More generally, hypergeometric series can be generalized to describe the symmetries of any symmetric space; in particular, hypergeometric series can be developed for any Lie group. The statement of the parity of spherical harmonics is then. H Basically, you can always think of a spherical harmonic in terms of the generalized polynomial. e^{-i m \phi} {\displaystyle \gamma } 2 m S In both classical and quantum mechanical systems, angular momentum (together with linear momentum and energy) is one of the three fundamental properties of motion. where Statements relating the growth of the Sff() to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. {\displaystyle r=0} They will be functions of \(0 \leq \theta \leq \pi\) and \(0 \leq \phi<2 \pi\), i.e. Essentially all the properties of the spherical harmonics can be derived from this generating function. Finally, the equation for R has solutions of the form R(r) = A r + B r 1; requiring the solution to be regular throughout R3 forces B = 0.[3]. The angular components of . 1 By polarization of A, there are coefficients The vector spherical harmonics are now defined as the quantities that result from the coupling of ordinary spherical harmonics and the vectors em to form states of definite J (the resultant of the orbital angular momentum of the spherical harmonic and the one unit possessed by the em ). , , z By definition, (382) where is an integer. The animation shows the time dependence of the stationary state i.e. c {\displaystyle m<0} Y Y http://titan.physx.u-szeged.hu/~mmquantum/videok/Gombfuggveny_fazis_idofejlodes.flv. &\hat{L}_{z}=-i \hbar \partial_{\phi} , since any such function is automatically harmonic. R : {\displaystyle \varphi } Chapters 1 and 2. ) m f r is replaced by the quantum mechanical spin vector operator as a function of are the Legendre polynomials, and they can be derived as a special case of spherical harmonics. {\displaystyle m>0} Spherical harmonics can be generalized to higher-dimensional Euclidean space C 2 See here for a list of real spherical harmonics up to and including | the one containing the time dependent factor \(e_{it/}\) as well given by the function \(Y_{1}^{3}(,)\). (the irregular solid harmonics and order = is just the 3-dimensional space of all linear functions m However, the solutions of the non-relativistic Schrdinger equation without magnetic terms can be made real. The geodesy[11] and magnetics communities never include the CondonShortley phase factor in their definitions of the spherical harmonic functions nor in the ones of the associated Legendre polynomials. Y In 1782, Pierre-Simon de Laplace had, in his Mcanique Cleste, determined that the gravitational potential ) C , For a given value of , there are 2 + 1 independent solutions of this form, one for each integer m with m . ( {\displaystyle Y_{\ell }^{m}} m terms (sines) are included: The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation. The expansion coefficients are the analogs of Fourier coefficients, and can be obtained by multiplying the above equation by the complex conjugate of a spherical harmonic, integrating over the solid angle , and utilizing the above orthogonality relationships. {\displaystyle \mathbf {J} } 0 Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle between x1 and x. R {\displaystyle \theta } they can be considered as complex valued functions whose domain is the unit sphere. . ( The tensor spherical harmonics 1 The Clebsch-Gordon coecients Consider a system with orbital angular momentum L~ and spin angular momentum ~S. Operators for the square of the angular momentum and for its zcomponent: To make full use of rotational symmetry and angular momentum, we will restrict our attention to spherically symmetric potentials, \begin {aligned} V (\vec {r}) = V (r). {\displaystyle {\mathcal {Y}}_{\ell }^{m}} {\displaystyle \ell } Y Y ( Y = l m ( ( Specifically, if, A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. m and Now we're ready to tackle the Schrdinger equation in three dimensions. {\displaystyle c\in \mathbb {C} } There are several different conventions for the phases of Nlm, so one has to be careful with them. , or alternatively where C as a function of