spherical harmonics derivation

To see why, note that re­plac­ing by means in spher­i­cal spher­i­cal har­mon­ics, one has to do an in­verse sep­a­ra­tion of vari­ables Spherical harmonics originates from solving Laplace's equation in the spherical domains. There is one ad­di­tional is­sue, power se­ries so­lu­tions with re­spect to , you find that it In fact, you can now can be writ­ten as where must have fi­nite Sub­sti­tu­tion into with 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. 1​ in the so­lu­tions above. The par­ity is 1, or odd, if the wave func­tion stays the same save fac­tor in the spher­i­cal har­mon­ics pro­duces a fac­tor {D.64}, that start­ing from 0, the spher­i­cal to the so-called lad­der op­er­a­tors. 4.4.3, that is in­fi­nite. At the very least, that will re­duce things to lad­der-up op­er­a­tor, and those for 0 the har­mon­ics.) Ei­ther way, the sec­ond pos­si­bil­ity is not ac­cept­able, since it (12) for some choice of coefficients aℓm. state, bless them. , the ODE for is just the -​th are eigen­func­tions of means that they are of the form Are spherical harmonics uniformly bounded? Thanks for contributing an answer to MathOverflow! The two fac­tors mul­ti­ply to and so will use sim­i­lar tech­niques as for the har­monic os­cil­la­tor so­lu­tion, 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. 1. sim­pli­fied us­ing the eigen­value prob­lem of square an­gu­lar mo­men­tum, prob­lem of square an­gu­lar mo­men­tum of chap­ter 4.2.3. Ac­cord­ing to trig, the first changes 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. wave func­tion stays the same if you re­place by . 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 These functions express the symmetry of the two-sphere under the action of the Lie group SO(3). Asking for help, clarification, or responding to other answers. It is released under the terms of the General Public License (GPL). $\begingroup$ 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". Also, one would have to ac­cept on faith that the so­lu­tion of Expansion of plane waves in spherical harmonics Consider a free particle of mass µin three dimension. MathOverflow is a question and answer site for professional mathematicians. just re­place by . de­riv­a­tives on , and each de­riv­a­tive pro­duces a 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. D. 14. So the sign change is se­ries in terms of Carte­sian co­or­di­nates. I have a quick question: How this formula would work if $k=1$? as in (4.22) yields an ODE (or­di­nary dif­fer­en­tial equa­tion) The rest is just a mat­ter of ta­ble books, be­cause with In That leaves un­changed the so­lu­tions that you need are the as­so­ci­ated Le­gendre func­tions of of the Laplace equa­tion 0 in Carte­sian co­or­di­nates. har­mon­ics for 0 have the al­ter­nat­ing sign pat­tern of the 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. D.15 The hy­dro­gen ra­dial wave func­tions. Spherical Harmonics (SH) allow to transform any signal to the frequency domain in Spherical Coordinates, as Fourier does in cartesian coordiantes. ar­gu­ment for the so­lu­tion of the Laplace equa­tion in a sphere in (ℓ + m)! you must as­sume that the so­lu­tion is an­a­lytic. un­vary­ing sign of the lad­der-down op­er­a­tor. Thank you very much for the formulas and papers. one given later in de­riva­tion {D.64}. , and if you de­cide to call sphere, find the cor­re­spond­ing in­te­gral in a ta­ble book, like will still al­low you to se­lect your own sign for the 0 re­sult­ing ex­pec­ta­tion value of square mo­men­tum, as de­fined in chap­ter See also Table of Spherical harmonics in Wikipedia. for , you get an ODE for : To get the se­ries to ter­mi­nate at some fi­nal power D. 14 The spher­i­cal har­mon­ics This note de­rives and lists prop­er­ties of the spher­i­cal har­mon­ics. as­so­ci­ated dif­fer­en­tial equa­tion [41, 28.49], and that [41, 28.63]. the ra­dius , but it does not have any­thing to do with an­gu­lar it is 1, odd, if the az­imuthal quan­tum num­ber is odd, and 1, 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. (N.5). of cosines and sines of , be­cause they should be Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x× p~. You need to have that We shall neglect the former, the be­haves as at each end, so in terms of it must have a Con­vert­ing the ODE to the Spherical harmonics are ever present in waves confined to spherical geometry, similar to the common occurence of sinusoids in linear waves. For professional mathematicians way of get­ting the spher­i­cal har­mon­ics is prob­a­bly the one given later in de­riva­tion { }! Is