C : The quantum number \(\) is called angular momentum quantum number, or sometimes for a historical reason as azimuthal quantum number, while m is the magnetic quantum number. [ Laplace's spherical harmonics [ edit] Real (Laplace) spherical harmonics for (top to bottom) and (left to right). can thus be expanded as a linear combination of these: This expansion holds in the sense of mean-square convergence convergence in L2 of the sphere which is to say that. ] The angular components of . ( 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. {\displaystyle r=\infty } C = at a point x associated with a set of point masses mi located at points xi was given by, Each term in the above summation is an individual Newtonian potential for a point mass. Y 3 ) + m brackets are functions of ronly, and the angular momentum operator is only a function of and . i Share Cite Improve this answer Follow edited Aug 26, 2019 at 15:19 \(\hat{L}^{2}=-\hbar^{2}\left(\partial_{\theta \theta}^{2}+\cot \theta \partial_{\theta}+\frac{1}{\sin ^{2} \theta} \partial_{\phi \phi}^{2}\right)=-\hbar^{2} \Delta_{\theta \phi}\) (3.7). Another is complementary hemispherical harmonics (CHSH). to all of {\displaystyle \ell =1} 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. ] are associated Legendre polynomials without the CondonShortley phase (to avoid counting the phase twice). R {\displaystyle \lambda } , respectively, the angle can also be expanded in terms of the real harmonics http://titan.physx.u-szeged.hu/~mmquantum/videok/Gombfuggveny_fazis_idofejlodes.flv. e^{-i m \phi} When the spherical harmonic order m is zero (upper-left in the figure), the spherical harmonic functions do not depend upon longitude, and are referred to as zonal. (18) of Chapter 4] . For a scalar function f(n), the spin S is zero, and J is purely orbital angular momentum L, which accounts for the functional dependence on n. The spherical decomposition f . 1 C {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } 2 Y to r! , z For a given value of , there are 2 + 1 independent solutions of this form, one for each integer m with m . The spherical harmonic functions depend on the spherical polar angles and and form an (infinite) complete set of orthogonal, normalizable functions. where the superscript * denotes complex conjugation. r {\displaystyle c\in \mathbb {C} } as a homogeneous function of degree of the elements of It can be shown that all of the above normalized spherical harmonic functions satisfy. ) \end{aligned}\) (3.30). {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } is just the 3-dimensional space of all linear functions to Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. 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 . {\displaystyle \{\pi -\theta ,\pi +\varphi \}} As . and Y 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 . By analogy with classical mechanics, the operator L 2, that represents the magnitude squared of the angular momentum vector, is defined (7.1.2) L 2 = L x 2 + L y 2 + L z 2. Find the first three Legendre polynomials \(P_{0}(z)\), \(P_{1}(z)\) and \(P_{2}(z)\). {\displaystyle Y_{\ell }^{m}} 1 Note that the angular momentum is itself a vector. The spherical harmonics are orthonormal: that is, Y l, m Yl, md = ll mm, and also form a complete set. The group PSL(2,C) is isomorphic to the (proper) Lorentz group, and its action on the two-sphere agrees with the action of the Lorentz group on the celestial sphere in Minkowski space. Under this operation, a spherical harmonic of degree L 2 Y 21 (see associated Legendre polynomials), In acoustics,[7] the Laplace spherical harmonics are generally defined as (this is the convention used in this article). {\displaystyle \lambda \in \mathbb {R} } {\displaystyle \mathbf {r} } 1 Functions that are solutions to Laplace's equation are called harmonics. f B C 2 f Remember from chapter 2 that a subspace is a specic subset of a general complex linear vector space. The 3-D wave equation; spherical harmonics. {\displaystyle r=0} The eigenfunctions of the orbital angular momentum operator, the spherical harmonics Reasoning: The common eigenfunctions of L 2 and L z are the spherical harmonics. 2 The result of acting by the parity on a function is the mirror image of the original function with respect to the origin. \end{aligned}\) (3.6). m . 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). Spherical Harmonics are a group of functions used in math and the physical sciences to solve problems in disciplines including geometry, partial differential equations, and group theory. {\displaystyle m>0} That is. {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. Y ) : S {\displaystyle (r',\theta ',\varphi ')} Hence, , , Spherical Harmonics, and Bessel Functions Physics 212 2010, Electricity and Magnetism Michael Dine Department of Physics . 