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