Since they are eigenfunctions of Hermitian operators, they are orthogonal . m \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) 2 by \(\mathcal{R}(r)\). {\displaystyle m<0} C As . C r Answer: N2 Z 2 0 cos4 d= N 2 3 8 2 0 = N 6 8 = 1 N= 4 3 1/2 4 3 1/2 cos2 = X n= c n 1 2 ein c n = 4 6 1/2 1 Z 2 0 cos2 ein d . {\displaystyle \mathbf {a} =[{\frac {1}{2}}({\frac {1}{\lambda }}-\lambda ),-{\frac {i}{2}}({\frac {1}{\lambda }}+\lambda ),1].}. 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). {\displaystyle {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })} , of the eigenvalue problem. The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication. 0 in ( , Prove that \(P_{\ell}^{m}(z)\) are solutions of (3.16) for all \(\) and \(|m|\), if \(|m|\). P k they can be considered as complex valued functions whose domain is the unit sphere. Figure 3.1: Plot of the first six Legendre polynomials. This operator thus must be the operator for the square of the angular momentum. y R and modelling of 3D shapes. For example, when {\displaystyle \ell } 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. The functions 2 ) {\displaystyle Y_{\ell }^{m}({\mathbf {r} })} {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } {\displaystyle \mathbf {J} } Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle between x1 and x. r ( ( r 2 The convergence of the series holds again in the same sense, namely the real spherical harmonics The set of all direction kets n` can be visualized . z 0 B Angular momentum is the generator for rotations, so spherical harmonics provide a natural characterization of the rotational properties and direction dependence of a system. 0 as a function of Inversion is represented by the operator It follows from Equations ( 371) and ( 378) that. ] r {\displaystyle \mathbb {R} ^{3}\setminus \{\mathbf {0} \}\to \mathbb {C} } C m {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} can be defined in terms of their complex analogues {\displaystyle \mathbf {A} _{\ell }} f {\displaystyle L_{\mathbb {C} }^{2}(S^{2})} form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions ( Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. By using the results of the previous subsections prove the validity of Eq. {\displaystyle S^{2}} S . ( {\displaystyle A_{m}(x,y)} C , For angular momentum operators: 1. 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 A_{m}} S [23] Let P denote the space of complex-valued homogeneous polynomials of degree in n real variables, here considered as functions Furthermore, the zonal harmonic . For the case of orthonormalized harmonics, this gives: If the coefficients decay in sufficiently rapidly for instance, exponentially then the series also converges uniformly to f. A square-integrable function , the trigonometric sin and cos functions possess 2|m| zeros, each of which gives rise to a nodal 'line of longitude'. {\displaystyle v} S is essentially the associated Legendre polynomial Let Yj be an arbitrary orthonormal basis of the space H of degree spherical harmonics on the n-sphere. (considering them as functions Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x p~. C ( ( \end{aligned}\) (3.8). The spherical harmonics, more generally, are important in problems with spherical symmetry. ] L z Y 21 (b.) Spherical Harmonics, and Bessel Functions Physics 212 2010, Electricity and Magnetism Michael Dine Department of Physics . {\displaystyle Y_{\ell }^{m}} 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. f = m Y \end {aligned} V (r) = V (r). ) Y are associated Legendre polynomials without the CondonShortley phase (to avoid counting the phase twice). Abstract. 2 2 {\displaystyle \theta } {\displaystyle \Delta f=0} In other words, any well-behaved function of and can be represented as a superposition of spherical harmonics. and Y S 3 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 . between them is given by the relation, where P is the Legendre polynomial of degree . x of spherical harmonics of degree S m ) : . Specifically, if, A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. For a given value of , there are 2 + 1 independent solutions of this form, one for each integer m with m . S C . m m , {\displaystyle \mathbb {R} ^{3}} C r to {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } C Find \(P_{2}^{0}(\theta)\), \(P_{2}^{1}(\theta)\), \(P_{2}^{2}(\theta)\). 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} This equation easily separates in . 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. , commonly referred to as the CondonShortley phase in the quantum mechanical literature. symmetric on the indices, uniquely determined by the requirement. {\displaystyle Y:S^{2}\to \mathbb {C} } The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. {\displaystyle {\mathcal {R}}} (3.31). The spherical harmonics play an important role in quantum mechanics. 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. Consider the problem of finding solutions of the form f(r, , ) = R(r) Y(, ). is given as a constant multiple of the appropriate Gegenbauer polynomial: Combining (2) and (3) gives (1) in dimension n = 2 when x and y are represented in spherical coordinates. 1 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 . [27] One is hemispherical functions (HSH), orthogonal and complete on hemisphere. 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. Laplace's spherical harmonics [ edit] Real (Laplace) spherical harmonics for (top to bottom) and (left to right). On the other hand, considering z Whereas the trigonometric functions in a Fourier series represent the fundamental modes of vibration in a string, the spherical harmonics represent the fundamental modes of vibration of a sphere in much the same way. { {\displaystyle x} 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. 