The convergence of the series holds again in the same sense, namely the real spherical harmonics is homogeneous of degree A specific set of spherical harmonics, denoted The 3-D wave equation; spherical harmonics. For convenience, we list the spherical harmonics for = 0,1,2 and non-negative values of m. = 0, Y0 0 (,) = 1 4 = 1, Y1 When you apply L 2 to an angular momentum eigenstate l, then you find L 2 l = [ l ( l + 1) 2] l. That is, l ( l + 1) 2 is the value of L 2 which is associated to the eigenstate l. : {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} {\displaystyle Y_{\ell }^{m}} e \end{aligned}\) (3.27). In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum[4]. That is, a polynomial p is in P provided that for any real f {\displaystyle \mathbb {R} ^{n}\to \mathbb {C} } Remember from chapter 2 that a subspace is a specic subset of a general complex linear vector space. r The solid harmonics were homogeneous polynomial solutions 1 The eigenfunctions of \(\hat{L}^{2}\) will be denoted by \(Y(,)\), and the angular eigenvalue equation is: \(\begin{aligned} m The classical definition of the angular momentum vector is, \(\mathcal{L}=\mathbf{r} \times \mathbf{p}\) (3.1), which depends on the choice of the point of origin where |r|=r=0|r|=r=0. It is common that the (cross-)power spectrum is well approximated by a power law of the form. R m The spherical harmonics with negative can be easily compute from those with positive . f There is no requirement to use the CondonShortley phase in the definition of the spherical harmonic functions, but including it can simplify some quantum mechanical operations, especially the application of raising and lowering operators. In that case, one needs to expand the solution of known regions in Laurent series (about Y T is essentially the associated Legendre polynomial ) An exception are the spin representation of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). \(\hat{L}^{2}=-\hbar^{2}\left(\partial_{\theta \theta}^{2}+\cot \theta \partial_{\theta}+\frac{1}{\sin ^{2} \theta} \partial_{\phi \phi}^{2}\right)=-\hbar^{2} \Delta_{\theta \phi}\) (3.7). Y If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. P {\displaystyle r^{\ell }Y_{\ell }^{m}(\mathbf {r} /r)} form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions The spherical harmonics have definite parity. {\displaystyle f_{\ell }^{m}} x &\Pi_{\psi_{+}}(\mathbf{r})=\quad \psi_{+}(-\mathbf{r})=\psi_{+}(\mathbf{r}) \\ r , and if. 's of degree are eigenfunctions of the square of the orbital angular momentum operator, Laplace's equation imposes that the Laplacian of a scalar field f is zero. {\displaystyle \ell =1} m m as a homogeneous function of degree ( There are several different conventions for the phases of \(\mathcal{N}_{l m}\), so one has to be careful with them. m l : {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } {\displaystyle \Re [Y_{\ell }^{m}]=0} As . ( They occur in . 3 S , are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. = r Such an expansion is valid in the ball. {\displaystyle Y_{\ell }^{m}({\mathbf {r} })} {\displaystyle \mathbf {r} } x , 2 {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } m 4 . R , the real and imaginary components of the associated Legendre polynomials each possess |m| zeros, each giving rise to a nodal 'line of latitude'. Your vector spherical harmonics are functions of in the vector space $$ \pmb{Y}_{j\ell m} \in V=\left\{ \mathbf f:\mathbb S^2 \to \mathbb C^3 : \int_{\mathbb S^2} |\mathbf f(\pmb\Omega)|^2 \mathrm d \pmb\Omega <\infty . These angular solutions Just as in one dimension the eigenfunctions of d 2 / d x 2 have the spatial dependence of the eigenmodes of a vibrating string, the spherical harmonics have the spatial dependence of the eigenmodes of a vibrating spherical . [ The Laplace spherical harmonics {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } {\displaystyle S^{n-1}\to \mathbb {C} } m : A R {\displaystyle P_{\ell }^{m}:[-1,1]\to \mathbb {R} } The first term depends only on \(\) while the last one is a function of only \(\). {\displaystyle Y_{\ell }^{m}} are composed of circles: there are |m| circles along longitudes and |m| circles along latitudes. 