DOCSLIB.ORG

Spherical Harmonics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spherical Harmonics 221A Lecture Notes Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, L~ = ~x × ~p. (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, [Li,Lj] = ih¯ ijkLk. (2) When using the position representation, the action of the angular mo- mentum on any state is given by a differential operator h¯ h¯ h~x|L~ |ψi = ~x × ∇h~ ~x|ψi = ~x × ∇~ ψ(~x). (3) i i Loosely, we can write h¯ L~ = ~x × ∇~ , (4) i but you have to be careful on what this differential operator is acting on. If you act on a position ket instead of a bra, h¯ L~ |~xi = −~x × ∇|~ ~xi. (5) i Whenever I write the orbital angular momentum operator as a differential operator in this note, it is understood that it acts on a position bra instead of ket. With this caveat in mind, we can rewrite the orbital angular momentum operator in the polar coordinates. Following the usual definitions x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ, (6) 1 we can rewrite the derivatives using the chain rule, ∂r ∂rx ∂ry ∂rz ∂x ∂θ = ∂θx ∂θy ∂θz ∂y ∂φ ∂φx ∂φy ∂φz ∂z sin θ cos φ sin θ sin φ cos θ ∂x = r cos θ cos φ r cos θ sin φ −r sin θ ∂y . (7) −r sin θ sin φ r sin θ cos φ 0 ∂z We invert the matrix and find ∂ r sin2 θ cos φ sin θ cos θ cos φ − sin φ ∂ x 1 r 2 ∂y = r sin θ sin φ sin θ cos θ sin φ cos φ ∂θ . (8) r sin θ 2 ∂z r sin θ cos θ − sin θ 0 ∂φ Now the orbital angular momentum operators can be written in terms of spherical coordinates, h¯ ∂ ∂ ! L = − sin φ − cot θ cos φ , (9) x i ∂θ ∂φ h¯ ∂ ∂ ! L = cos φ − cot θ sin φ , (10) y i ∂θ ∂φ h¯ ∂ L = . (11) z i ∂φ It is useful to take the combinations h¯ ∂ ∂ ! L = e±iφ ±i − cot θ . (12) ± i ∂θ ∂φ Finally, " 1 ∂ ∂ ! 1 ∂2 # L~ 2 = −h¯2 sin θ + . (13) sin θ ∂θ ∂θ sin2 θ ∂φ2 The free-particle Hamiltonian is ~p2 −h¯2 −h¯2 ∂2 2 ∂ ! L~ 2 = (∂2 + ∂2 + ∂2) = + + . (14) 2m 2m x y z 2m ∂r2 r ∂r 2mr2 Looking at the expression for Lz, you can see that it is of the same form as the momentum operator of a particle on a circle, whose eigenvalues are quantized as nh¯ for n ∈ Z. Therefore, Lz is also quantized as mh¯ for m ∈ Z. An immediate consequence is that no half-odd values are allowed for Lz, and hence half-odd j cannot be obtained for orbital angular momentum. Only integer j is possible. 2 2 Spherical Harmonics ~ 2 Now we look for eigenstates of L and Lz to find the Hilbert space for the orbital angular momentum. In any spherically symmetric systems, energy eigenstates can be given by produce wave functions of the form∗ m ψ(~x) = R(r)Yl (θ, φ). (15) In other words, we are parameterizing the position eigenbasis in terms of polar rather than Cartesian coordinates, h~x| = hr, θ, φ|. (16) The spherical harmonics are defined as the wave functions of angular mo- mentum eigenstates m Yl (θ, φ) = hθ, φ|l, mi. (17) Sakurai uses the notation h~n| and call them “direction eigenkets.” Clearly, the defintions of angular momemum eigenstates L~ 2|l, mi = l(l + 1)¯h2|l, mi, (18) Lz|l, mi = mh¯|l, mi, (19) translate to the differential equations " 1 ∂ ∂ ! 1 ∂2 # − h¯2 sin θ + Y m = l(l + 1)¯h2Y m, (20) sin θ ∂θ ∂θ sin2 θ ∂φ2 l l h¯ ∂ Y m = mhY¯ m. (21) i ∂φ l l The latter equation is easy to solve: the azimuth dependence of the spherical harmonics must be eimφ. But figuring out the polar angle dependence needs more work. The rest of the discussion here is on this issue. Both in the case of the harmonic oscillator and the Landau levels (energy levels of a charged particle in a uniform magnetic field), it was useful to write ∗This corresponds to the separation of variables in Hailton–Jacobi theory, where the action S is written as a sum of different pieces S(r) + S(θ, φ). Because the wave function corresponds to eiS/h¯ , it is natural for the wave function to be given by a product of different pieces. 