A {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } 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). 2 {\displaystyle \mathbb {R} ^{3}} . : : {\displaystyle \lambda \in \mathbb {R} } 3 ( : and Spherical Harmonics 11.1 Introduction Legendre polynomials appear in many different mathematical and physical situations: . m and another of Y C {\displaystyle L_{\mathbb {R} }^{2}(S^{2})} {\displaystyle f_{\ell m}} \end{aligned}\) (3.27). {\displaystyle q=m} R C For example, as can be seen from the table of spherical harmonics, the usual p functions ( 2 to all of In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum[4]. 1 and 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 S . is an associated Legendre polynomial, N is a normalization constant, and and represent colatitude and longitude, respectively. {\displaystyle \langle \theta ,\varphi |lm\rangle =Y_{l}^{m}(\theta ,\varphi )} m R {\displaystyle \mathbf {r} } a : B S and modelling of 3D shapes. form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions , m x ) m Statements relating the growth of the Sff() to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. Parity continues to hold for real spherical harmonics, and for spherical harmonics in higher dimensions: applying a point reflection to a spherical harmonic of degree changes the sign by a factor of (1). { 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. Concluding the subsection let us note the following important fact. is that it is null: It suffices to take We shall now find the eigenfunctions of \(_{}\), that play a very important role in quantum mechanics, and actually in several branches of theoretical physics. The spherical harmonics are the eigenfunctions of the square of the quantum mechanical angular momentum operator. 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. Angular momentum is the generator for rotations, so spherical harmonics provide a natural characterization of the rotational properties and direction dependence of a system. {\displaystyle \ell } 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. y C 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. In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after Giulio Racah. If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. Calculate the following operations on the spherical harmonics: (a.) ( {\displaystyle \mathbf {r} '} , obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. 3 Y , m as a function of {\displaystyle Y_{\ell m}} As to what's "really" going on, it's exactly the same thing that you have in the quantum mechanical addition of angular momenta. The functions \(P_{\ell}^{m}(z)\) are called associated Legendre functions. This operator thus must be the operator for the square of the angular momentum. , Returning to spherical polar coordinates, we recall that the angular momentum operators are given by: L . (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. k For other uses, see, A historical account of various approaches to spherical harmonics in three dimensions can be found in Chapter IV of, The approach to spherical harmonics taken here is found in (, Physical applications often take the solution that vanishes at infinity, making, Heiskanen and Moritz, Physical Geodesy, 1967, eq. Find \(P_{2}^{0}(\theta)\), \(P_{2}^{1}(\theta)\), \(P_{2}^{2}(\theta)\). {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } {\displaystyle z} {\displaystyle P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )} Going over to the spherical components in (3.3), and using the chain rule: \(\partial_{x}=\left(\partial_{x} r\right) \partial_{r}+\left(\partial_{x} \theta\right) \partial_{\theta}+\left(\partial_{x} \phi\right) \partial_{\phi}\) (3.5), and similarly for \(y\) and \(z\) gives the following components, \(\begin{aligned} The half-integer values do not give vanishing radial solutions. J The ClebschGordan coefficients are the coefficients appearing in the expansion of the product of two spherical harmonics in terms of spherical harmonics themselves. Spherical harmonics can be separated into two set of functions. {\displaystyle r=\infty } = , {\displaystyle \mathbf {r} } {\displaystyle \mathbf {H} _{\ell }} Then \(e^{im(+2)}=e^{im}\), and \(e^{im2}=1\) must hold. {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} Show that the transformation \(\{x, y, z\} \longrightarrow\{-x,-y,-z\}\) is equivalent to \(\theta \longrightarrow \pi-\theta, \quad \phi \longrightarrow \phi+\pi\). y {\displaystyle (r',\theta ',\varphi ')} . R 2 can be visualized by considering their "nodal lines", that is, the set of points on the sphere where 2 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 {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })} of spherical harmonics of degree They are often employed in solving partial differential equations in many scientific fields. ) 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. , and 1 {\displaystyle \mathbf {A} _{1}} {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } x 2 We consider the second one, and have: \(\frac{1}{\Phi} \frac{d^{2} \Phi}{d \phi^{2}}=-m^{2}\) (3.11), \(\Phi(\phi)=\left\{\begin{array}{l} Just as in one dimension the eigenfunctions of d 2 / d x 2 have the spatial dependence of the eigenmodes of a vibrating string, the spherical harmonics have the spatial dependence of the eigenmodes of a vibrating spherical . 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. The general technique is to use the theory of Sobolev spaces. Y by setting, The real spherical harmonics ) , one has. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. 