1 {\displaystyle \mathbf {r} } The reason why we consider parity in connection with the angular momentum is that the simultaneous eigenfunctions of \(\hat{L}^{2}\) and \(\hat{L}_{z}\) the spherical harmonics times any function of the radial variable r are eigenfunctions of \(\) as well, and the corresponding eigenvalues are \((1)^{}\). A and Spherical Harmonics 11.1 Introduction Legendre polynomials appear in many different mathematical and physical situations: . { f S In both definitions, the spherical harmonics are orthonormal, The disciplines of geodesy[10] and spectral analysis use, The magnetics[10] community, in contrast, uses Schmidt semi-normalized harmonics. y Like the sines and cosines in Fourier series, the spherical harmonics may be organized by (spatial) angular frequency, as seen in the rows of functions in the illustration on the right. ) {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } ), In 1867, William Thomson (Lord Kelvin) and Peter Guthrie Tait introduced the solid spherical harmonics in their Treatise on Natural Philosophy, and also first introduced the name of "spherical harmonics" for these functions. is the operator analogue of the solid harmonic S As these are functions of points in real three dimensional space, the values of \(()\) and \((+2)\) must be the same, as these values of the argument correspond to identical points in space. This is because a plane wave can actually be written as a sum over spherical waves: \[ e^{i\vec{k}\cdot\vec{r}}=e^{ikr\cos\theta}=\sum_l i^l(2l+1)j_l(kr)P_l(\cos\theta) \label{10.2.2}\] Visualizing this plane wave flowing past the origin, it is clear that in spherical terms the plane wave contains both incoming and outgoing spherical waves. Y The total angular momentum of the system is denoted by ~J = L~ + ~S. ) 3 , {\displaystyle (2\ell +1)} Show that the transformation \(\{x, y, z\} \longrightarrow\{-x,-y,-z\}\) is equivalent to \(\theta \longrightarrow \pi-\theta, \quad \phi \longrightarrow \phi+\pi\). C 1 Given two vectors r and r, with spherical coordinates ) m The first few functions are the following, with one of the usual phase (sign) conventions: \(Y_{0}^{0}(\theta, \phi)=\frac{1}{\sqrt{4} \pi}\) (3.25), \(Y_{1}^{0}(\theta, \phi)=\sqrt{\frac{3}{4 \pi}} \cos \theta, \quad Y_{1}^{1}(\theta, \phi)=-\sqrt{\frac{3}{8 \pi}} \sin \theta e^{i \phi}, \quad Y_{1}^{-1}(\theta, \phi)=\sqrt{\frac{3}{8 \pi}} \sin \theta e^{-i \phi}\) (3.26). + ( r {\displaystyle Y_{\ell }^{m}} The function \(P_{\ell}^{m}(z)\) is a polynomial in z only if \(|m|\) is even, otherwise it contains a term \(\left(1-z^{2}\right)^{|m| / 2}\) which is a square root. , When = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. R ) m Y {\displaystyle {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })} Y {\displaystyle P_{\ell }^{m}} While the standard spherical harmonics are a basis for the angular momentum operator, the spinor spherical harmonics are a basis for the total angular momentum operator (angular momentum plus spin ). {4\pi (l + |m|)!} In this chapter we will discuss the basic theory of angular momentum which plays an extremely important role in the study of quantum mechanics. \(\begin{aligned} {\displaystyle \ell } The spherical harmonic functions depend on the spherical polar angles and and form an (infinite) complete set of orthogonal, normalizable functions. {\displaystyle S^{2}} ), instead of the Taylor series (about ) r , which can be seen to be consistent with the output of the equations above. in their expansion in terms of the . 2 by \(\mathcal{R}(r)\). Another is complementary hemispherical harmonics (CHSH). Laplace equation. = {\displaystyle Y_{\ell }^{m}} : directions respectively. m 2 Y That is. C m m Y provide a basis set of functions for the irreducible representation of the group SO(3) of dimension transforms into a linear combination of spherical harmonics of the same degree. C m = i 2 ) C Find \(P_{2}^{0}(\theta)\), \(P_{2}^{1}(\theta)\), \(P_{2}^{2}(\theta)\). \end{aligned}\) (3.8). Historically the spherical harmonics with the labels \(=0,1,2,3,4\) are called \(s, p, d, f, g \ldots\) functions respectively, the terminology is coming from spectroscopy. {\displaystyle f_{\ell }^{m}\in \mathbb {C} } In order to satisfy this equation for all values of \(\) and \(\) these terms must be separately equal to a constant with opposite signs. It follows from Equations ( 371) and ( 378) that. Y {\displaystyle \ell =4} specified by these angles. \end{array}\right.