3 down an equation for the ground state of the form a|0i = 0. It was useful because it gave us a linear differential equation instead of a quadratic one (e.g., Schr¨odinger equation). The linear differential equations are far easier to solve. We can take the same strategy for the spherical harmonics. 2.1 Derivation of Spherical Harmonics We know from the general representation theory of angular momenta that L−|l, −li = 0 because it cannot be lowered any more. Sakurai starts with |l, li instead. Of course you get the same result, but I find this way somewhat less confusing. In the position representation, we find h¯ ∂ ∂ ! 0 = hθ, φ|L |l, −li = e−iφ −i − cot θ Y −l(θ, φ) = 0. (22) − i ∂θ ∂φ l On the other hand, we know already that h¯ ∂ − lhY¯ −l = hθ, φ|L |l, −li = Y −l (23) l z i ∂φ l −l −l −ilφ and hence the azimuth dependence of Yl is Yl (θ, φ) = f(θ)e . Therefore, Eq. (22) becomes d ! −i + il cot θ f(θ) = 0. (24) dθ This equation is solved easily, by writing it as 1 df = l cot θdθ, (25) f and integrating both sides to log f = l log sin θ + const. (26) Therefore, f(θ) = c sinl θ (27) with an overall normalization constant c. Putting things together, we find −l l −ilφ Yl (θ, φ) = c sin θe . (28) 4 The absolute value of c can be fixed by the normalization Z Z 1 Z 2π l 2 2 2l 1 = dΩ|Yl | = d cos θ dφ|c| sin θ. (29) −1 0 φ integral trivially gives a factor of 2π. The θ integral is most conveniently done using the variable x = cos θ, Z 1 1 = 2π|c|2 dx(1 − x2)l.. (30) −1 Writing 1−x2 = (1−x)(1+x), and further change the variable to x = −1+2t, Z 1 Z 1 1 = 2π|c|2 2dt (2t(2 − 2t))l = 2π|c|222l+1 dt tl(1 − t)l. (31) 0 0 The integral is nothing but the Beta function Z 1 Γ(p)Γ(q) B(p, q) = dt tp−1(1 − t)q−1 = . (32) 0 Γ(p + q) Therefore, Γ(l + 1)Γ(l + 1) l!l! 1 = 2π|c|222l+1 = 4π|c|222l . (33) Γ(2l + 2) (2l + 1)! We find s 1 (2l + 1)! |c| = . (34) 2ll! 4π The phase of c is fixed by picking a convention. The commonly used convention (the same as Sakurai’s) is not to have an additional phase factor s 1 (2l + 1)! Y −l(θ, φ) = sinl θe−ilφ. (35) l 2ll! 4π −l m Now that we know Yl , we can keep acting L+ on it to obtain all Yl . Recall that the general discussion of angular momentum taught us that q q L+|l, mi = l(l + 1) − m(m + 1)|l, m + 1i = (l − m)(l + m + 1)|l, m + 1i. (36) 5 Then we find m 1 1 Yl (θ, φ) = q q h¯ (l + m)(l − m + 1) h¯ (l + m − 1)(l − m + 2) " !#l+m 1 1 h¯ iφ ∂ ∂ −l ··· q q e i − cot θ Yl (θ, φ) h¯ (2)(l + l − 1) h¯ (1)(l + l) i ∂θ ∂φ v u " !#l+m u (l − m)! ∂ ∂ = t eiφ + i cot θ Y −l(θ, φ) (37) (2l)!(l + m)! ∂θ ∂φ l Now the question is to bring it to a more usable form. The φ derivatives always give eigenvalues. But each time L+ acts, there is a factor of e+iφ and makes the eigenvalue of −i∂/∂φ increase one by one. Therefore, v s u ! 1 (2l + 1)!u (l − m)! d Y m(θ, φ) = t eimφ − (m − 1) cot θ l 2ll! 4π (2l)!(l + m)! dθ d ! d ! d ! − (m − 2) cot θ ··· + (l − 1) cot θ + l cot θ sinl θ. dθ dθ dθ v u ! 1 u(2l + 1)(l − m)! d = t eimφ − (m − 1) cot θ 2ll! 4π(l + m)! dθ d ! d ! d ! − (m − 2) cot θ ··· + (l − 1) cot θ + l cot θ sinl θ. dθ dθ dθ (38) Here we have cancelled factors ofh ¯ and i. The next trick we need is to write d ! 1 d + k cot θ = sink θ (39) dθ sink θ dθ Then v u ! 1 u(2l + 1)(l − m)! 1 d Y m(θ, φ) = t eimφ sin−(m−1) θ l 2ll! 4π(l + m)! sin−(m−1) θ dθ 1 d ! 1 d ! 1 d ! sin−(m−2) θ ··· sinl−1 θ sinl θ sinl θ sin−(m−2) θ dθ sinl−1 θ dθ sinl θ dθ 6 v u !l+m 1 u(2l + 1)(l − m)! 1 d = t eimφ sinm θ sin2l θ 2ll! 4π(l + m)! sin θ dθ v u !l+m 1 u(2l + 1)(l − m)! d = t eimφ sinm θ − sin2l θ 2ll! 4π(l + m)! d cos θ v l+m u l+m (−1) u(2l + 1)(l − m)! d = t eimφ(1 − x2)m/2 (1 − x2)2l.
