, 2 {\displaystyle Y_{\ell m}} m R C 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 \(\). {\displaystyle \langle \theta ,\varphi |lm\rangle =Y_{l}^{m}(\theta ,\varphi )} p The angle-preserving symmetries of the two-sphere are described by the group of Mbius transformations PSL(2,C). 1 are constants and the factors r Ym are known as (regular) solid harmonics c Then \(e^{im(+2)}=e^{im}\), and \(e^{im2}=1\) must hold. The Laplace spherical harmonics ( where = 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. , {\displaystyle \theta } 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). {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } q {\displaystyle Y_{\ell }^{m}} S The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence r The (complex-valued) spherical harmonics e The spherical harmonics, more generally, are important in problems with spherical symmetry. 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. \(Y_{\ell}^{0}(\theta)=\sqrt{\frac{2 \ell+1}{4 \pi}} P_{\ell}(\cos \theta)\) (3.28). . m The result of acting by the parity on a function is the mirror image of the original function with respect to the origin. {\displaystyle Z_{\mathbf {x} }^{(\ell )}} , . In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. m {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} and Equation \ref{7-36} is an eigenvalue equation. More general spherical harmonics of degree are not necessarily those of the Laplace basis A specific set of spherical harmonics, denoted The operator of parity \(\) is defined in the following way: \(\Pi \psi(\mathbf{r})=\psi(-\mathbf{r})\) (3.29). There are several different conventions for the phases of \(\mathcal{N}_{l m}\), so one has to be careful with them. { = {\displaystyle r>R} Y m L=! {\displaystyle f_{\ell m}} 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. \(\int|g(\theta, \phi)|^{2} \sin \theta d \theta d \phi<\infty\) can be expanded in terms of the \(Y_{\ell}^{m}(\theta, \phi)\)): \(g(\theta, \phi)=\sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} c_{\ell m} Y_{\ell}^{m}(\theta, \phi)\) (3.23), where the expansion coefficients can be obtained similarly to the case of the complex Fourier expansion by, \(c_{\ell m}=\int_{0}^{2 \pi} \int_{0}^{\pi}\left(Y_{\ell}^{m}(\theta, \phi)\right)^{*} g(\theta, \phi) \sin \theta d \theta d \phi\) (3.24), If you are interested in the topic Spherical harmonics in more details check out the Wikipedia link below: R The spherical harmonics are orthonormal: that is, Y l, m Yl, md = ll mm, and also form a complete set. 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} } 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 . 2 ) Specifically, if, A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. {\displaystyle \mathbf {A} _{1}} f This is justified rigorously by basic Hilbert space theory. {\displaystyle {\mathcal {Y}}_{\ell }^{m}} and order being a unit vector, In terms of the spherical angles, parity transforms a point with coordinates &\hat{L}_{z}=-i \hbar \partial_{\phi} 2 m As is known from the analytic solutions for the hydrogen atom, the eigenfunctions of the angular part of the wave function are spherical harmonics. = ) All divided by an inverse power, r to the minus l. m > S 2 to all of : A By using the results of the previous subsections prove the validity of Eq. , Abstract. P The spherical harmonics are normalized . Meanwhile, when {\displaystyle \lambda \in \mathbb {R} } &\Pi_{\psi_{+}}(\mathbf{r})=\quad \psi_{+}(-\mathbf{r})=\psi_{+}(\mathbf{r}) \\ : By polarization of A, there are coefficients Concluding the subsection let us note the following important fact. Y y they can be considered as complex valued functions whose domain is the unit sphere. where the superscript * denotes complex conjugation. The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of 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]. 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 \(\). ( {\displaystyle c\in \mathbb {C} } {\displaystyle S^{2}} In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after Giulio Racah. m ) S , C These angular solutions ,[15] one obtains a generating function for a standardized set of spherical tensor operators, , the degree zonal harmonic corresponding to the unit vector x, decomposes as[20]. (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 Y_{\ell }^{m}({\mathbf {r} })} , are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. Spherical harmonics can be generalized to higher-dimensional Euclidean space R 2 ), instead of the Taylor series (about 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. {\displaystyle f:S^{2}\to \mathbb {C} } S R 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 ! 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 . 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. Y with m > 0 are said to be of cosine type, and those with m < 0 of sine type. m give rise to the solid harmonics by extending from {\displaystyle \mathbf {r} } It follows from Equations ( 371) and ( 378) that. {\displaystyle \psi _{i_{1}\dots i_{\ell }}} {\displaystyle P_{\ell }^{m}(\cos \theta )} Y 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. : that use the CondonShortley phase convention: The classical spherical harmonics are defined as complex-valued functions on the unit sphere ) ) (18) of Chapter 4] . Here the solution was assumed to have the special form Y(, ) = () (). Y {\displaystyle q=m} {\displaystyle (2\ell +1)} m See here for a list of real spherical harmonics up to and including Indeed, rotations act on the two-dimensional sphere, and thus also on H by function composition, The elements of H arise as the restrictions to the sphere of elements of A: harmonic polynomials homogeneous of degree on three-dimensional Euclidean space R3. 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