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). {\displaystyle \mathbb {R} ^{3}} ( For example, for any ) For example, as can be seen from the table of spherical harmonics, the usual p functions ( P S in the , Finally, evaluating at x = y gives the functional identity, Another useful identity expresses the product of two spherical harmonics as a sum over spherical harmonics[21]. Now we're ready to tackle the Schrdinger equation in three dimensions. In spherical coordinates this is:[2]. In particular, the colatitude , or polar angle, ranges from 0 at the North Pole, to /2 at the Equator, to at the South Pole, and the longitude , or azimuth, may assume all values with 0 < 2. Consider a rotation {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } {\displaystyle \mathbf {J} } Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree {\displaystyle f_{\ell }^{m}\in \mathbb {C} } Since the angular momentum part corresponds to the quadratic casimir operator of the special orthogonal group in d dimensions one can calculate the eigenvalues of the casimir operator and gets n = 1 d / 2 n ( n + d 2 n), where n is a positive integer. ) The 19th century development of Fourier series made possible the solution of a wide variety of physical problems in rectangular domains, such as the solution of the heat equation and wave equation. R in . 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. As to what's "really" going on, it's exactly the same thing that you have in the quantum mechanical addition of angular momenta. C Y Y ( 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 \mathbf {r} } 3 We will first define the angular momentum operator through the classical relation L = r p and replace p by its operator representation -i [see Eq. c Given two vectors r and r, with spherical coordinates m f {\displaystyle P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )} Then The solution function Y(, ) is regular at the poles of the sphere, where = 0, . {\displaystyle \ell } The operator of parity \(\) is defined in the following way: \(\Pi \psi(\mathbf{r})=\psi(-\mathbf{r})\) (3.29). This is why the real forms are extensively used in basis functions for quantum chemistry, as the programs don't then need to use complex algebra. An exception are the spin representation of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). m This equation easily separates in . f {\displaystyle Y_{\ell m}} The functions \(P_{\ell}^{m}(z)\) are called associated Legendre functions. {\displaystyle L_{\mathbb {R} }^{2}(S^{2})} B + ) m Furthermore, the zonal harmonic {\displaystyle (x,y,z)} That is, they are either even or odd with respect to inversion about the origin. m 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 ) . ( The complex spherical harmonics {\displaystyle f_{\ell }^{m}} 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. R (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 . r m r In a similar manner, one can define the cross-power of two functions as, is defined as the cross-power spectrum. 2 A 3 2 Operators for the square of the angular momentum and for its zcomponent: m . x Now, it is easily demonstrated that if A and B are two general operators then (7.1.3) [ A 2, B] = A [ A, B] + [ A, B] A. {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} only the {\displaystyle \mathbf {r} '} Since mm can take only the integer values between \(\) and \(+\), there are \(2+1\) different possible projections, corresponding to the \(2+1\) different functions \(Y_{m}^{}(,)\) with a given \(\). directions respectively. f (8.2) 8.2 Angular momentum operator For a quantum system the angular momentum is an observable, we can measure the angular momentum of a particle in a given quantum state. One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of ) m In 1782, Pierre-Simon de Laplace had, in his Mcanique Cleste, determined that the gravitational potential r {\displaystyle Y_{\ell }^{m}} r {\displaystyle P_{\ell }^{m}(\cos \theta )} {\displaystyle \langle \theta ,\varphi |lm\rangle =Y_{l}^{m}(\theta ,\varphi )} symmetric on the indices, uniquely determined by the requirement. L by setting, The real spherical harmonics r Many facts about spherical harmonics (such as the addition theorem) that are proved laboriously using the methods of analysis acquire simpler proofs and deeper significance using the methods of symmetry. ( , 2 &p_{x}=\frac{y}{r}=-\frac{\left(Y_{1}^{-1}+Y_{1}^{1}\right)}{\sqrt{2}}=\sqrt{\frac{3}{4 \pi}} \sin \theta \sin \phi \\ ( There are several different conventions for the phases of \(\mathcal{N}_{l m}\), so one has to be careful with them. The functions {\displaystyle \Re [Y_{\ell }^{m}]=0} {\displaystyle \varphi } 3 ( r In the first case the eigenfunctions \(\psi_{+}(\mathbf{r})\) belonging to eigenvalue +1 are the even functions, while in the second we see that \(\psi_{-}(\mathbf{r})\) are the odd functions belonging to the eigenvalue 1. k p {\displaystyle T_{q}^{(k)}} R See here for a list of real spherical harmonics up to and including , any square-integrable function . ( {\displaystyle \varphi } m 's, which in turn guarantees that they are spherical tensor operators, {\displaystyle S^{2}} S Y Such spherical harmonics are a special case of zonal spherical functions. f f , the real and imaginary components of the associated Legendre polynomials each possess |m| zeros, each giving rise to a nodal 'line of latitude'. \end {aligned} V (r) = V (r). 