Load more

Recommended publications
  • Quaternion Zernike Spherical Polynomials
    MATHEMATICS OF COMPUTATION Volume 84, Number 293, May 2015, Pages 1317–1337 S 0025-5718(2014)02888-3 Article electronically published on August 29, 2014 QUATERNION ZERNIKE SPHERICAL POLYNOMIALS J. MORAIS AND I. CAC¸ AO˜ Abstract. Over the past few years considerable attention has been given to the role played by the Zernike polynomials (ZPs) in many different fields of geometrical optics, optical engineering, and astronomy. The ZPs and their applications to corneal surface modeling played a key role in this develop- ment. These polynomials are a complete set of orthogonal functions over the unit circle and are commonly used to describe balanced aberrations. In the present paper we introduce the Zernike spherical polynomials within quater- nionic analysis ((R)QZSPs), which refine and extend the Zernike moments (defined through their polynomial counterparts). In particular, the underlying functions are of three real variables and take on either values in the reduced and full quaternions (identified, respectively, with R3 and R4). (R)QZSPs are orthonormal in the unit ball. The representation of these functions in terms of spherical monogenics over the unit sphere are explicitly given, from which several recurrence formulae for fast computer implementations can be derived. A summary of their fundamental properties and a further second or- der homogeneous differential equation are also discussed. As an application, we provide the reader with plot simulations that demonstrate the effectiveness of our approach. (R)QZSPs are new in literature and have some consequences that are now under investigation. 1. Introduction 1.1. The Zernike spherical polynomials. The complex Zernike polynomials (ZPs) have long been successfully used in many different fields of optics.
    [Show full text]
  • Laplacian Eigenmodes for the Three Sphere
    Laplacian eigenmodes for the three sphere M. Lachi`eze-Rey Service d’Astrophysique, C. E. Saclay 91191 Gif sur Yvette cedex, France October 10, 2018 Abstract The vector space k of the eigenfunctions of the Laplacian on the 3 V three sphere S , corresponding to the same eigenvalue λk = k (k + 2 − k 2), has dimension (k + 1) . After recalling the standard bases for , V we introduce a new basis B3, constructed from the reductions to S3 of peculiar homogeneous harmonic polynomia involving null vectors. We give the transformation laws between this basis and the usual hyper-spherical harmonics. Thanks to the quaternionic representations of S3 and SO(4), we are able to write explicitely the transformation properties of B3, and thus of any eigenmode, under an arbitrary rotation of SO(4). This offers the possibility to select those functions of k which remain invariant under V a chosen rotation of SO(4). When the rotation is an holonomy transfor- mation of a spherical space S3/Γ, this gives a method to calculates the eigenmodes of S3/Γ, which remains an open probleme in general. We illustrate our method by (re-)deriving the eigenmodes of lens and prism space. In a forthcoming paper, we present the derivation for dodecahedral space. 1 Introduction 3 The eigenvalues of the Laplacian ∆ of S are of the form λk = k (k + 2), + − k where k IN . For a given value of k, they span the eigenspace of ∈ 2 2 V dimension (k +1) . This vector space constitutes the (k +1) dimensional irreductible representation of SO(4), the isometry group of S3.