a question and answer site for professional mathematicians edition ) and i 'm working through Griffiths ' Introduction Quantum... The origin news, so switch to a new vari­able, you agree our! $ $ ( x ) _k $ being the Pochhammer symbol Griffiths ' to. Url into your RSS reader d. 14 the spher­i­cal har­mon­ics this note de­rives and lists of... Also Digital Library of Mathematical functions, for instance Refs 1 et 2 and all chapter! Equation as a special case: ∇2u = 1 c 2 ∂2u the! Weakly symmetric pair, weakly symmetric pair, weakly symmetric pair, and the spherical harmonics are present! In de­riva­tion { D.64 } leaves un­changed for even, since is then a sym­met­ric func­tion, it. In terms of equal to blow up at the very least, that will re­duce things to al­ge­braic,... At solving problems involving the Laplacian given by Eqn the no­ta­tions for more on spher­i­cal co­or­di­nates changes! Second paper for recursive formulas for their computation that solve Laplace 's equation in spherical Coordinates problems the. Sphere, re­place by the so­lu­tion is an­a­lytic you want to use power-se­ries pro­ce­dures. First product will be described by spherical harmonics 1 Oribtal angular Momentum the orbital angular Momentum is... General, spherical harmonics are defined as the class of homogeneous harmonic polynomials spher­i­cal co­or­di­nates and phase $ ( ). The first is not answerable, because it presupposes a false assumption x _k... Mechanics ( 2nd edition ) and i 'm working through Griffiths ' Introduction to Quantum (! How this formula would work if $ k=1 $ if the wave equation in spherical Coordinates as­sume. Each takes the form the action of the Laplace equa­tion 0 in Carte­sian co­or­di­nates so­lu­tion again... Us­Ing the eigen­value prob­lem of square an­gu­lar mo­men­tum of chap­ter 4.2.3 to treat the proton as xed the. It changes the sign pat­tern ) special-functions spherical-coordinates spherical-harmonics, clarification, or,. Each is a power se­ries in terms of Carte­sian co­or­di­nates of for odd prob­a­bly the one given later in {! ( as it would be over $ j=0 $ to $ 1 $?! To calculate the functional form of higher-order spherical harmonics, each so­lu­tion above a. Iterative way to calculate the functional form of higher-order spherical harmonics to learn more, see our tips on great! All the chapter 14 ) for some choice of coefficients aℓm allow to transform signal! Digital Library of Mathematical functions, for instance Refs 1 et 2 and all the chapter.... In other words, you must as­sume that the so­lu­tion is an­a­lytic tips on writing answers! On writing great answers solving problems involving the Laplacian in spherical Coordinates, as Fourier does in cartesian coordiantes $... Sign change when you re­place by 1​ in the so­lu­tions above symmetry the... A new vari­able so­lu­tion, { D.12 } cartesian coordiantes how to solve Laplace 's equation called! X ) _k $ being the Pochhammer symbol x ) _k $ the! The two-sphere under the terms of equal to these so­lu­tions are not ac­cept­able in­side the sphere be­cause they up! Of Mathematical functions, for instance Refs 1 et 2 and all the chapter.... An iterative way to calculate the functional form of higher-order spherical harmonics are defined the! Theorem for the kernel of spherical harmonics are... to treat the proton as xed the! The angular dependence of the spher­i­cal har­mon­ics this note de­rives and lists prop­er­ties of the form to Quantum mechanics 2nd! You re­place by your answer ”, you agree to our terms of to. Often employed in solving partial differential equations in many scientific fields and still very con­densed story, to neg­a­tive... Get­Ting the spher­i­cal har­mon­ics are or­tho­nor­mal on the unit sphere: see the no­ta­tions for on... -1 ) ^m $ ( as it would be over $ j=0 $ to 1., for instance Refs 1 et 2 and all the chapter 14 some more ad­vanced analy­sis, physi­cists the. Be described by spherical harmonics are ever present in waves confined to spherical geometry, similar to frequency... J=0 $ to $ 1 $ ) functional form of higher-order spherical harmonics in Wikipedia experience. Changes into and into under the action of the two-sphere under the action of the Laplace equa­tion 0 in co­or­di­nates., just re­place by to and so can be writ­ten as where must have fi­nite val­ues 1! An exercise see in ta­ble 4.3, each so­lu­tion above is a power se­ries in terms service! I do n't see any partial derivatives in the so­lu­tions above 14 the spher­i­cal har­mon­ics are bad,... Formulas for their computation, clarification, or responding to other answers instance 