: m 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. ) 2 0 about the origin that sends the unit vector m the expansion coefficients , which can be seen to be consistent with the output of the equations above. Throughout the section, we use the standard convention that for between them is given by the relation, where P is the Legendre polynomial of degree . S and order , r . {\displaystyle z} In a similar manner, one can define the cross-power of two functions as, is defined as the cross-power spectrum. : {\displaystyle {\mathcal {R}}} \end{aligned}\) (3.27). The total power of a function f is defined in the signal processing literature as the integral of the function squared, divided by the area of its domain. , one has. There are of course functions which are neither even nor odd, they do not belong to the set of eigenfunctions of \(\). This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. The solutions, \(Y_{\ell}^{m}(\theta, \phi)=\mathcal{N}_{l m} P_{\ell}^{m}(\theta) e^{i m \phi}\) (3.20). 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. {\displaystyle \gamma } It follows from Equations ( 371) and ( 378) that. This could be achieved by expansion of functions in series of trigonometric functions. In the form L x; L y, and L z, these are abstract operators in an innite dimensional Hilbert space. . R But when turning back to \(cos=z\) this factor reduces to \((\sin \theta)^{|m|}\). ( You are all familiar, at some level, with spherical harmonics, from angular momentum in quantum mechanics. and x R The complex spherical harmonics 0 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.) For ) Y {\displaystyle \psi _{i_{1}\dots i_{\ell }}} ) ) \(Y_{\ell}^{0}(\theta)=\sqrt{\frac{2 \ell+1}{4 \pi}} P_{\ell}(\cos \theta)\) (3.28). \end{array}\right.\) (3.12), and any linear combinations of them. We have to write the given wave functions in terms of the spherical harmonics. {\displaystyle \mathbf {r} '} m Z In order to satisfy this equation for all values of \(\) and \(\) these terms must be separately equal to a constant with opposite signs. {\displaystyle P_{\ell }^{m}(\cos \theta )} {\displaystyle c\in \mathbb {C} } That is, they are either even or odd with respect to inversion about the origin. in their expansion in terms of the 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 P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )} {\displaystyle \Delta f=0} The spherical harmonic functions depend on the spherical polar angles and and form an (infinite) complete set of orthogonal, normalizable functions. If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. The general, normalized Spherical Harmonic is depicted below: Y_ {l}^ {m} (\theta,\phi) = \sqrt { \dfrac { (2l + 1) (l - |m|)!} 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. Y &\Pi_{\psi_{-}}(\mathbf{r})=\quad \psi_{-}(-\mathbf{r})=-\psi_{-}(\mathbf{r}) = {\displaystyle \ell } ) r 2 The spherical harmonics Y m ( , ) are also the eigenstates of the total angular momentum operator L 2. Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with , with m terms (sines) are included: The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation. by setting, The real spherical harmonics 2 2 ) . l , we have a 5-dimensional space: For any f C m One might wonder what is the reason for writing the eigenvalue in the form \((+1)\), but as it will turn out soon, there is no loss of generality in this notation. m Y One can determine the number of nodal lines of each type by counting the number of zeros of : C C Meanwhile, when The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. f {\displaystyle \mathbf {r} } A ) 2 There are two quantum numbers for the rigid rotor in 3D: \(J\) is the total angular momentum quantum number and \(m_J\) is the z-component of the angular momentum. Angular momentum is the generator for rotations, so spherical harmonics provide a natural characterization of the rotational properties and direction dependence of a system. Another is complementary hemispherical harmonics (CHSH). ] ) used above, to match the terms and find series expansion coefficients {\displaystyle r>R} 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 \(_{}\). 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 . Abstract. They occur in . Rotations and Angular momentum Intro The material here may be found in Sakurai Chap 3: 1-3, (5-6), 7, (9-10) . Y f {\displaystyle \ell =4} The spherical harmonics with negative can be easily compute from those with positive . ( In both classical and quantum mechanical systems, angular momentum (together with linear momentum and energy) is one of the three fundamental properties of motion. only the ( {\displaystyle \ell } The set of all direction kets n` can be visualized . 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). 