    [Show full text]
  • A User's Guide to Spherical Harmonics
    A User's Guide to Spherical Harmonics Martin J. Mohlenkamp∗ Version: October 18, 2016 This pamphlet is intended for the scientist who is considering using Spherical Harmonics for some appli- cation. It is designed to introduce the Spherical Harmonics from a theoretical perspective and then discuss those practical issues necessary for their use in applications. I expect great variability in the backgrounds of the readers. In order to be informative to those who know less, without insulting those who know more, I have adopted the strategy \State even basic facts, but briefly." My dissertation work was a fast transform for Spherical Harmonics [6]. After its publication, I started receiving questions about Spherical Harmonics in general, rather than about my work specifically. This pamphlet aims to answer such general questions. If you find mistakes, or feel that important material is unclear or missing, please inform me. 1 A Theory of Spherical Harmonics In this section we give a development of Spherical Harmonics. There are other developments from other perspectives. The one chosen here has the benefit of being very concrete. 1.1 Mathematical Preliminaries We define the L2 inner product of two functions to be Z hf; gi = f(s)¯g(s)ds (1) R 2π whereg ¯ denotes complex conjugation and the integral is over the space of interest, for example 0 dθ on the R 2π R π 2 p 2 circle or 0 0 sin φdφdθ on the sphere. We define the L norm by jjfjj = hf; fi. The space L consists of all functions such that jjfjj < 1.
    [Show full text]
  • Spherical-Harmonic Tensors Jm
    Spherical-harmonic tensors Francisco Gonzalez Ledesma1 and Matthew Mewes2 1Department of Physics, Florida State University, Tallahassee, Florida 32306, USA 2Physics Department, California Polytechnic State University, San Luis Obispo, California 93407, USA The connection between spherical harmonics and symmetric tensors is explored. For each spher- ical harmonic, a corresponding traceless symmetric tensor is constructed. These tensors are then extended to include nonzero traces, providing an orthonormal angular-momentum eigenbasis for symmetric tensors of any rank. The relationship between the spherical-harmonic tensors and spin- weighted spherical harmonics is derived. The results facilitate the spherical-harmonic expansion of a large class of tensor-valued functions. Several simple illustrative examples are discussed, and the formalism is used to derive the leading-order effects of violations of Lorentz invariance in Newtonian gravity. I. INTRODUCTION Spherical harmonics Yjm provide an orthonormal basis for scalar functions on the 2-sphere and have numerous applications in physics and related fields. While they are commonly written in terms of the spherical-coordinate polar angle θ and azimuthal angle φ, spherical harmonics can be expressed in terms of cartesian coordinates, which is convenient in certain applications. The cartesian versions involve rank-j symmetric trace-free tensors jm [1–5]. These form a basis for traceless tensors and provide a link between functions on the sphere and symmetricY traceless tensors in three dimensions. This work builds on the above understanding in several ways. We first develop a new method for calculating the scalar spherical harmonics Yjm in terms of components of the direction unit vector n = sin θ cos φ ex + sin θ sin φ ey + cos θ ez .
    [Show full text]
  • Electrodynamic Spherical Harmonic: Defi- Nition and Properties
    Electrodynamic spherical harmonic A V Novitsky Department of Theoretical Physics, Belarusian State University, Nezavisimosti Avenue 4, 220050 Minsk, Belarus [email protected] Abstract Electrodynamic spherical harmonic is a second rank tensor in three- dimensional space. It allows to separate the radial and angle variables in vector solutions of Maxwell’s equations. Using the orthonormaliza- tion for electrodynamic spherical harmonic, a boundary problem on a sphere can be easily solved. 1 Introduction In this paper we introduce new function — electrodynamic spherical har- monic. It is represented as a second rank tensor in three-dimensional space. But the function differs from tensor spherical harmonic [1–5]. By its defini- tion the electrodynamic spherical function possesses a number of properties of usual scalar and vector harmonics and includes them as component parts. Why electrodynamic spherical harmonic? The word “electrodynamic” im- arXiv:0803.4093v1 [math-ph] 28 Mar 2008 plies that the function is applied for solution of vector field problems. We use it in electrodynamics, however one can apply the function for description of spin fields in quantum field theory. Electrodynamic spherical harmonic is not a simple designation of the well- known functions. It satisfies the Maxwell equations and describes the angular dependence of vector fields. The introduced function separates the variables (radial coordinate and angles) in the fields. Moreover, the notation of the fields in terms of electrodynamic spherical harmonic noticeably simplifies the solution of a boundary problem on spherical interface. Just application of 1 the ortonormalization condition allows to find the coefficients of spherical function expansion (e.g. scattering field amplitudes).
