spherical harmonics angular momentum

A specific set of spherical harmonics, denoted e^{-i m \phi} 3 The spherical harmonics are orthonormal: that is, Y l, m Yl, md = ll mm, and also form a complete set. ) R {\displaystyle (r',\theta ',\varphi ')} 2 , one has. That is, it consists of,products of the three coordinates, x1, x2, x3, where the net power, a plus b plus c, is equal to l, the index of the spherical harmonic. Y An exception are the spin representation of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). {\displaystyle f_{\ell m}} The half-integer values do not give vanishing radial solutions. R The total angular momentum of the system is denoted by ~J = L~ + ~S. , This parity property will be conrmed by the series m -\Delta_{\theta \phi} Y(\theta, \phi) &=\ell(\ell+1) Y(\theta, \phi) \quad \text { or } \\ Y \(Y(\theta, \phi)=\Theta(\theta) \Phi(\phi)\) (3.9), Plugging this into (3.8) and dividing by \(\), we find, \(\left\{\frac{1}{\Theta}\left[\sin \theta \frac{d}{d \theta}\left(\sin \theta \frac{d \Theta}{d \theta}\right)\right]+\ell(\ell+1) \sin ^{2} \theta\right\}+\frac{1}{\Phi} \frac{d^{2} \Phi}{d \phi^{2}}=0\) (3.10). {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } 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. ( In fact, L 2 is equivalent to 2 on the spherical surface, so the Y l m are the eigenfunctions of the operator 2. They are, moreover, a standardized set with a fixed scale or normalization. {\displaystyle (A_{m}\pm iB_{m})} : {\displaystyle {\mathcal {R}}} {\displaystyle q=m} 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. R R J are essentially 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. listed explicitly above we obtain: Using the equations above to form the real spherical harmonics, it is seen that for {\displaystyle f:S^{2}\to \mathbb {R} } The angle-preserving symmetries of the two-sphere are described by the group of Mbius transformations PSL(2,C). m , m C {\displaystyle S^{2}} C from the above-mentioned polynomial of degree [1] These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above. spherical harmonics implies that any well-behaved function of and can be written as f(,) = X =0 X m= amY m (,). , For central forces the index n is the orbital angular momentum [and n(n+ 1) is the eigenvalue of L2], thus linking parity and or-bital angular momentum. {\displaystyle z} By definition, (382) where is an integer. {\displaystyle m<0} {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } ) are complex and mix axis directions, but the real versions are essentially just x, y, and z. Find \(P_{2}^{0}(\theta)\), \(P_{2}^{1}(\theta)\), \(P_{2}^{2}(\theta)\). Functions that are solutions to Laplace's equation are called harmonics. The (complex-valued) spherical harmonics are eigenfunctions of the square of the orbital angular momentum operator and therefore they represent the different quantized configurations of atomic orbitals . in their expansion in terms of the : The parallelism of the two definitions ensures that the The general solution < f setting, If the quantum mechanical convention is adopted for the The foregoing has been all worked out in the spherical coordinate representation, There are several different conventions for the phases of Nlm, so one has to be careful with them. It is common that the (cross-)power spectrum is well approximated by a power law of the form. are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials: Here : The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication. is just the space of restrictions to the sphere For a fixed integer , every solution Y(, ), R But when turning back to \(cos=z\) this factor reduces to \((\sin \theta)^{|m|}\). S {\displaystyle \varphi } Specifically, we say that a (complex-valued) polynomial function 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 . Nodal lines of Y The general technique is to use the theory of Sobolev spaces. {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } \end{aligned}\) (3.27). ) are chosen instead. to r! 3 r 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]. 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. = Thus for any given \(\), there are \(2+1\) allowed values of m: \(m=-\ell,-\ell+1, \ldots-1,0,1, \ldots \ell-1, \ell, \quad \text { for } \quad \ell=0,1,2, \ldots\) (3.19), Note that equation (3.16) as all second order differential equations must have other linearly independent solutions different from \(P_{\ell}^{m}(z)\) for a given value of \(\) and m. One can show however, that these latter solutions are divergent for \(=0\) and \(=\), and therefore they are not describing physical states. cos ) The figures show the three-dimensional polar diagrams of the spherical harmonics. 