( 2 {\displaystyle \ell } r [12], A real basis of spherical harmonics Since they are eigenfunctions of Hermitian operators, they are orthogonal . is that it is null: It suffices to take of Laplace's equation. x ) {\displaystyle \ell } The animation shows the time dependence of the stationary state i.e. Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner D-matrix. Now, it is easily demonstrated that if A and B are two general operators then (7.1.3) [ A 2, B] = A [ A, B] + [ A, B] A. 2 A A R \left(\partial_{\theta \theta}^{2}+\cot \theta \partial_{\theta}+\frac{1}{\sin ^{2} \theta} \partial_{\phi \phi}^{2}\right) Y(\theta, \phi) &=-\ell(\ell+1) Y(\theta, \phi) {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } One can choose \(e^{im}\), and include the other one by allowing mm to be negative. {\displaystyle Y_{\ell m}} In that case, one needs to expand the solution of known regions in Laurent series (about 3 : 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. Now we're ready to tackle the Schrdinger equation in three dimensions. i 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. ( are constants and the factors r Ym are known as (regular) solid harmonics 3 The function \(P_{\ell}^{m}(z)\) is a polynomial in z only if \(|m|\) is even, otherwise it contains a term \(\left(1-z^{2}\right)^{|m| / 2}\) which is a square root. [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. = Throughout the section, we use the standard convention that for The absolute value of the function in the direction given by \(\) and \(\) is equal to the distance of the point from the origin, and the argument of the complex number is obtained by the colours of the surface according to the phase code of the complex number in the chosen direction. Spherical Harmonics, and Bessel Functions Physics 212 2010, Electricity and Magnetism Michael Dine Department of Physics . {\displaystyle S^{2}\to \mathbb {C} } ; the remaining factor can be regarded as a function of the spherical angular coordinates R a (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. ) (see associated Legendre polynomials), In acoustics,[7] the Laplace spherical harmonics are generally defined as (this is the convention used in this article). These angular solutions Rotations and Angular momentum Intro The material here may be found in Sakurai Chap 3: 1-3, (5-6), 7, (9-10) . : they can be considered as complex valued functions whose domain is the unit sphere. 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. Y k , This could be achieved by expansion of functions in series of trigonometric functions. B f y Another way of using these functions is to create linear combinations of functions with opposite m-s. However, the solutions of the non-relativistic Schrdinger equation without magnetic terms can be made real. [ spherical harmonics implies that any well-behaved function of and can be written as f(,) = X =0 X m= amY m (,). m C = Legal. Any function of and can be expanded in the spherical harmonics . . x In the quantum mechanics community, it is common practice to either include this phase factor in the definition of the associated Legendre polynomials, or to append it to the definition of the spherical harmonic functions. {\displaystyle p:\mathbb {R} ^{3}\to \mathbb {C} } : Y and The spherical harmonics are orthogonal functions, and are properly normalized with respect to integration over the entire solid angle: (381) The spherical harmonics also form a complete set for representing general functions of and . {\displaystyle {\mathcal {Y}}_{\ell }^{m}} r The general, normalized Spherical Harmonic is depicted below: Y_ {l}^ {m} (\theta,\phi) = \sqrt { \dfrac { (2l + 1) (l - |m|)!} Y The Laplace spherical harmonics S {\displaystyle v} above as a sum. This system is also a complete one, which means that any complex valued function \(g(,)\) that is square integrable on the unit sphere, i.e. &\hat{L}_{z}=-i \hbar \partial_{\phi} {\displaystyle A_{m}(x,y)} The condition on the order of growth of Sff() is related to the order of differentiability of f in the next section. m The spherical harmonics are normalized . The ClebschGordan coefficients are the coefficients appearing in the expansion of the product of two spherical harmonics in terms of spherical harmonics themselves. By using the results of the previous subsections prove the validity of Eq. R Let Yj be an arbitrary orthonormal basis of the space H of degree spherical harmonics on the n-sphere. ( Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. f : {\displaystyle \Im [Y_{\ell }^{m}]=0} Given two vectors r and r, with spherical coordinates x 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. {\displaystyle L_{\mathbb {R} }^{2}(S^{2})} 3 \(Y_{\ell}^{0}(\theta)=\sqrt{\frac{2 \ell+1}{4 \pi}} P_{\ell}(\cos \theta)\) (3.28). are the Legendre polynomials, and they can be derived as a special case of spherical harmonics. provide a basis set of functions for the irreducible representation of the group SO(3) of dimension 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. and Spherical Harmonics 11.1 Introduction Legendre polynomials appear in many different mathematical and physical situations: . Y = The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication. ( 2 m . r {\displaystyle \mathbf {r} } Figure 3.1: Plot of the first six Legendre polynomials. . ( This parity property will be conrmed by the series , or alternatively where {\displaystyle \mathbf {r} } m A The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. 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). As . This is valid for any orthonormal basis of spherical harmonics of degree, Applications of Legendre polynomials in physics, Learn how and when to remove this template message, "Symmetric tensor spherical harmonics on the N-sphere and their application to the de Sitter group SO(N,1)", "Zernike like functions on spherical cap: principle and applications in optical surface fitting and graphics rendering", "On nodal sets and nodal domains on S and R", https://en.wikipedia.org/w/index.php?title=Spherical_harmonics&oldid=1146217720, D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, This page was last edited on 23 March 2023, at 13:52. \end {aligned} V (r) = V (r). 