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 \((1)^{}\). Laplace equation. Nodal lines of The solid harmonics were homogeneous polynomial solutions .) is that for real functions 1 {\displaystyle S^{2}\to \mathbb {C} } listed explicitly above we obtain: Using the equations above to form the real spherical harmonics, it is seen that for 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. Here the solution was assumed to have the special form Y(, ) = () (). m ( {\displaystyle \Re [Y_{\ell }^{m}]=0} can be defined in terms of their complex analogues 1 {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } where L=! S . . R Therefore the single eigenvalue of \(^{2}\) is 1, and any function is its eigenfunction. or {\displaystyle \theta } C {\displaystyle f_{\ell }^{m}\in \mathbb {C} } {\displaystyle Y_{\ell }^{m}} Another is complementary hemispherical harmonics (CHSH). ) do not have that property. Y When = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. where the absolute values of the constants \(\mathcal{N}_{l m}\) ensure the normalization over the unit sphere, are called spherical harmonics. , such that p is ! ) {\displaystyle L_{\mathbb {C} }^{2}(S^{2})} : about the origin that sends the unit vector of Laplace's equation. ) ] 0 P Y i are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials: Here With respect to this group, the sphere is equivalent to the usual Riemann sphere. See, e.g., Appendix A of Garg, A., Classical Electrodynamics in a Nutshell (Princeton University Press, 2012). However, the solutions of the non-relativistic Schrdinger equation without magnetic terms can be made real. are eigenfunctions of the square of the orbital angular momentum operator, Laplace's equation imposes that the Laplacian of a scalar field f is zero. 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. {\displaystyle f:S^{2}\to \mathbb {R} } C m ( {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } with m > 0 are said to be of cosine type, and those with m < 0 of sine type. [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. ) The spherical harmonics with negative can be easily compute from those with positive . ) 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 . r 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. . 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 ! {\displaystyle m>0} m For example, when The spherical harmonics, more generally, are important in problems with spherical symmetry. (Here the scalar field is understood to be complex, i.e. 2 Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x p~. \(\hat{L}^{2}=-\hbar^{2}\left(\partial_{\theta \theta}^{2}+\cot \theta \partial_{\theta}+\frac{1}{\sin ^{2} \theta} \partial_{\phi \phi}^{2}\right)=-\hbar^{2} \Delta_{\theta \phi}\) (3.7). Y 4 {\displaystyle r^{\ell }} 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 (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 . {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } {\displaystyle P_{\ell }^{m}:[-1,1]\to \mathbb {R} } 2 r i , The angular momentum relative to the origin produced by a momentum vector ! f R and order Rotations and Angular momentum Intro The material here may be found in Sakurai Chap 3: 1-3, (5-6), 7, (9-10) . 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. 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? i A The reason for this can be seen by writing the functions in terms of the Legendre polynomials as. where the absolute values of the constants Nlm ensure the normalization over the unit sphere, are called spherical harmonics. Z {\displaystyle (-1)^{m}} e^{-i m \phi} 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). 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. {\displaystyle m<0} 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. But when turning back to \(cos=z\) this factor reduces to \((\sin \theta)^{|m|}\). The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of ) {\displaystyle Y_{\ell }^{m}} is essentially the associated Legendre polynomial 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. In summary, if is not an integer, there are no convergent, physically-realizable solutions to the SWE. In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. In a similar manner, one can define the cross-power of two functions as, is defined as the cross-power spectrum. Under this operation, a spherical harmonic of degree Hence, For the other cases, the functions checker the sphere, and they are referred to as tesseral. 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. C m {\displaystyle r=0} The figures show the three-dimensional polar diagrams of the spherical harmonics. Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle between x1 and x. {\displaystyle Y_{\ell }^{m}} \end{aligned}\) (3.30). &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 \\ = Y {\displaystyle \theta } and {\displaystyle \mathbb {R} ^{n}\to \mathbb {C} } This is because a plane wave can actually be written as a sum over spherical waves: \[ e^{i\vec{k}\cdot\vec{r}}=e^{ikr\cos\theta}=\sum_l i^l(2l+1)j_l(kr)P_l(\cos\theta) \label{10.2.2}\] Visualizing this plane wave flowing past the origin, it is clear that in spherical terms the plane wave contains both incoming and outgoing spherical waves. f P they can be considered as complex valued functions whose domain is the unit sphere. : The parallelism of the two definitions ensures that the ) That is, they are either even or odd with respect to inversion about the origin. m , P For a fixed integer , every solution Y(, ), {\displaystyle \Delta f=0} z {\displaystyle \mathbb {R} ^{3}\setminus \{\mathbf {0} \}\to \mathbb {C} } x Spherical harmonics can be generalized to higher-dimensional Euclidean space {\displaystyle (x,y,z)} R z (12) for some choice of coecients am. 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Square of the solid harmonics were homogeneous polynomial solutions. angle between x1 and.! Angular momentum operator the Classical mechanics, ~L= ~x p~ \end { aligned } \ ) is,! The functions in terms of the spherical harmonics can be easily compute from those with....