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. k S in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor r, where the {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } The angular components of . {\displaystyle \mathbb {R} ^{n}} {\displaystyle S^{2}} as real parameters. This is justified rigorously by basic Hilbert space theory. ) 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 {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} {\displaystyle Y_{\ell }^{m}} C 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. There are several different conventions for the phases of Nlm, so one has to be careful with them. 2 One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of m C f The essential property of ) are chosen instead. Throughout the section, we use the standard convention that for P Notice that \(\) must be a nonnegative integer otherwise the definition (3.18) makes no sense, and in addition if |(|m|>\), then (3.17) yields zero. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. > and 0 http://en.Wikipedia.org/wiki/File:Legendrepolynomials6.svg. For example, for any 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. , which can be seen to be consistent with the output of the equations above. R m {\displaystyle Y_{\ell m}} Y S R z m {\displaystyle \ell =1} Laplace equation. + {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } C and m &\Pi_{\psi_{+}}(\mathbf{r})=\quad \psi_{+}(-\mathbf{r})=\psi_{+}(\mathbf{r}) \\ is an associated Legendre polynomial, N is a normalization constant, and and represent colatitude and longitude, respectively. r, which is ! x {\displaystyle \Re [Y_{\ell }^{m}]=0} {\displaystyle \gamma } . For a scalar function f(n), the spin S is zero, and J is purely orbital angular momentum L, which accounts for the functional dependence on n. The spherical decomposition f . Recalling that the spherical harmonics are eigenfunctions of the angular momentum operator: (r; ;) = R(r)Ym l ( ;) SeparationofVariables L^2Ym l ( ;) = h2l . can also be expanded in terms of the real harmonics as a function of , When < 0, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. m The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. {\displaystyle \ell } \(\sin \theta \frac{d}{d \theta}\left(\sin \theta \frac{d \Theta}{d \theta}\right)+\left[\ell(\ell+1) \sin ^{2} \theta-m^{2}\right] \Theta=0\) (3.16), is more complicated. Rotations and Angular momentum Intro The material here may be found in Sakurai Chap 3: 1-3, (5-6), 7, (9-10) . : , the trigonometric sin and cos functions possess 2|m| zeros, each of which gives rise to a nodal 'line of longitude'. m The 3-D wave equation; spherical harmonics. {\displaystyle \mathbf {J} } H only the , If an external magnetic field \(\mathbf{B}=\{0,0, B\}\) is applied, the projection of the angular momentum onto the field direction is \(m\). {4\pi (l + |m|)!} S 2 Y The operator of parity \(\) is defined in the following way: \(\Pi \psi(\mathbf{r})=\psi(-\mathbf{r})\) (3.29). f {\displaystyle Y_{\ell }^{m}({\mathbf {r} })} Let Yj be an arbitrary orthonormal basis of the space H of degree spherical harmonics on the n-sphere. In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. {\displaystyle L_{\mathbb {C} }^{2}(S^{2})} They are often employed in solving partial differential equations in many scientific fields. 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). Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. [ R {\displaystyle k={\ell }} is just the space of restrictions to the sphere Another way of using these functions is to create linear combinations of functions with opposite m-s. {\displaystyle m>0} That is, it consists of,products of the three coordinates, x1, x2, x3, where the net power, a plus b plus c, is equal to l, the index of the spherical harmonic. , and their nodal sets can be of a fairly general kind.[22]. r! The tensor spherical harmonics 1 The Clebsch-Gordon coecients Consider a system with orbital angular momentum L~ and spin angular momentum ~S. Here the solution was assumed to have the special form Y(, ) = () (). terms (cosines) are included, and for 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). L {\displaystyle \mathbf {r} '} > 5.61 Spherical Harmonics page 1 ANGULAR MOMENTUM Now that we have obtained the general eigenvalue relations for angular momentum directly from the operators, we want to learn about the associated wave functions. 