\) (3.12), and any linear combinations of them. {\displaystyle q=m} Share Cite Improve this answer Follow edited Aug 26, 2019 at 15:19 For other uses, see, A historical account of various approaches to spherical harmonics in three dimensions can be found in Chapter IV of, The approach to spherical harmonics taken here is found in (, Physical applications often take the solution that vanishes at infinity, making, Heiskanen and Moritz, Physical Geodesy, 1967, eq. S The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. a {\displaystyle f:S^{2}\to \mathbb {C} \supset \mathbb {R} } One can choose \(e^{im}\), and include the other one by allowing mm to be negative. ( ( Just prior to that time, Adrien-Marie Legendre had investigated the expansion of the Newtonian potential in powers of r = |x| and r1 = |x1|. By using the results of the previous subsections prove the validity of Eq. \left(\partial_{\theta \theta}^{2}+\cot \theta \partial_{\theta}+\frac{1}{\sin ^{2} \theta} \partial_{\phi \phi}^{2}\right) Y(\theta, \phi) &=-\ell(\ell+1) Y(\theta, \phi) . This constant is traditionally denoted by \(m^{2}\) and \(m^{2}\) (note that this is not the mass) and we have two equations: one for \(\), and another for \(\). and In this chapter we discuss the angular momentum operator one of several related operators analogous to classical angular momentum. Laplace's spherical harmonics [ edit] Real (Laplace) spherical harmonics for (top to bottom) and (left to right). S 2 Show that \(P_{}(z)\) are either even, or odd depending on the parity of \(\). P ( is that it is null: It suffices to take C Angular momentum and spherical harmonics The angular part of the Laplace operator can be written: (12.1) Eliminating (to solve for the differential equation) one needs to solve an eigenvalue problem: (12.2) where are the eigenvalues, subject to the condition that the solution be single valued on and . listed explicitly above we obtain: Using the equations above to form the real spherical harmonics, it is seen that for That is, they are either even or odd with respect to inversion about the origin. 2 Recalling that the spherical harmonics are eigenfunctions of the angular momentum operator: (r; ;) = R(r)Ym l ( ;) SeparationofVariables L^2Ym l ( ;) = h2l . Prove that \(P_{\ell}^{m}(z)\) are solutions of (3.16) for all \(\) and \(|m|\), if \(|m|\). {\displaystyle k={\ell }} about the origin that sends the unit vector + n r k C of the elements of 's, which in turn guarantees that they are spherical tensor operators, , respectively, the angle {\displaystyle p:\mathbb {R} ^{3}\to \mathbb {C} } = R , the solid harmonics with negative powers of The spherical harmonics are the eigenfunctions of the square of the quantum mechanical angular momentum operator. One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of For a fixed integer , every solution Y(, ), Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). Let us also note that the \(m=0\) functions do not depend on \(\), and they are proportional to the Legendre polynomials in \(cos\). S One sees at once the reason and the advantage of using spherical coordinates: the operators in question do not depend on the radial variable r. This is of course also true for \(\hat{L}^{2}=\hat{L}_{x}^{2}+\hat{L}_{y}^{2}+\hat{L}_{z}^{2}\) which turns out to be \(^{2}\) times the angular part of the Laplace operator \(_{}\). being a unit vector, In terms of the spherical angles, parity transforms a point with coordinates . {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } to all of We consider the second one, and have: \(\frac{1}{\Phi} \frac{d^{2} \Phi}{d \phi^{2}}=-m^{2}\) (3.11), \(\Phi(\phi)=\left\{\begin{array}{l} This operator thus must be the operator for the square of the angular momentum. In quantum mechanics the constants \(\ell\) and \(m\) are called the azimuthal quantum number and magnetic quantum number due to their association with rotation and how the energy of an . i The spherical harmonics called \(J_J^{m_J}\) are functions whose probability \(|Y_J^{m_J}|^2\) has the well known shapes of the s, p and d orbitals etc learned in general chemistry. Spherical harmonics originate from solving Laplace's equation in the spherical domains. . The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry. R 1 is that for real functions ) : {\displaystyle m} i There are of course functions which are neither even nor odd, they do not belong to the set of eigenfunctions of \(\). 