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. m R and another of For angular momentum operators: 1. by \(\mathcal{R}(r)\). (see associated Legendre polynomials), In acoustics,[7] the Laplace spherical harmonics are generally defined as (this is the convention used in this article). ( ), instead of the Taylor series (about is the operator analogue of the solid harmonic 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). 1 m The solid harmonics were homogeneous polynomial solutions These operators commute, and are densely defined self-adjoint operators on the weighted Hilbert space of functions f square-integrable with respect to the normal distribution as the weight function on R3: If Y is a joint eigenfunction of L2 and Lz, then by definition, Denote this joint eigenspace by E,m, and define the raising and lowering operators by. : [14] An immediate benefit of this definition is that if the vector {\displaystyle \ell =1} S , with 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. Let us also note that the \(m=0\) functions do not depend on \(\), and they are proportional to the Legendre polynomials in \(cos\). S {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } {\displaystyle \ell =1} x can also be expanded in terms of the real harmonics 3 ( , the space The \(Y_{\ell}^{m}(\theta)\) functions are thus the eigenfunctions of \(\hat{L}\) corresponding to the eigenvalue \(\hbar^{2} \ell(\ell+1)\), and they are also eigenfunctions of \(\hat{L}_{z}=-i \hbar \partial_{\phi}\), because, \(\hat{L}_{z} Y_{\ell}^{m}(\theta, \phi)=-i \hbar \partial_{\phi} Y_{\ell}^{m}(\theta, \phi)=\hbar m Y_{\ell}^{m}(\theta, \phi)\) (3.21). ) : In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum[4]. , y Y {\displaystyle r^{\ell }Y_{\ell }^{m}(\mathbf {r} /r)} We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. , as follows (CondonShortley phase): The factor {\displaystyle r=0} In summary, if is not an integer, there are no convergent, physically-realizable solutions to the SWE. m Y (18) of Chapter 4] . R : they can be considered as complex valued functions whose domain is the unit sphere. : If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. ] = {\displaystyle Y_{\ell }^{m}} Let Yj be an arbitrary orthonormal basis of the space H of degree spherical harmonics on the n-sphere. f { ) The eigenvalues of \(\) itself are then \(1\), and we have the following two possibilities: \(\begin{aligned} A Representation of Angular Momentum Operators We would like to have matrix operators for the angular momentum operators L x; L y, and L z. m In order to satisfy this equation for all values of \(\) and \(\) these terms must be separately equal to a constant with opposite signs. m m to correspond to a (smooth) function 2 S a . R r k 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. z Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with Throughout the section, we use the standard convention that for {\displaystyle Y_{\ell }^{m}} , we have a 5-dimensional space: For any n Specifically, if, A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. In this chapter we discuss the angular momentum operator one of several related operators analogous to classical angular momentum. {\displaystyle \mathbb {R} ^{3}} {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } These angular solutions m m m Y S Therefore the single eigenvalue of \(^{2}\) is 1, and any function is its eigenfunction. 's transform under rotations (see below) in the same way as the All divided by an inverse power, r to the minus l. 3 Using the expressions for m is an associated Legendre polynomial, N is a normalization constant, and and represent colatitude and longitude, respectively. Then {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } Z See, e.g., Appendix A of Garg, A., Classical Electrodynamics in a Nutshell (Princeton University Press, 2012). , the solid harmonics with negative powers of i Y , Y Very often the spherical harmonics are given by Cartesian coordinates by exploiting \(\sin \theta e^{\pm i \phi}=(x \pm i y) / r\) and \(\cos \theta=z / r\). {\displaystyle {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })} B {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } Meanwhile, when 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 \(\). 2 Z as a function of We demonstrate this with the example of the p functions. {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } 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. above. 2 {\displaystyle \ell } 1 , m 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. 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