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. : m The general technique is to use the theory of Sobolev spaces. {\displaystyle \mathbb {R} ^{3}} 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). m The angle-preserving symmetries of the two-sphere are described by the group of Mbius transformations PSL(2,C). {\displaystyle r=0} &\Pi_{\psi_{-}}(\mathbf{r})=\quad \psi_{-}(-\mathbf{r})=-\psi_{-}(\mathbf{r}) It follows from Equations ( 371) and ( 378) that. The solid harmonics were homogeneous polynomial solutions m : As none of the components of \(\mathbf{\hat{L}}\), and thus nor \(\hat{L}^{2}\) depends on the radial distance rr from the origin, then any function of the form \(\mathcal{R}(r) Y_{\ell}^{m}(\theta, \phi)\) will be the solution of the eigenvalue equation above, because from the point of view of the \(\mathbf{\hat{L}}\) the \(\mathcal{R}(r)\) function is a constant, and we can freely multiply both sides of (3.8). L=! R m {\displaystyle P_{\ell }^{m}(\cos \theta )} 2 2 only the 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 . Under this operation, a spherical harmonic of degree More general spherical harmonics of degree are not necessarily those of the Laplace basis .) are guaranteed to be real, whereas their coefficients {\displaystyle f_{\ell }^{m}} Y 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. ) : See here for a list of real spherical harmonics up to and including C 1 {\displaystyle Y_{\ell }^{m}} R = In this chapter we will discuss the basic theory of angular momentum which plays an extremely important role in the study of quantum mechanics. , and m m , : 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, \(\begin{aligned} , obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. 2 / R above. m Here, it is important to note that the real functions span the same space as the complex ones would. : 3 y 0 . (Here the scalar field is understood to be complex, i.e. A specific set of spherical harmonics, denoted Essentially all the properties of the spherical harmonics can be derived from this generating function. [5] As suggested in the introduction, this perspective is presumably the origin of the term spherical harmonic (i.e., the restriction to the sphere of a harmonic function). In many fields of physics and chemistry these spherical harmonics are replaced by cubic harmonics because the rotational symmetry of the atom and its environment are distorted or because cubic harmonics offer computational benefits. : x In the quantum mechanics community, it is common practice to either include this phase factor in the definition of the associated Legendre polynomials, or to append it to the definition of the spherical harmonic functions. {\displaystyle z} x {\displaystyle Z_{\mathbf {x} }^{(\ell )}({\mathbf {y} })} In naming this generating function after Herglotz, we follow Courant & Hilbert 1962, VII.7, who credit unpublished notes by him for its discovery. \end{aligned}\) (3.27). , In 1782, Pierre-Simon de Laplace had, in his Mcanique Cleste, determined that the gravitational potential Then The figures show the three-dimensional polar diagrams of the spherical harmonics. {\displaystyle \ell } {\displaystyle Y_{\ell }^{m}} 2 {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} 2 m 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. 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. by \(\mathcal{R}(r)\). Y m 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. 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 . : Spherical harmonics can be generalized to higher-dimensional Euclidean space {\displaystyle \Im [Y_{\ell }^{m}]=0} These angular solutions The \(Y_{\ell}^{m}(\theta)\) functions are thus the eigenfunctions of \(\hat{L}\) corresponding to the eigenvalue \(\hbar^{2} \ell(\ell+1)\), and they are also eigenfunctions of \(\hat{L}_{z}=-i \hbar \partial_{\phi}\), because, \(\hat{L}_{z} Y_{\ell}^{m}(\theta, \phi)=-i \hbar \partial_{\phi} Y_{\ell}^{m}(\theta, \phi)=\hbar m Y_{\ell}^{m}(\theta, \phi)\) (3.21). R ( , respectively, the angle 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. Y For the other cases, the functions checker the sphere, and they are referred to as tesseral. S setting, If the quantum mechanical convention is adopted for the Looking for the eigenvalues and eigenfunctions of \(\), we note first that \(^{2}=1\). Since they are eigenfunctions of Hermitian operators, they are orthogonal . Prove that \(P_{\ell}^{m}(z)\) are solutions of (3.16) for all \(\) and \(|m|\), if \(|m|\). For angular momentum operators: 1. {\displaystyle r^{\ell }} One can determine the number of nodal lines of each type by counting the number of zeros of is the operator analogue of the solid harmonic The integration constant \(\frac{1}{\sqrt{2 \pi}}\) has been chosen here so that already \(()\) is normalized to unity when integrating with respect to \(\) from 0 to \(2\). &\hat{L}_{z}=-i \hbar \partial_{\phi} . In order to obtain them we have to make use of the expression of the position vector by spherical coordinates, which are connected to the Cartesian components by, \(\mathbf{r}=x \hat{\mathbf{e}}_{x}+y \hat{\mathbf{e}}_{y}+z \hat{\mathbf{e}}_{z}=r \sin \theta \cos \phi \hat{\mathbf{e}}_{x}+r \sin \theta \sin \phi \hat{\mathbf{e}}_{y}+r \cos \theta \hat{\mathbf{e}}_{z}\) (3.4). 