(18) of Chapter 4] . Y Angular momentum is not a property of a wavefunction at a point; it is a property of a wavefunction as a whole. ) For a fixed integer , every solution Y(, ), ) S Basically, you can always think of a spherical harmonic in terms of the generalized polynomial. R {\displaystyle f_{\ell m}} can also be expanded in terms of the real harmonics S {\displaystyle (-1)^{m}} 2 m {\displaystyle \ell =1} 1 S Specifically, if, A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. ( Here, it is important to note that the real functions span the same space as the complex ones would. Share Cite Improve this answer Follow edited Aug 26, 2019 at 15:19 Y is replaced by the quantum mechanical spin vector operator C \end{aligned}\) (3.8). 2 Spherical harmonics can be generalized to higher-dimensional Euclidean space (the irregular solid harmonics m Many facts about spherical harmonics (such as the addition theorem) that are proved laboriously using the methods of analysis acquire simpler proofs and deeper significance using the methods of symmetry. (12) for some choice of coecients am. . . A 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]. Thus, the wavefunction can be written in a form that lends to separation of variables. . only the the formula, Several different normalizations are in common use for the Laplace spherical harmonic functions , and R {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} ) y In this chapter we discuss the angular momentum operator one of several related operators analogous to classical angular momentum. Inversion is represented by the operator {\displaystyle m} The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry. The vector spherical harmonics are now defined as the quantities that result from the coupling of ordinary spherical harmonics and the vectors em to form states of definite J (the resultant of the orbital angular momentum of the spherical harmonic and the one unit possessed by the em ). . r, which is ! 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. ] , where ( [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). {\displaystyle f_{\ell }^{m}\in \mathbb {C} } C On the other hand, considering m : , are the Legendre polynomials, and they can be derived as a special case of spherical harmonics. 1 . 3 ( m r! . = 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 S The are essentially For S ( , Chapters 1 and 2. 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 \((1)^{}\). provide a basis set of functions for the irreducible representation of the group SO(3) of dimension {\displaystyle e^{\pm im\varphi }} k Y This is justified rigorously by basic Hilbert space theory. ) m C One concludes that the spherical harmonics in the solution for the electron wavefunction in the hydrogen atom identify the angular momentum of the electron. , such that 2 {\displaystyle \varphi } , 2 2 = ( In particular, if Sff() decays faster than any rational function of as , then f is infinitely differentiable. J {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } 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. can be visualized by considering their "nodal lines", that is, the set of points on the sphere where S {\displaystyle \mathbb {R} ^{3}} m The result of acting by the parity on a function is the mirror image of the original function with respect to the origin. Thus, p2=p r 2+p 2 can be written as follows: p2=pr 2+ L2 r2. 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. : C r 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. : The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation. 2 The real spherical harmonics m m m ( ) are complex and mix axis directions, but the real versions are essentially just x, y, and z. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence r 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. symmetric on the indices, uniquely determined by the requirement. ) \end{aligned}\) (3.27). C , respectively, the angle The half-integer values do not give vanishing radial solutions. {\displaystyle \ell =2} Y 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). p is ! 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. 2 {\displaystyle \langle \theta ,\varphi |lm\rangle =Y_{l}^{m}(\theta ,\varphi )} m {\displaystyle (A_{m}\pm iB_{m})} Looking for the eigenvalues and eigenfunctions of \(\), we note first that \(^{2}=1\). \(\begin{aligned} Concluding the subsection let us note the following important fact. {\displaystyle \theta } m [14] An immediate benefit of this definition is that if the vector f m {\displaystyle Y_{\ell }^{m}} m {\displaystyle \varphi } The spherical harmonics, more generally, are important in problems with spherical symmetry. Functions that are solutions to Laplace's equation are called harmonics. i 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). {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } {\displaystyle P_{\ell }^{m}:[-1,1]\to \mathbb {R} } 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). The (complex-valued) spherical harmonics {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} e^{i m \phi} \\ m , obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. \end{array}\right.\) (3.12), and any linear combinations of them. : Y 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 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. 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. x ( 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 Therefore the single eigenvalue of \(^{2}\) is 1, and any function is its eigenfunction. Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner D-matrix. r ( P = setting, If the quantum mechanical convention is adopted for the ] f Considering {\displaystyle c\in \mathbb {C} } {\displaystyle m<0} {\displaystyle \ell } ) by setting, The real spherical harmonics 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 \\ m Legal. 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 . 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? : Let us also note that the \(m=0\) functions do not depend on \(\), and they are proportional to the Legendre polynomials in \(cos\). Y m The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. i 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. Clebsch Gordon coecients allow us to express the total angular momentum basis |jm; si in terms of the direct product {\displaystyle \mathbb {R} ^{n}\to \mathbb {C} } y Calculate the following operations on the spherical harmonics: (a.) &\hat{L}_{z}=-i \hbar \partial_{\phi} m R {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } : In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. S {\displaystyle \mathbf {H} _{\ell }} It follows from Equations ( 371) and ( 378) that. > (Here the scalar field is understood to be complex, i.e. 1 {\displaystyle \ell } n Spherical Harmonics, and Bessel Functions Physics 212 2010, Electricity and Magnetism Michael Dine Department of Physics . R {\displaystyle r} : When you apply L 2 to an angular momentum eigenstate l, then you find L 2 l = [ l ( l + 1) 2] l. That is, l ( l + 1) 2 is the value of L 2 which is associated to the eigenstate l. 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