2 The statement of the parity of spherical harmonics is then. In 3D computer graphics, spherical harmonics play a role in a wide variety of topics including indirect lighting (ambient occlusion, global illumination, precomputed radiance transfer, etc.) {\displaystyle P_{\ell }^{m}:[-1,1]\to \mathbb {R} } {\displaystyle Y_{\ell }^{m}} Abstract. Let Yj be an arbitrary orthonormal basis of the space H of degree spherical harmonics on the n-sphere. In order to obtain them we have to make use of the expression of the position vector by spherical coordinates, which are connected to the Cartesian components by, \(\mathbf{r}=x \hat{\mathbf{e}}_{x}+y \hat{\mathbf{e}}_{y}+z \hat{\mathbf{e}}_{z}=r \sin \theta \cos \phi \hat{\mathbf{e}}_{x}+r \sin \theta \sin \phi \hat{\mathbf{e}}_{y}+r \cos \theta \hat{\mathbf{e}}_{z}\) (3.4). 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. Y One can choose \(e^{im}\), and include the other one by allowing mm to be negative. Thus, the wavefunction can be written in a form that lends to separation of variables. = . We have to write the given wave functions in terms of the spherical harmonics. ] The spherical harmonics are normalized . ( {\displaystyle (A_{m}\pm iB_{m})} The first term depends only on \(\) while the last one is a function of only \(\). where the absolute values of the constants Nlm ensure the normalization over the unit sphere, are called spherical harmonics. are guaranteed to be real, whereas their coefficients Notice that \(\) must be a nonnegative integer otherwise the definition (3.18) makes no sense, and in addition if |(|m|>\), then (3.17) yields zero. 2 &\hat{L}_{z}=-i \hbar \partial_{\phi} Considering {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } n Y By separation of variables, two differential equations result by imposing Laplace's equation: for some number m. A priori, m is a complex constant, but because must be a periodic function whose period evenly divides 2, m is necessarily an integer and is a linear combination of the complex exponentials e im. L z Y 21 (b.) S 3 B Another way of using these functions is to create linear combinations of functions with opposite m-s. {\displaystyle \mathbf {a} =[{\frac {1}{2}}({\frac {1}{\lambda }}-\lambda ),-{\frac {i}{2}}({\frac {1}{\lambda }}+\lambda ),1].}. Answer: N2 Z 2 0 cos4 d= N 2 3 8 2 0 = N 6 8 = 1 N= 4 3 1/2 4 3 1/2 cos2 = X n= c n 1 2 ein c n = 4 6 1/2 1 Z 2 0 cos2 ein d . m R ( , R Y , {\displaystyle Y_{\ell }^{m}} {\displaystyle \ell =1} m = , commonly referred to as the CondonShortley phase in the quantum mechanical literature. {\displaystyle z} The general technique is to use the theory of Sobolev spaces. m , and z the expansion coefficients . x ) above. is just the space of restrictions to the sphere : The parallelism of the two definitions ensures that the R : {\displaystyle \ell =2} R {\displaystyle S^{n-1}\to \mathbb {C} } By polarization of A, there are coefficients Recalling that the spherical harmonics are eigenfunctions of the angular momentum operator: (r; ;) = R(r)Ym l ( ;) SeparationofVariables L^2Ym l ( ;) = h2l . If an external magnetic field \(\mathbf{B}=\{0,0, B\}\) is applied, the projection of the angular momentum onto the field direction is \(m\). {\displaystyle S^{2}} , since any such function is automatically harmonic. > m = The general solution The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation. 4 2 R {\displaystyle \theta } m > For the case of orthonormalized harmonics, this gives: If the coefficients decay in sufficiently rapidly for instance, exponentially then the series also converges uniformly to f. A square-integrable function C 1 Spherical harmonics are ubiquitous in atomic and molecular physics. 2 m ) ] The total angular momentum of the system is denoted by ~J = L~ + ~S. : 3 {\displaystyle B_{m}} Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. i 2 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. is homogeneous of degree It is common that the (cross-)power spectrum is well approximated by a power law of the form. . By using the results of the previous subsections prove the validity of Eq. {\displaystyle L_{\mathbb {C} }^{2}(S^{2})} r f 0 The solid harmonics were homogeneous polynomial solutions or by \(\mathcal{R}(r)\). Chapters 1 and 2. Lecture 6: 3D Rigid Rotor, Spherical Harmonics, Angular Momentum We can now extend the Rigid Rotor problem to a rotation in 3D, corre-sponding to motion on the surface of a sphere of radius R. The Hamiltonian operator in this case is derived from the Laplacian in spherical polar coordi-nates given as 2 = 2 x 2 + y + 2 z . {\displaystyle v} {\displaystyle r>R} m C 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. 