1 i Equation \ref{7-36} is an eigenvalue equation. and modelling of 3D shapes. 2 2 2 A {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } m ( 1.1 Orbital Angular Momentum - Spherical Harmonics Classically, the angular momentum of a particle is the cross product of its po-sition vector r =(x;y;z) and its momentum vector p =(p x;p y;p z): L = rp: The quantum mechanical orbital angular momentum operator is dened in the same way with p replaced by the momentum operator p!ihr . or . S to Laplace's equation S Considering from the above-mentioned polynomial of degree Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section Higher dimensions below). A variety of techniques are available for doing essentially the same calculation, including the Wigner 3-jm symbol, the Racah coefficients, and the Slater integrals. m m 2 , {\displaystyle f_{\ell }^{m}\in \mathbb {C} } Furthermore, a change of variables t = cos transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial Pm(cos ) . S m above. The reason for this can be seen by writing the functions in terms of the Legendre polynomials as. (1) From this denition 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 . {\displaystyle r>R} The functions \(P_{\ell}^{m}(z)\) are called associated Legendre functions. R ] C As to what's "really" going on, it's exactly the same thing that you have in the quantum mechanical addition of angular momenta. r {\displaystyle r} In many fields of physics and chemistry these spherical harmonics are replaced by cubic harmonics because the rotational symmetry of the atom and its environment are distorted or because cubic harmonics offer computational benefits. The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } S L 2 Y 21 That is: Spherically symmetric means that the angles range freely through their full domains each of which is finite leading to a universal set of discrete separation constants for the angular part of all spherically symmetric problems. Angular momentum is not a property of a wavefunction at a point; it is a property of a wavefunction as a whole. 3 Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. This is valid for any orthonormal basis of spherical harmonics of degree, Applications of Legendre polynomials in physics, Learn how and when to remove this template message, "Symmetric tensor spherical harmonics on the N-sphere and their application to the de Sitter group SO(N,1)", "Zernike like functions on spherical cap: principle and applications in optical surface fitting and graphics rendering", "On nodal sets and nodal domains on S and R", https://en.wikipedia.org/w/index.php?title=Spherical_harmonics&oldid=1146217720, D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, This page was last edited on 23 March 2023, at 13:52. In this setting, they may be viewed as the angular portion of a set of solutions to Laplace's equation in three dimensions, and this viewpoint is often taken as an alternative definition. The essential property of More generally, the analogous statements hold in higher dimensions: the space H of spherical harmonics on the n-sphere is the irreducible representation of SO(n+1) corresponding to the traceless symmetric -tensors. The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication. Since they are eigenfunctions of Hermitian operators, they are orthogonal . m m y . where \(P_{}(z)\) is the \(\)-th Legendre polynomial, defined by the following formula, (called the Rodrigues formula): \(P_{\ell}(z):=\frac{1}{2^{\ell} \ell ! L~ spherical harmonics angular momentum ~S. 378 ) that being a unit vector, in terms of the Legendre polynomials appear many! 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