    [Show full text]
  • Computation of Real-Valued Basis Functions Which Transform As Irreducible Representations of the Polyhedral Groups
    COMPUTATION OF REAL-VALUED BASIS FUNCTIONS WHICH TRANSFORM AS IRREDUCIBLE REPRESENTATIONS OF THE POLYHEDRAL GROUPS NAN XU∗ AND PETER C. DOERSCHUKy Abstract. Basis functions which are invariant under the operations of a rotational point group G are able to describe any 3-D object which exhibits the rotational point group symmetry. However, in order to characterize the spatial statistics of an ensemble of objects in which each object is different but the statistics exhibit the symmetry, a complete set of basis functions is required. In particular, for each irreducible representation (irrep) of G, it is necessary to include basis functions that transform according to that irrep. This complete set of basis functions is a basis for square- integrable functions on the surface of the sphere in 3-D. Because the objects are real-valued, it is convenient to have real-valued basis functions. In this paper the existence of such real-valued bases is proven and an algorithm for their computation is provided for the icosahedral I and the octahedral O symmetries. Furthermore, it is proven that such a real-valued basis does not exist for the tetrahedral T symmetry because some irreps of T are essentially complex. The importance of these basis functions to computations in single-particle cryo electron microscopy is described. Key words. polyhedral groups, real-valued matrix irreducible representations, numerical com- putation of similarity transformations, basis functions that transform as a row of an irreducible representation of a rotation group, Classical groups (02.20.Hj), Special functions (02.30.Gp). AMS subject classifications. 2604, 2004, 57S17, 57S25 1.
    [Show full text]
  • Spherical Harmonics and Homogeneous Har- Monic Polynomials
    SPHERICAL HARMONICS AND HOMOGENEOUS HAR- MONIC POLYNOMIALS 1. The spherical Laplacean. Denote by S ½ R3 the unit sphere. For a function f(!) de¯ned on S, let f~ denote its extension to an open neighborhood N of S, constant along normals to S (i.e., constant along rays from the origin). We say f 2 C2(S) if f~ is a C2 function in N , and for such functions de¯ne a di®erential operator ¢S by: ¢Sf := ¢f;~ where ¢ on the right-hand side is the usual Laplace operator in R3. With a little work (omitted here) one may derive the expression for ¢ in polar coordinates (r; !) in R2 (r > 0;! 2 S): 2 1 ¢u = u + u + ¢ u: rr r r r2 S (Here ¢Su(r; !) is the operator ¢S acting on the function u(r; :) in S, for each ¯xed r.) A homogeneous polynomial of degree n ¸ 0 in three variables (x; y; z) is a linear combination of `monomials of degree n': d1 d2 d3 x y z ; di ¸ 0; d1 + d2 + d3 = n: This de¯nes a vector space (over R) denoted Pn. A simple combinatorial argument (involving balls and separators, as most of them do), seen in class, yields the dimension: 1 d := dim(P ) = (n + 1)(n + 2): n n 2 Writing a polynomial p 2 Pn in polar coordinates, we necessarily have: n p(r; !) = r f(!); f = pjS; where f is the restriction of p to S. This is an injective linear map p 7! f, but the functions on S so obtained are rather special (a dn-dimensional subspace of the in¯nite-dimensional space C(S) of continuous functions-let's call it Pn(S)) We are interested in the subspace Hn ½ Pn of homogeneous harmonic polynomials of degree n (¢p = 0).