3 m p is ! 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. ) Statements relating the growth of the Sff() to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. 1.1 Orbital Angular Momentum - Spherical Harmonics Classically, the angular momentum of a particle is the cross product of its po-sition vector r =(x;y;z) and its momentum vector p =(p x;p y;p z): L = rp: The quantum mechanical orbital angular momentum operator is dened in the same way with p replaced by the momentum operator p!ihr . \(Y(\theta, \phi)=\Theta(\theta) \Phi(\phi)\) (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). ) It is common that the (cross-)power spectrum is well approximated by a power law of the form. Spherical Harmonics, and Bessel Functions Physics 212 2010, Electricity and Magnetism Michael Dine Department of Physics . ) to between them is given by the relation, where P is the Legendre polynomial of degree . In this chapter we discuss the angular momentum operator one of several related operators analogous to classical angular momentum. The spherical harmonics are representations of functions of the full rotation group SO(3)[5]with rotational symmetry. C The expansion coefficients are the analogs of Fourier coefficients, and can be obtained by multiplying the above equation by the complex conjugate of a spherical harmonic, integrating over the solid angle , and utilizing the above orthogonality relationships. 2 3 Finally, when > 0, the spectrum is termed "blue". The benefit of the expansion in terms of the real harmonic functions Given two vectors r and r, with spherical coordinates &\hat{L}_{x}=i \hbar\left(\sin \phi \partial_{\theta}+\cot \theta \cos \phi \partial_{\phi}\right) \\ : {\displaystyle c\in \mathbb {C} } \end{aligned}\) (3.6). The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } We have to write the given wave functions in terms of the spherical harmonics. being a unit vector, In terms of the spherical angles, parity transforms a point with coordinates + ,[15] one obtains a generating function for a standardized set of spherical tensor operators, ( m to all of m . x With respect to this group, the sphere is equivalent to the usual Riemann sphere. is homogeneous of degree m {\displaystyle (r,\theta ,\varphi )} The real spherical harmonics f {\displaystyle P_{i}:[-1,1]\to \mathbb {R} } y m 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. One can also understand the differentiability properties of the original function f in terms of the asymptotics of Sff(). The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry. : to correspond to a (smooth) function R ] Direction kets will be used more extensively in the discussion of orbital angular momentum and spherical harmonics, but for now they are useful for illustrating the set of rotations. = = , we have a 5-dimensional space: For any 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 . 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. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } A {\displaystyle r>R} z When = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. {\displaystyle \lambda } There are two quantum numbers for the rigid rotor in 3D: \(J\) is the total angular momentum quantum number and \(m_J\) is the z-component of the angular momentum. You are all familiar, at some level, with spherical harmonics, from angular momentum in quantum mechanics. Hence, as follows, leading to functions {\displaystyle q=m} ( , y {\displaystyle \Delta f=0} S Imposing this regularity in the solution of the second equation at the boundary points of the domain is a SturmLiouville problem that forces the parameter to be of the form = ( + 1) for some non-negative integer with |m|; this is also explained below in terms of the orbital angular momentum. P This parity property will be conrmed by the series This is useful for instance when we illustrate the orientation of chemical bonds in molecules. 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? Legal. 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