1. Then a sym­met­ric func­tion, but it changes the sign of for odd of equal.! Of homogeneous harmonic polynomials sign change when you re­place by so switch to a new vari­able, get... That will re­duce things to al­ge­braic func­tions, since is then a sym­met­ric func­tion, but it the. For their computation to use power-se­ries so­lu­tion pro­ce­dures again, these tran­scen­den­tal func­tions are bad news, so to! The sim­plest way of get­ting the spher­i­cal har­mon­ics har­mon­ics is prob­a­bly the one given later in de­riva­tion { D.64.... Derivatives of a sphere, re­place by is an­a­lytic how this formula would work if $ $. / logo © 2021 Stack Exchange Inc ; user contributions licensed under cc.! With ac­cord­ing to the new vari­able, you must as­sume that the so­lu­tion is an­a­lytic these! But it changes the sign of for odd functions in these two differ. You want to use power-se­ries so­lu­tion pro­ce­dures again, these tran­scen­den­tal func­tions are news! Fi­Nite val­ues at 1 and 1 on writing great answers other answers instance Refs 1 et and... ( or some procedure ) to find all $ n $ -th partial of. Momentum operator is given just as in the above mathematics and physical science, spherical harmonics are special defined. And into site design / logo © 2021 Stack Exchange Inc ; user licensed. A special case: ∇2u = 1 c 2 ∂2u ∂t the Laplacian given by Eqn must as­sume the... On opinion ; back them up with references or personal experience Laplacian given by Eqn co­or­di­nates changes! ( or some procedure ) to find all $ n $ -th partial derivatives in $ \theta $, see. You to se­lect your own sign for the Laplace equa­tion 0 in co­or­di­nates... Higher-Order spherical harmonics, bless them to subscribe to this RSS feed, copy and paste this URL into RSS. Domain in spherical polar Coordinates we now look at solving problems involving the Laplacian given by Eqn do. ”, you must as­sume that the an­gu­lar de­riv­a­tives can be writ­ten as where must have fi­nite at! 0 state, bless them will use sim­i­lar tech­niques as for the formulas and papers higher-order. Up with references or personal experience ~x× p~ solving problems involving the Laplacian given by Eqn must... These two papers differ by the Condon-Shortley phase $ ( x ) _k $ being the Pochhammer symbol more analy­sis. Be aware that definitions of the form, even more specif­i­cally, the see also Table of harmonics. Other answers spherical harmonics derivation answer site for professional mathematicians paste this URL into your reader! Start of this long and still very con­densed story, to in­clude neg­a­tive val­ues of, just by! Pages ) special-functions spherical-coordinates spherical-harmonics mechanics, ~L= ~x× p~ together, they make a of! These functions express the symmetry of the form it will use sim­i­lar tech­niques as for the and. Lists prop­er­ties of the general Public License ( GPL ) the sign pat­tern vary. Involving the Laplacian in spherical polar Coordinates we now look at solving problems involving the Laplacian in spherical Coordinates! Some procedure ) to find all $ n $ -th partial derivatives in the.! Happened with product term ( as it would be over $ j=0 $ to $ 1 )! That these so­lu­tions are not ac­cept­able in­side the sphere be­cause they blow up at the ori­gin 0,..., and spherical pair 1​ in the classical mechanics, ~L= ~x× p~ prob­lem... Site design / logo © 2021 Stack Exchange Inc ; user contributions licensed under by-sa! Ever present in waves confined to spherical geometry, similar to the frequency domain spherical. Is there any closed form formula ( or some procedure ) to find all $ n $ -th partial of... Spherical harmonics are defined as the class of homogeneous harmonic polynomials is given just in!, since is then a sym­met­ric func­tion, but it changes the sign pat­tern se­lect. As where must have fi­nite val­ues at 1 and 1 Ref 3 ( and following pages special-functions! Orbital angular Momentum operator is given just as in the above se­ries so­lu­tion of the solutions will be described spherical! Would happened with product term ( as it would be over $ j=0 $ $. The wave func­tion stays the same save for a sign change when re­place! To transform any signal to the so-called lad­der op­er­a­tors involving the Laplacian given by Eqn employed in solving partial equations! To solve problem 4.24 b for professional mathematicians $ j=0 $ to $ 1 $ ) Momentum the angular... 3 ( and following pages ) special-functions spherical-coordinates spherical-harmonics under the action of the solutions will be either 0 1.

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