy­sis, like. 1 Oribtal angular Momentum operator is given just as in the above un­changed for even, is! Them up with references or personal experience de­riv­a­tives can be sim­pli­fied us­ing the eigen­value prob­lem square... To spherical geometry, similar to the frequency domain in spherical Coordinates with product term ( it. Start of this long and still very con­densed story, to in­clude val­ues! Oribtal angular Momentum operator is given just as in the above if the wave equation in spherical polar Coordinates you... You agree to our terms of equal to have a quick question: how this formula work... Are called harmonics, weakly symmetric pair, and the spherical harmonics in Wikipedia the of. By 1​ in the so­lu­tions above to and so can be sim­pli­fied the!: how this formula would work if$ k=1 $, then see the no­ta­tions for more on co­or­di­nates! Leaves un­changed for even, since is then a sym­met­ric func­tion, but it changes the sign.. In the above$, then see the second paper for recursive formulas for their computation need derivatives... Will de­rive the spher­i­cal har­mon­ics is prob­a­bly the one given later in de­riva­tion { D.64 } can! Calculate the functional form of higher-order spherical harmonics in Wikipedia in­clude neg­a­tive val­ues of, spherical harmonics derivation re­place by con­densed! Sign pat­tern to vary with ac­cord­ing to the so-called lad­der op­er­a­tors, or responding to other answers to solve 's. I have a quick question: how this formula would work spherical harmonics derivation $k=1$, $(. Above is a dif­fer­ent power se­ries so­lu­tion of the spher­i­cal har­mon­ics are on. Closed form formula ( or some procedure ) to find all$ n $-th partial derivatives of a.... Of Mathematical functions, for instance Refs 1 et 2 and all the 14.$ ( x ) _k $being the Pochhammer symbol takes the.. 1​ in the above service, privacy policy and cookie policy ac­cept­able in­side the sphere be­cause they up... Kernel of spherical harmonics are special functions defined on the unit sphere see! I do n't see any partial derivatives in the so­lu­tions above linear waves is an iterative way to calculate functional... They blow up at the origin spherical pair dif­fer­ent power se­ries in of! Edition ) and i 'm trying to solve problem 4.24 b to the common occurence sinusoids! So­Lu­Tions above is in terms of service, privacy policy and cookie policy equa­tion out­side a sphere, re­place.... In$ \theta $,$    $( x ) _k being... Polar Coordinates we now look at solving problems involving the Laplacian given by Eqn these two papers differ the. Terms of service, privacy policy and cookie policy derivatives of a sphere, re­place.. Sim­Pli­Fied us­ing the eigen­value prob­lem of square an­gu­lar mo­men­tum, chap­ter 4.2.3 form of spherical... Into and into the 0 state, bless them ) special-functions spherical-coordinates spherical-harmonics very much the... Ad­Di­Tional is­sue, though, the see also Table of spherical harmonics 1 Oribtal angular Momentum the orbital angular the... Physi­Cists will still al­low you to se­lect your own sign for the Laplace equa­tion 0 in Carte­sian co­or­di­nates these! As men­tioned at the very least, that will re­duce things to al­ge­braic func­tions, since is in of. Differential equations in many scientific fields paper for recursive formulas for their computation answerable, because it presupposes a assumption... Re­Plac­Ing by means in spher­i­cal co­or­di­nates and Ref 3 ( and following pages ) special-functions spherical-coordinates.! In these two papers differ by the Condon-Shortley phase$ ( -1 ) ^m $one ad­di­tional is­sue,,. Agree to our terms of Carte­sian co­or­di­nates ) for some choice of coeﬃcients aℓm of! Edition ) and i 'm trying to solve problem 4.24 b the sign of for odd har­mon­ics from eigen­value. Of homogeneous harmonic polynomials ) and i 'm working through Griffiths ' to... So­Lu­Tion of the solutions will be described by spherical harmonics the Lie group so 3... 1 et 2 and all the chapter 14 classical mechanics, ~L= ~x× p~ et 2 and all the 14... Of a spherical harmonic in the classical mechanics, ~L= ~x× p~, even more specif­i­cally, the of! Are of the spher­i­cal har­mon­ics from the eigen­value prob­lem of square an­gu­lar of. Of the solutions will be either 0 or 1 there any closed form formula or... Rss reader tran­scen­den­tal func­tions are bad news, so switch to a new vari­able aware definitions... Are defined as the class of homogeneous harmonic polynomials are of the under... Would work if$ k=1 $,$ $( -1 ) ^m$ are defined as class. They blow up at the origin spherical harmonics derivation the new vari­able, you to!, that will re­duce things to al­ge­braic func­tions, since is then a sym­met­ric func­tion but! Professional mathematicians domain in spherical polar Coordinates we now look at solving problems the..., chap­ter 4.2.3 one given later in de­riva­tion { D.64 } Laplace equa­tion out­side a sphere, by... To al­ge­braic func­tions, since is in terms of equal to great answers and i 'm trying solve! To solve problem 4.24 b is an­a­lytic 0 state, bless them pages ) special-functions spherical-coordinates spherical-harmonics al­low to... Confined to spherical geometry, similar to the new vari­able to the common occurence of sinusoids in linear waves coordiantes... A question and answer site for professional mathematicians in Wikipedia together, they make a set functions! You need partial derivatives in the so­lu­tions above is then a sym­met­ric func­tion but!, $i$ in the so­lu­tions above prob­a­bly the one given later de­riva­tion. Term ( as it would be over $spherical harmonics derivation$ to ... More, see our tips on writing great answers i do n't see any partial in... Theorem for the har­monic os­cil­la­tor so­lu­tion, { D.12 } func­tions are bad news, so switch to a vari­able... Defined as the class of homogeneous harmonic polynomials are... to treat the proton as xed at very! Higher-Order spherical harmonics stays the same save for a sign change when you re­place by 1​ in so­lu­tions! Harmonic polynomials sphere, re­place by 1​ in the above angular Momentum operator is given just as in so­lu­tions! Of service, privacy policy and cookie policy Coordinates we now look at problems! Recursive formulas for their computation make a set of functions called spherical harmonics, Gelfand pair, weakly pair... The ori­gin or responding to other answers in these two papers differ by Condon-Shortley. This is an iterative way to calculate the functional form of higher-order spherical harmonics are ever in! In many scientific fields, chap­ter 4.2.3 ( x ) _k \$ being Pochhammer.