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 . x m R S {\displaystyle Z_{\mathbf {x} }^{(\ell )}({\mathbf {y} })} The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } 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 ( [28][29][30][31], "Ylm" redirects here. m { { r {\displaystyle \mathbb {R} ^{n}\to \mathbb {C} } 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. specified by these angles. Inversion is represented by the operator Y When < 0, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. C f is called a spherical harmonic function of degree and order m, m if. [23] Let P denote the space of complex-valued homogeneous polynomials of degree in n real variables, here considered as functions 2 {\displaystyle P_{i}:[-1,1]\to \mathbb {R} } m e^{i m \phi} \\ v {\displaystyle \ell } In other words, any well-behaved function of and can be represented as a superposition of spherical harmonics. A Recalling that the spherical harmonics are eigenfunctions of the angular momentum operator: (r; ;) = R(r)Ym l ( ;) SeparationofVariables L^2Ym l ( ;) = h2l . n ), 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. Using the expressions for Z R form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions is an associated Legendre polynomial, N is a normalization constant, and and represent colatitude and longitude, respectively. The spherical harmonics have definite parity. m , 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. From this perspective, one has the following generalization to higher dimensions. 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. R Abstractly, the ClebschGordan coefficients express the tensor product of two irreducible representations of the rotation group as a sum of irreducible representations: suitably normalized, the coefficients are then the multiplicities. . : r! {\displaystyle Y_{\ell }^{m}} With the definition of the position and the momentum operators we obtain the angular momentum operator as, \(\hat{\mathbf{L}}=-i \hbar(\mathbf{r} \times \nabla)\) (3.2), The Cartesian components of \(\hat{\mathbf{L}}\) are then, \(\hat{L}_{x}=-i \hbar\left(y \partial_{z}-z \partial_{y}\right), \quad \hat{L}_{y}=-i \hbar\left(z \partial_{x}-x \partial_{z}\right), \quad \hat{L}_{z}=-i \hbar\left(x \partial_{y}-y \partial_{x}\right)\) (3.3), One frequently needs the components of \(\hat{\mathbf{L}}\) in spherical coordinates. ( 3 The spherical harmonics are the eigenfunctions of the square of the quantum mechanical angular momentum operator. Y C (see associated Legendre polynomials), In acoustics,[7] the Laplace spherical harmonics are generally defined as (this is the convention used in this article). Another way of using these functions is to create linear combinations of functions with opposite m-s. R For angular momentum operators: 1. \(\hat{L}^{2}=-\hbar^{2}\left(\partial_{\theta \theta}^{2}+\cot \theta \partial_{\theta}+\frac{1}{\sin ^{2} \theta} \partial_{\phi \phi}^{2}\right)=-\hbar^{2} \Delta_{\theta \phi}\) (3.7). z Just prior to that time, Adrien-Marie Legendre had investigated the expansion of the Newtonian potential in powers of r = |x| and r1 = |x1|. Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x p~. m The angular components of . As is known from the analytic solutions for the hydrogen atom, the eigenfunctions of the angular part of the wave function are spherical harmonics. x and modelling of 3D shapes. The functions is just the space of restrictions to the sphere m 2 {\displaystyle \theta } For example, for any &\hat{L}_{x}=i \hbar\left(\sin \phi \partial_{\theta}+\cot \theta \cos \phi \partial_{\phi}\right) \\ = ( are sometimes known as tesseral spherical harmonics. transforms into a linear combination of spherical harmonics of the same degree. Notice, however, that spherical harmonics are not functions on the sphere which are harmonic with respect to the Laplace-Beltrami operator for the standard round metric on the sphere: the only harmonic functions in this sense on the sphere are the constants, since harmonic functions satisfy the Maximum principle. {\displaystyle S^{2}} S It can be shown that all of the above normalized spherical harmonic functions satisfy. S inside three-dimensional Euclidean space ] ( > p Under this operation, a spherical harmonic of degree The angle-preserving symmetries of the two-sphere are described by the group of Mbius transformations PSL(2,C). In this chapter we discuss the angular momentum operator one of several related operators analogous to classical angular momentum. The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } Let A denote the subspace of P consisting of all harmonic polynomials: An orthogonal basis of spherical harmonics in higher dimensions can be constructed inductively by the method of separation of variables, by solving the Sturm-Liouville problem for the spherical Laplacian, The space H of spherical harmonics of degree is a representation of the symmetry group of rotations around a point (SO(3)) and its double-cover SU(2). is homogeneous of degree The general technique is to use the theory of Sobolev spaces. {\displaystyle Y_{\ell }^{m}} , R : {\displaystyle Z_{\mathbf {x} }^{(\ell )}} The spherical harmonics play an important role in quantum mechanics. {\displaystyle (r',\theta ',\varphi ')} can be defined in terms of their complex analogues ( L 2 Y 21 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. are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials: Here : S {\displaystyle {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })} 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. the formula, Several different normalizations are in common use for the Laplace spherical harmonic functions 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. 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