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. {\displaystyle f:S^{2}\to \mathbb {R} } m > m R {\displaystyle (r,\theta ,\varphi )} are associated Legendre polynomials without the CondonShortley phase (to avoid counting the phase twice). They are often employed in solving partial differential equations in many scientific fields. v If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. 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 . : 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. S to { Details of the calculation: ( r) = (x + y - 3z)f (r) = (rsincos + rsinsin - 3rcos)f (r) ) From this it follows that mm must be an integer, \(\Phi(\phi)=\frac{1}{\sqrt{2 \pi}} e^{i m \phi} \quad m=0, \pm 1, \pm 2 \ldots\) (3.15). give rise to the solid harmonics by extending from An exception are the spin representation of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). f &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 \\ [ {\displaystyle Z_{\mathbf {x} }^{(\ell )}({\mathbf {y} })} {\displaystyle r^{\ell }} S Angular momentum and its conservation in classical mechanics. {\displaystyle m>0} This page titled 3: Angular momentum in quantum mechanics is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Mihly Benedict via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. (Here the scalar field is understood to be complex, i.e. , of the eigenvalue problem. {\displaystyle k={\ell }} , For a given value of , there are 2 + 1 independent solutions of this form, one for each integer m with m . . {\displaystyle \theta } f ( Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree Angular momentum and spherical harmonics The angular part of the Laplace operator can be written: (12.1) Eliminating (to solve for the differential equation) one needs to solve an eigenvalue problem: (12.2) where are the eigenvalues, subject to the condition that the solution be single valued on and . 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). {\displaystyle L_{\mathbb {C} }^{2}(S^{2})} 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. terms (cosines) are included, and for , , and The operator of parity \(\) is defined in the following way: \(\Pi \psi(\mathbf{r})=\psi(-\mathbf{r})\) (3.29). 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. , 0 , are essentially Hence, transforms into a linear combination of spherical harmonics of the same degree. The functions \(P_{\ell}^{m}(z)\) are called associated Legendre functions. {\displaystyle S^{2}} Therefore the single eigenvalue of \(^{2}\) is 1, and any function is its eigenfunction. ) 0 {4\pi (l + |m|)!} z Y In 1782, Pierre-Simon de Laplace had, in his Mcanique Cleste, determined that the gravitational potential , R {\displaystyle m} / When < 0, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. In order to satisfy this equation for all values of \(\) and \(\) these terms must be separately equal to a constant with opposite signs. The figures show the three-dimensional polar diagrams of the spherical harmonics. Y {\displaystyle r>R} , obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. {\displaystyle \varphi } We will use the actual function in some problems. 0 (the irregular solid harmonics It follows from Equations ( 371) and ( 378) that. ) directions respectively. 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. He discovered that if r r1 then, where is the angle between the vectors x and x1. 3 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 Y_{\ell }^{m}} S ( {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } 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. 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. = ) &\Pi_{\psi_{-}}(\mathbf{r})=\quad \psi_{-}(-\mathbf{r})=-\psi_{-}(\mathbf{r}) If the functions f and g have a zero mean (i.e., the spectral coefficients f00 and g00 are zero), then Sff() and Sfg() represent the contributions to the function's variance and covariance for degree , respectively. The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. ) {\displaystyle \ell =1} = {\displaystyle \psi _{i_{1}\dots i_{\ell }}} Looking for the eigenvalues and eigenfunctions of \(\), we note first that \(^{2}=1\). From this perspective, one has the following generalization to higher dimensions. R That is, a polynomial p is in P provided that for any real Y They will be functions of \(0 \leq \theta \leq \pi\) and \(0 \leq \phi<2 \pi\), i.e. 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). The tensor spherical harmonics 1 The Clebsch-Gordon coecients Consider a system with orbital angular momentum L~ and spin angular momentum ~S. } p The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. They are, moreover, a standardized set with a fixed scale or normalization. to Laplace's equation m C We shall now find the eigenfunctions of \(_{}\), that play a very important role in quantum mechanics, and actually in several branches of theoretical physics. S Show that \(P_{}(z)\) are either even, or odd depending on the parity of \(\). S {\displaystyle f:S^{2}\to \mathbb {R} } = R p , so the magnitude of the angular momentum is L=rp . . listed explicitly above we obtain: Using the equations above to form the real spherical harmonics, it is seen that for is the operator analogue of the solid harmonic setting, If the quantum mechanical convention is adopted for the {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } m {\displaystyle \{\theta ,\varphi \}} Your vector spherical harmonics are functions of in the vector space $$ \pmb{Y}_{j\ell m} \in V=\left\{ \mathbf f:\mathbb S^2 \to \mathbb C^3 : \int_{\mathbb S^2} |\mathbf f(\pmb\Omega)|^2 \mathrm d \pmb\Omega <\infty . 2 {\displaystyle \mathbf {a} } 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. {\displaystyle r=\infty } Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section Higher dimensions below). {\displaystyle \{\pi -\theta ,\pi +\varphi \}} ) m C the expansion coefficients . = 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 . : Y ( 's transform under rotations (see below) in the same way as the : Equation in three dimensions will use the actual function in some problems \... 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