    [Show full text]
  • Arxiv:1805.12125V2 [Quant-Ph] 27 Sep 2018 Calculations in Quantum Mechanics? Even Quantum Experts Will find Something New in the Way We Compute the Spherical Harmonics
    Condensed MatTER Physics, 2018, Vol. 21, No 3, 33002: 1–12 DOI: 10.5488/CMP.21.33002 HTtp://www.icmp.lviv.ua/journal Calculating SPHERICAL HARMONICS WITHOUT DERIVATIVES M. WEITZMAN1, J.K. Freericks2 1 3515 E. Rochelle Ave., Las Vegas, NV 89121, USA 2 Department OF Physics, GeorGETOWN University, 37th AND O Sts. NW, Washington, DC 20057, USA Received May 30, 2018 The DERIVATION OF SPHERICAL HARMONICS IS THE SAME IN NEARLY EVERY QUANTUM MECHANICS TEXTBOOK AND classroom. It IS FOUND TO BE DIffiCULT TO follow, HARD TO understand, AND CHALLENGING TO REPRODUCE BY MOST students. In THIS work, WE SHOW HOW ONE CAN DETERMINE SPHERICAL HARMONICS IN A MORE NATURAL WAY BASED ON OPERATORS AND A POWERFUL IDENTITY CALLED THE EXPONENTIAL DISENTANGLING OPERATOR IDENTITY (known IN QUANTUM optics, BUT LITTLE USED elsewhere). This NEW STRATEGY FOLLOWS NATURALLY AFTER ONE HAS INTRODUCED DirAC notation, COMPUTED THE ANGULAR MOMENTUM ALGEBRa, AND DETERMINED THE ACTION OF THE ANGULAR MOMENTUM RAISING AND LOWERING 2 OPERATORS ON THE SIMULTANEOUS ANGULAR MOMENTUM EIGENSTATES (under Lˆ AND Lˆ z). Key words: ANGULAR momentum, SPHERICAL harmonics, OPERATOR METHODS PACS: 01.40.gb, 02.20.Qs, 03.65.Fd, 31.15.-p 1. Introduction One of us (JKF), first met Prof. Stasyuk in the early 2000s when visiting Ukraine for a conference and then subsequently became involved in a long-standing collaboration with Andrij Shvaika, who is also at the Institute for Condensed Matter Physics in Lviv. It is always a pleasure to meet Prof. Stasyuk on subsequent trips to Lviv as his friendly smile and enthusiasm for physics is contagious.
    [Show full text]
  • Appendix A. Angular Momentum and Spherical Harmonics
    Appendix A. Angular Momentum and Spherical Harmonics The total angular momentum of an isolated system is conserved; it is a constant of the motion. If we change to a new, perhaps more convenient, coordinate system related to the old by a rotation, the wave function for the system trans- forms in a defmite way depending upon the value of the angular momentum. If the system is made up of two or more parts, each of them may have angular momentum, and we need to know the quantum rules for combining these to a resultant. We cannot do more here than introduce and summarise a few salient points about these matters. More detailed discussions of the quantum theory of angular momentum are available in several treatises (for example, Brink and Satchler, 1968; see also Messiah, 1962). A general word of caution is in order. The literature on this subject presents a wide variety of conventions for notation, normalisation and phase of the various quantities which occur. Sometimes these conventions represent largely arbitrary choices, sometimes they emphasise different aspects of the quantities involved. If the reader is aware of this possibility, and if he adopts a consistent set of formulae, little difficulty should be encountered. AI ANGULAR MOMENTUM IN QUANTUM THEORY An isolated system will have a total angular momentum whose square has a discrete value given by j(j + 1)1i 2 , where1i is Planck's constant divided by 21T and where j is an integer or integer-plus-one-half according to whether the 283 284 INTRODUCTION TO NUCLEAR REACTIONS system as a whole is a boson or a fermion.* That is to say, the wave function of the system will be an eigenfunction of 12 , the operator for the square of the angular momentum J2 lj) = j(j + l)lj) (AI) (Henceforth, we shall use natural units so that1i = 1.) Note that we often speak of 'the angular momentum j' when strictly we mean that the square of the angu- lar momentum isj(j + 1).