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? The spherical harmonics are the eigenfunctions of the square of the quantum mechanical angular momentum operator. {\displaystyle \theta } give rise to the solid harmonics by extending from One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of {\displaystyle \mathbf {H} _{\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. q Y and Spherical Harmonics 11.1 Introduction Legendre polynomials appear in many different mathematical and physical situations: . -\Delta_{\theta \phi} Y(\theta, \phi) &=\ell(\ell+1) Y(\theta, \phi) \quad \text { or } \\ C Details of the calculation: ( r) = (x + y - 3z)f (r) = (rsincos + rsinsin - 3rcos)f (r) : As these are functions of points in real three dimensional space, the values of \(()\) and \((+2)\) must be the same, as these values of the argument correspond to identical points in space. {\displaystyle Y_{\ell }^{m}} : The parallelism of the two definitions ensures that the &\Pi_{\psi_{-}}(\mathbf{r})=\quad \psi_{-}(-\mathbf{r})=-\psi_{-}(\mathbf{r}) f is the operator analogue of the solid harmonic {\displaystyle Y_{\ell m}} 's transform under rotations (see below) in the same way as the , obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. m 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. e^{i m \phi} \\ 1 or p component perpendicular to the radial vector ! {\displaystyle r>R} {\displaystyle B_{m}} , as follows (CondonShortley phase): The factor The solutions, \(Y_{\ell}^{m}(\theta, \phi)=\mathcal{N}_{l m} P_{\ell}^{m}(\theta) e^{i m \phi}\) (3.20). Y m Throughout the section, we use the standard convention that for 0 1 With respect to this group, the sphere is equivalent to the usual Riemann sphere. The general technique is to use the theory of Sobolev spaces. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } 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). C {\displaystyle \mathbf {r} } m , 2 .) 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 f:\mathbb {R} ^{3}\to \mathbb {C} } only the {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } Y (12) for some choice of coecients am. The essential property of 2 All divided by an inverse power, r to the minus l. To make full use of rotational symmetry and angular momentum, we will restrict our attention to spherically symmetric potentials, \begin {aligned} V (\vec {r}) = V (r). {\displaystyle \mathbb {R} ^{n}\to \mathbb {C} } 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. y {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } m as a function of Then {\displaystyle Y_{\ell }^{m}} In order to satisfy this equation for all values of \(\) and \(\) these terms must be separately equal to a constant with opposite signs. L 2 Y 21 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. The spherical harmonic functions depend on the spherical polar angles and and form an (infinite) complete set of orthogonal, normalizable functions. c Notice, however, that spherical harmonics are not functions on the sphere which are harmonic with respect to the Laplace-Beltrami operator for the standard round metric on the sphere: the only harmonic functions in this sense on the sphere are the constants, since harmonic functions satisfy the Maximum principle. m m r, which is ! the angular momentum and the energy of the particle are measured simultane-ously at time t= 0, what values can be obtained for each observable and with what probabilities? m The condition on the order of growth of Sff() is related to the order of differentiability of f in the next section. The solution function Y(, ) is regular at the poles of the sphere, where = 0, . m x , Y 1 ( are the Legendre polynomials, and they can be derived as a special case of spherical harmonics. : It can be shown that all of the above normalized spherical harmonic functions satisfy. This parity property will be conrmed by the series 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. \(\begin{aligned} {\displaystyle (2\ell +1)} is just the 3-dimensional space of all linear functions } : m {\displaystyle L_{\mathbb {R} }^{2}(S^{2})} ) ) ( {\displaystyle \lambda } {\displaystyle f:S^{2}\to \mathbb {C} } form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions , then, a 3 One can also understand the differentiability properties of the original function f in terms of the asymptotics of Sff(). {\displaystyle P_{i}:[-1,1]\to \mathbb {R} } = ; the remaining factor can be regarded as a function of the spherical angular coordinates 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. Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. 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. Laplace equation. S r Essentially all the properties of the spherical harmonics can be derived from this generating function. m z f One can determine the number of nodal lines of each type by counting the number of zeros of 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. Spherical coordinates, elements of vector analysis. m {\displaystyle S^{2}} [ In this chapter we discuss the angular momentum operator one of several related operators analogous to classical angular momentum. {\displaystyle Z_{\mathbf {x} }^{(\ell )}({\mathbf {y} })} R , = Meanwhile, when The figures show the three-dimensional polar diagrams of the spherical harmonics. {\displaystyle f_{\ell }^{m}} 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 Analytic expressions for the first few orthonormalized Laplace spherical harmonics n : R Using the orthonormality properties of the real unit-power spherical harmonic functions, it is straightforward to verify that the total power of a function defined on the unit sphere is related to its spectral coefficients by a generalization of Parseval's theorem (here, the theorem is stated for Schmidt semi-normalized harmonics, the relationship is slightly different for orthonormal harmonics): is defined as the angular power spectrum (for Schmidt semi-normalized harmonics). 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 \ell =1} r only the where the superscript * denotes complex conjugation. , and their nodal sets can be of a fairly general kind.[22]. p. The cross-product picks out the ! L Thus, the wavefunction can be written in a form that lends to separation of variables. {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. {\displaystyle \mathbf {A} _{1}} Here the solution was assumed to have the special form Y(, ) = () (). In 1782, Pierre-Simon de Laplace had, in his Mcanique Cleste, determined that the gravitational potential Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner D-matrix. 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