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. , and 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. ( 3 In this chapter we will discuss the basic theory of angular momentum which plays an extremely important role in the study of quantum mechanics. . 1-62. : A {\displaystyle \mathbb {R} ^{3}} Y Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. The {\displaystyle \mathbf {r} } For a given value of , there are 2 + 1 independent solutions of this form, one for each integer m with m . The result of acting by the parity on a function is the mirror image of the original function with respect to the origin. cos they can be considered as complex valued functions whose domain is the unit sphere. m {\displaystyle r=\infty } C 2 The geodesy[11] and magnetics communities never include the CondonShortley phase factor in their definitions of the spherical harmonic functions nor in the ones of the associated Legendre polynomials. 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. See, e.g., Appendix A of Garg, A., Classical Electrodynamics in a Nutshell (Princeton University Press, 2012). ) 1 Y 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. The analog of the spherical harmonics for the Lorentz group is given by the hypergeometric series; furthermore, the spherical harmonics can be re-expressed in terms of the hypergeometric series, as SO(3) = PSU(2) is a subgroup of PSL(2,C). q ( When < 0, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. Y In 1782, Pierre-Simon de Laplace had, in his Mcanique Cleste, determined that the gravitational potential 0 } Essentially all the properties of the spherical harmonics can be derived from this generating function. 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. {\displaystyle \ell } and another of , , The tensor spherical harmonics 1 The Clebsch-Gordon coecients Consider a system with orbital angular momentum L~ and spin angular momentum ~S. 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. S m R The operator of parity \(\) is defined in the following way: \(\Pi \psi(\mathbf{r})=\psi(-\mathbf{r})\) (3.29). {\displaystyle S^{n-1}\to \mathbb {C} } The complex spherical harmonics m that obey Laplace's equation. The solutions, \(Y_{\ell}^{m}(\theta, \phi)=\mathcal{N}_{l m} P_{\ell}^{m}(\theta) e^{i m \phi}\) (3.20). 2 If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. i and Spherical Harmonics 11.1 Introduction Legendre polynomials appear in many different mathematical and physical situations: . We shall now find the eigenfunctions of \(_{}\), that play a very important role in quantum mechanics, and actually in several branches of theoretical physics. Spherical harmonics are ubiquitous in atomic and molecular physics. , = The spherical harmonics are normalized . A The foregoing has been all worked out in the spherical coordinate representation, and modelling of 3D shapes. n S For example, when = ( , m v S from the above-mentioned polynomial of degree 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. , i.e. 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. Y For angular momentum operators: 1. ] : In other words, any well-behaved function of and can be represented as a superposition of spherical harmonics. The convergence of the series holds again in the same sense, namely the real spherical harmonics where the absolute values of the constants \(\mathcal{N}_{l m}\) ensure the normalization over the unit sphere, are called spherical harmonics. 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. {\displaystyle y} 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 . only, or equivalently of the orientational unit vector f , the solid harmonics with negative powers of It can be shown that all of the above normalized spherical harmonic functions satisfy. The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation. {\displaystyle m<0} {\displaystyle r} S Spherical Harmonics are a group of functions used in math and the physical sciences to solve problems in disciplines including geometry, partial differential equations, and group theory. = as real parameters. 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 Y_{\ell m}} S : [14] An immediate benefit of this definition is that if the vector One can also understand the differentiability properties of the original function f in terms of the asymptotics of Sff(). {\displaystyle \Delta f=0} R : Spherical harmonics originate from solving Laplace's equation in the spherical domains. Using the expressions for Remember from chapter 2 that a subspace is a specic subset of a general complex linear vector space. , r! 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.) {\displaystyle \ell =1} {\displaystyle k={\ell }} 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 \(\). 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. 