The spherical harmonics are orthonormal: that is, Y l, m Yl, md = ll mm, and also form a complete set. However, the solutions of the non-relativistic Schrdinger equation without magnetic terms can be made real. Y Essentially all the properties of the spherical harmonics can be derived from this generating function. &p_{x}=\frac{x}{r}=\frac{\left(Y_{1}^{-1}-Y_{1}^{1}\right)}{\sqrt{2}}=\sqrt{\frac{3}{4 \pi}} \sin \theta \cos \phi \\ R { m 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. For example, for any 's, which in turn guarantees that they are spherical tensor operators, ) Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle between x1 and x. = {\displaystyle S^{2}} of spherical harmonics of degree Concluding the subsection let us note the following important fact. It follows from Equations ( 371) and ( 378) that. = R With \(\cos \theta=z\) the solution is, \(P_{\ell}^{m}(z):=\left(1-z^{2}\right)^{|m| 2}\left(\frac{d}{d z}\right)^{|m|} P_{\ell}(z)\) (3.17). ) 2 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 . 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. 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. S {\displaystyle q=m} 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. {\displaystyle \langle \theta ,\varphi |lm\rangle =Y_{l}^{m}(\theta ,\varphi )} are composed of circles: there are |m| circles along longitudes and |m| circles along latitudes. Recalling that the spherical harmonics are eigenfunctions of the angular momentum operator: (r; ;) = R(r)Ym l ( ;) SeparationofVariables L^2Ym l ( ;) = h2l . 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 By polarization of A, there are coefficients , and Y This is well known in quantum mechanics, since [ L 2, L z] = 0, the good quantum numbers are and m. Would it be possible to find another solution analogous to the spherical harmonics Y m ( , ) such that [ L 2, L x or y] = 0? m Y S ) S S For example, when m Laplace equation. Prove that \(P_{}(z)\) are solutions of (3.16) for \(m=0\). n R 2 There are of course functions which are neither even nor odd, they do not belong to the set of eigenfunctions of \(\). The tensor spherical harmonics 1 The Clebsch-Gordon coecients Consider a system with orbital angular momentum L~ and spin angular momentum ~S. 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. [17] The result can be proven analytically, using the properties of the Poisson kernel in the unit ball, or geometrically by applying a rotation to the vector y so that it points along the z-axis, and then directly calculating the right-hand side. a The result of acting by the parity on a function is the mirror image of the original function with respect to the origin. ) Y where the absolute values of the constants \(\mathcal{N}_{l m}\) ensure the normalization over the unit sphere, are called spherical harmonics. (the irregular solid harmonics symmetric on the indices, uniquely determined by the requirement. On the unit sphere C 2 Y , respectively, the angle ) are complex and mix axis directions, but the real versions are essentially just x, y, and z. The spherical harmonics with negative can be easily compute from those with positive . Spherical harmonics are ubiquitous in atomic and molecular physics. The animation shows the time dependence of the stationary state i.e. = , of the eigenvalue problem. and Y C R 1 ) 3 Rotations and Angular momentum Intro The material here may be found in Sakurai Chap 3: 1-3, (5-6), 7, (9-10) . 2 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. 2 : S m in the , the degree zonal harmonic corresponding to the unit vector x, decomposes as[20]. [ } ( Z S ( ) &p_{z}=\frac{z}{r}=Y_{1}^{0}=\sqrt{\frac{3}{4 \pi}} \cos \theta Just prior to that time, Adrien-Marie Legendre had investigated the expansion of the Newtonian potential in powers of r = |x| and r1 = |x1|. [ The 3-D wave equation; spherical harmonics. , which can be seen to be consistent with the output of the equations above. {\displaystyle \ell =4} S C {\displaystyle \mathbf {r} } Clebsch Gordon coecients allow us to express the total angular momentum basis |jm; si in terms of the direct product ( Y are associated Legendre polynomials without the CondonShortley phase (to avoid counting the phase twice). ( setting, If the quantum mechanical convention is adopted for the , the solid harmonics with negative powers of The Laplace spherical harmonics Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. l Such spherical harmonics are a special case of zonal spherical functions. : 3 T 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. to all of , R They are eigenfunctions of the operator of orbital angular momentum and describe the angular distribution of particles which move in a spherically-symmetric field with the orbital angular momentum l and projection m. ) {\displaystyle \{\pi -\theta ,\pi +\varphi \}} Angular momentum is not a property of a wavefunction at a point; it is a property of a wavefunction as a whole. By definition, (382) where is an integer. 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. 2 2 When = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. Y x Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x p~. 2 {\displaystyle (2\ell +1)} k The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions The angle-preserving symmetries of the two-sphere are described by the group of Mbius transformations PSL(2,C). . Spherical Harmonics, and Bessel Functions Physics 212 2010, Electricity and Magnetism Michael Dine Department of Physics . {\displaystyle Y_{\ell }^{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). When = 0, the spectrum is "white" as each degree possesses equal power. 2 ( Y Y 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. ) p {\displaystyle S^{2}\to \mathbb {C} } {\displaystyle e^{\pm im\varphi }} , are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials: Here 1 p , so the magnitude of the angular momentum is L=rp . : {\displaystyle S^{n-1}\to \mathbb {C} } is an associated Legendre polynomial, N is a normalization constant, and and represent colatitude and longitude, respectively. p. The cross-product picks out the ! : .) An exception are the spin representation of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). \(Y(\theta, \phi)=\Theta(\theta) \Phi(\phi)\) (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). Spherical harmonics are important in many theoretical and practical applications, including the representation of multipole electrostatic and electromagnetic fields, electron configurations, gravitational fields, geoids, the magnetic fields of planetary bodies and stars, and the cosmic microwave background radiation. , as follows (CondonShortley phase): The factor Then R 3 2 m 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 ! 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. The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation. (18) of Chapter 4] . e Specifically, if, A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. v B , then, a Y One can determine the number of nodal lines of each type by counting the number of zeros of {\displaystyle A_{m}(x,y)} {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } 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). (See Applications of Legendre polynomials in physics for a more detailed analysis. 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