    [Show full text]
  • Spherical Harmonics
    B Spherical Harmonics PHERICAL harmonics are a frequency-space basis for representing functions defined over S the sphere. They are the spherical analogue of the 1D Fourier series. Spherical harmonics arise in many physical problems ranging from the computation of atomic electron configurations to the representation of gravitational and magnetic fields of planetary bodies. They also appear in the solutions of the Schrödinger equation in spherical coordinates. Spherical harmonics are therefore often covered in textbooks from these fields [MacRobert and Sneddon, 1967; Tinkham, 2003]. Spherical harmonics also have direct applicability in computer graphics. Light transport involves many quantities defined over the spherical and hemispherical domains, making spherical harmonics a natural basis for representing these functions. Early applications of spherical har- monics to computer graphics include the work by Cabral et al.[1987] and Sillion et al.[1991]. More recently, several in-depth introductions have appeared in the graphics literature [Ramamoorthi, 2002; Green, 2003; Wyman, 2004; Sloan, 2008]. In this appendix, we briefly review the spherical harmonics as they relate to computer graphics. We define the basis and examine several important properties arising from this defini- tion. 167 168 B.1 Definition A harmonic is a function that satisfies Laplace’s equation: 2 f 0. (B.1) r Æ As their name suggests, the spherical harmonics are an infinite set of harmonic functions defined on the sphere. They arise from solving the angular portion of Laplace’s equation in spherical coordinates using separation of variables. The spherical harmonic basis functions derived in this fashion take on complex values, but a complementary, strictly real-valued, set of harmonics can also be defined.
    [Show full text]
  • Spectral Geometry Processing with Manifold Harmonics Bruno Vallet, Bruno Lévy
    Spectral Geometry Processing with Manifold Harmonics Bruno Vallet, Bruno Lévy To cite this version: Bruno Vallet, Bruno Lévy. Spectral Geometry Processing with Manifold Harmonics. [Technical Report] 2007. inria-00186931 HAL Id: inria-00186931 https://hal.inria.fr/inria-00186931 Submitted on 13 Nov 2007 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Spectral Geometry Processing with Manifold Harmonics Bruno Vallet and Bruno Levy´ Tech Report ALICE-2007-001 Figure 1: Processing pipeline of our method: A: Compute the Manifold Harmonic Basis (MHB) of the input triangulated mesh. B: Transform the geometry into frequency space by computing the Manifold Harmonic Transform (MHT). C: Apply the frequency space filter on the transformed geometry. D: Transform back into geometric space by computing the inverse MHT. Abstract Introduction We present a new method to convert the geometry of a mesh into frequency space. The eigenfunctions of the Laplace-Beltrami op- 3D scanning technology easily produces computer representations erator are used to define Fourier-like function basis and transform. from real objects. However, the acquired geometry often presents Since this generalizes the classical Spherical Harmonics to arbitrary some noise that needs to be filtered out.
    [Show full text]
  • * Appendix A: Spherical Tensor Formalism
    * Appendix A: Spherical Tensor Formalism The geometrical aspects of the quantum-mechanical properties of molecular systems are described in terms of quantities familiar from angular momentum theory and the reader is referred to the text books existing on this subject. 1- 3 In this appendix we collect together the quantities and relations most fre­ quently used in calculations concerning the properties of solid hydrogen and other systems of interacting molecules. A.1 Spherical Harmonics The spherical harmonics Y,m(O,<jJ) are defined as the angular parts of the simultaneous eigenfunctions of the square and the :: component of the orbital angular momentum of a single particle, measured in units h (A.1) where m = I, I - 1, ... , -I and I = 0, 1,2, .... These functions are mu­ tually orthogonal and normalized on the unit sphere, (1m I I'm') == SOh S: Y;';" Y,'m,sine de d<jJ = ()ll'()mm' (A.2) The Y,m are determined uniquely by eqs. (A.1)-(A.2) up to arbitrary phase factors which we fix by choosing the so-called Condon and Shortley phases defined by the conditions that the nonvanishing matrix elements of the operators L, ± iLy and the values of the Y,o at 0 = °are real and positive (Lx ± iL)Y,m = [(I ± m + 1)(1 =+= m)]1'2y,.m±1 (A.3) and 21 + 1)1.12 Y,o(O,<jJ) = ( ~ (AA) 277 278 Appendix A • Spherical Tensor Formalism F or this choice of phases Y'tr,(8,c/J) = (-lrr;m(8,c/J) (A.5) where m == -m.
    [Show full text]