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. Find \(P_{2}^{0}(\theta)\), \(P_{2}^{1}(\theta)\), \(P_{2}^{2}(\theta)\). {\displaystyle Y_{\ell }^{m}} x {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } 1 = Y , {\displaystyle Z_{\mathbf {x} }^{(\ell )}} y m {\displaystyle e^{\pm im\varphi }} The half-integer values do not give vanishing radial solutions. 2 &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 \\ m {\displaystyle \varphi } 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. Chapters 1 and 2. The angular components of . e^{-i m \phi} n \end{aligned}\) (3.27). {\displaystyle S^{2}} {\displaystyle f_{\ell }^{m}} y He discovered that if r r1 then, where is the angle between the vectors x and x1. 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. ( {\displaystyle \Im [Y_{\ell }^{m}]=0} One can determine the number of nodal lines of each type by counting the number of zeros 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. about the origin that sends the unit vector ] 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. Looking for the eigenvalues and eigenfunctions of \(\), we note first that \(^{2}=1\). r {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} 1 ) 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. L 1 Y The spherical harmonic functions depend on the spherical polar angles and and form an (infinite) complete set of orthogonal, normalizable functions. , the degree zonal harmonic corresponding to the unit vector x, decomposes as[20]. : , and their nodal sets can be of a fairly general kind.[22]. m The real spherical harmonics 2 In summary, if is not an integer, there are no convergent, physically-realizable solutions to the SWE. {\displaystyle P_{\ell }^{m}(\cos \theta )} \end{aligned}\) (3.6). C {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } 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. Y Therefore the single eigenvalue of \(^{2}\) is 1, and any function is its eigenfunction. \(Y(\theta, \phi)=\Theta(\theta) \Phi(\phi)\) (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). Y . Y {\displaystyle Y_{\ell }^{m}} {\displaystyle \ell =1} In order to satisfy this equation for all values of \(\) and \(\) these terms must be separately equal to a constant with opposite signs. i Very often the spherical harmonics are given by Cartesian coordinates by exploiting \(\sin \theta e^{\pm i \phi}=(x \pm i y) / r\) and \(\cos \theta=z / r\). : m 2 {\displaystyle (r,\theta ,\varphi )} { S 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. , {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } k r Y {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } n L=! When = 0, the spectrum is "white" as each degree possesses equal power. {\displaystyle A_{m}(x,y)} The benefit of the expansion in terms of the real harmonic functions Here the solution was assumed to have the special form Y(, ) = () (). C ) m , 2 C , with m There are several different conventions for the phases of Nlm, so one has to be careful with them. and They are, moreover, a standardized set with a fixed scale or normalization. [ Thus, p2=p r 2+p 2 can be written as follows: p2=pr 2+ L2 r2. z Details of the calculation: ( r) = (x + y - 3z)f (r) = (rsincos + rsinsin - 3rcos)f (r) The functions only the 3 The eigenvalues of \(\) itself are then \(1\), and we have the following two possibilities: \(\begin{aligned} = Y can be visualized by considering their "nodal lines", that is, the set of points on the sphere where [ C For example, for any For z , of the eigenvalue problem. being a unit vector, In terms of the spherical angles, parity transforms a point with coordinates That is. ) {\displaystyle Y_{\ell }^{m}} c , can be defined in terms of their complex analogues The eigenfunctions of \(\hat{L}^{2}\) will be denoted by \(Y(,)\), and the angular eigenvalue equation is: \(\begin{aligned} Finally, the equation for R has solutions of the form R(r) = A r + B r 1; requiring the solution to be regular throughout R3 forces B = 0.[3]. ) Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. However, whereas every irreducible tensor representation of SO(2) and SO(3) is of this kind, the special orthogonal groups in higher dimensions have additional irreducible representations that do not arise in this manner. {\displaystyle z} Specifically, we say that a (complex-valued) polynomial function We will use the actual function in some problems. They will be functions of \(0 \leq \theta \leq \pi\) and \(0 \leq \phi<2 \pi\), i.e. R ) transforms into a linear combination of spherical harmonics of the same degree. ( r of the elements of as follows, leading to functions and can also be expanded in terms of the real harmonics f 2 m One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of However, the solutions of the non-relativistic Schrdinger equation without magnetic terms can be made real. The same sine and cosine factors can be also seen in the following subsection that deals the. 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