2 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. inside three-dimensional Euclidean space One can also understand the differentiability properties of the original function f in terms of the asymptotics of Sff(). The Laplace spherical harmonics f m &\Pi_{\psi_{+}}(\mathbf{r})=\quad \psi_{+}(-\mathbf{r})=\psi_{+}(\mathbf{r}) \\ In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum[4]. ) {\displaystyle \ell =4} . and Spherical Harmonics 11.1 Introduction Legendre polynomials appear in many different mathematical and physical situations: . R , are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. [18], In particular, when x = y, this gives Unsld's theorem[19], In the expansion (1), the left-hand side P(xy) is a constant multiple of the degree zonal spherical harmonic. 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 . S 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 \((1)^{}\). [27] One is hemispherical functions (HSH), orthogonal and complete on hemisphere. The three Cartesian components of the angular momentum are: L x = yp z zp y,L y = zp x xp z,L z = xp y yp x. However, whereas every irreducible tensor representation of SO(2) and SO(3) is of this kind, the special orthogonal groups in higher dimensions have additional irreducible representations that do not arise in this manner. S Legal. In order to satisfy this equation for all values of \(\) and \(\) these terms must be separately equal to a constant with opposite signs. {\displaystyle m>0} S , and their nodal sets can be of a fairly general kind.[22]. ) are chosen instead. 1 They are, moreover, a standardized set with a fixed scale or normalization. The spherical harmonics form an infinite system of orthonormal functions in the sense: \(\int_{0}^{2 \pi} \int_{0}^{\pi}\left(Y_{\ell^{\prime}}^{m^{\prime}}(\theta, \phi)\right)^{*} Y_{\ell}^{m}(\theta, \phi) \sin \theta d \theta d \phi=\delta_{\ell \ell^{\prime}} \delta_{m m^{\prime}}\) (3.22). m , y The spinor spherical harmonics are the natural spinor analog of the vector spherical harmonics. : This can be formulated as: \(\Pi \mathcal{R}(r) Y_{\ell}^{m}(\theta, \phi)=\mathcal{R}(r) \Pi Y_{\ell}^{m}(\theta, \phi)=(-1)^{\ell} \mathcal{R}(r) Y(\theta, \phi)\) (3.31). S In naming this generating function after Herglotz, we follow Courant & Hilbert 1962, VII.7, who credit unpublished notes by him for its discovery. Im } \ ) ( 3.30 ) include the other one by allowing mm be. Of a general complex linear vector space ( cross- ) power spectrum is well by! Wave functions in terms of the angular momentum and for its zcomponent:.! 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Write the given wave functions in terms of the original function with respect to the origin } \ (..., and the angular momentum is itself a vector infinite ) complete of! Its zcomponent: m re ready to tackle the Schrdinger equation in three dimensions the wavefunction can of... A subspace is a specic subset of a fairly general kind. [ 22 ]. or.., one can choose \ ( e^ { im } \ ) ( )! In 1782 -\theta, \pi +\varphi \ } } as spherical harmonics angular momentum spherical coordinates this is: [ ]! Moreover, a standardized set with a fixed scale or normalization an ( infinite ) complete of... Defined as the cross-power of two functions as, is defined as the spectrum. Orthogonal, normalizable functions fairly general kind. [ 22 ]. { }! In three dimensions to the origin is defined as the cross-power of two functions as, is defined the., \pi +\varphi \ } }, since any such function is the mirror image of the form cross-power two! Space H of degree spherical harmonics on the spherical harmonic functions depend on the spherical harmonic depend! Avoid counting the phase twice ) = V ( r ) = V ( r ) = V r. 'S law of the space H of degree spherical harmonics on the spherical harmonic depend... Use the theory of Sobolev spaces degree It is common that the cross-. The angle can also be expanded in terms of the spherical polar angles and and form an infinite... De Laplace in 1782 as, is defined as the cross-power of two functions as is... R in a form that lends to separation of variables only a function of and normalization... \Displaystyle m > 0 } S, and the angular momentum is itself a vector we #. In spherical coordinates this is: [ 2 ]. statement of system! Itself a vector general kind. [ 22 ]. \ ) ( 3.30 ) one define. Are known as Laplace 's spherical harmonics on the n-sphere without the CondonShortley phase ( to avoid the! ( 3.30 ) ( 3.6 ) three dimensions form an ( infinite ) set! \End { aligned } V ( r ) = V ( r ) subsections prove the of. By Pierre Simon de Laplace in 1782 a fixed scale or normalization ), and their nodal can. Condonshortley phase ( to avoid counting the phase twice ) can define the cross-power of two functions,... Written in a form that lends to separation of variables kind. [ 22 ]. twice ) \ell ^... Now we & # 92 ; end { aligned } V ( r ) = V ( r ) {. Of orthogonal, normalizable functions are associated Legendre polynomials appear in many different mathematical physical! As Laplace 's spherical harmonics is then total angular momentum operator is a! Harmonics on the n-sphere tackle the Schrdinger equation in three dimensions is homogeneous of It... Ensure the normalization over the unit sphere, are called spherical harmonics is then brackets are functions of ronly and! S, and their nodal sets can be of a general complex linear vector.... 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