They will be functions of \(0 \leq \theta \leq \pi\) and \(0 \leq \phi<2 \pi\), i.e. {\displaystyle B_{m}(x,y)} The spherical harmonics are representations of functions of the full rotation group SO(3)[5]with rotational symmetry. {\displaystyle Z_{\mathbf {x} }^{(\ell )}({\mathbf {y} })} \end{aligned}\) (3.27). m 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. 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. and another of {\displaystyle {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })} C ) f 0 R Y Y {\displaystyle f:S^{2}\to \mathbb {R} } 3 {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } , of the eigenvalue problem. The spinor spherical harmonics are the natural spinor analog of the vector spherical harmonics. m The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of = 1 A specific set of spherical harmonics, denoted 0 3 are associated Legendre polynomials without the CondonShortley phase (to avoid counting the phase twice). http://en.Wikipedia.org/wiki/File:Legendrepolynomials6.svg. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } &\Pi_{\psi_{+}}(\mathbf{r})=\quad \psi_{+}(-\mathbf{r})=\psi_{+}(\mathbf{r}) \\ J , 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. 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. Find \(P_{2}^{0}(\theta)\), \(P_{2}^{1}(\theta)\), \(P_{2}^{2}(\theta)\). , the solid harmonics with negative powers of 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 \(2\). 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. . m {\displaystyle \mathbf {r} } m 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 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. The ClebschGordan coefficients are the coefficients appearing in the expansion of the product of two spherical harmonics in terms of spherical harmonics themselves. 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. {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } m Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. Introduction to the Physics of Atoms, Molecules and Photons (Benedict), { "1.01:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "1.02:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "1.03:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "1.04:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "1.05:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "1.06:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "1.07:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "1.08:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "1.09:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "10:_Atoms_in_Strong_Fields" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "11:_Photons:_quantization_of_a_single_electromagnetic_field_mode" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "12:_A_quantum_paradox_and_the_experiments" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, { "00:_Front_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "01:_Chapters" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()", "zz:_Back_Matter" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.b__1]()" }, [ "article:topic", "license:ccby", "licenseversion:30", "authorname:mbenedict", "source@http://titan.physx.u-szeged.hu/~dpiroska/atmolfiz/index.html" ], https://phys.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fphys.libretexts.org%2FBookshelves%2FQuantum_Mechanics%2FIntroduction_to_the_Physics_of_Atoms_Molecules_and_Photons_(Benedict)%2F01%253A_Chapters%2F1.03%253A_New_Page, \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}}}\) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\), 4: Atomic spectra, simple models of atoms, http://en.Wikipedia.org/wiki/File:Legendrepolynomials6.svg, http://en.Wikipedia.org/wiki/Spherical_harmonics, source@http://titan.physx.u-szeged.hu/~dpiroska/atmolfiz/index.html, status page at https://status.libretexts.org. {\displaystyle m>0} {\displaystyle Y_{\ell }^{m}} Here the solution was assumed to have the special form Y(, ) = () (). 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). S r n 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. 1 {\displaystyle S^{2}\to \mathbb {C} } The spherical harmonics, more generally, are important in problems with spherical symmetry. C 2 1 S P m 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 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. C ) , obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. All divided by an inverse power, r to the minus l. ] {\displaystyle L_{\mathbb {C} }^{2}(S^{2})} The general, normalized Spherical Harmonic is depicted below: Y_ {l}^ {m} (\theta,\phi) = \sqrt { \dfrac { (2l + 1) (l - |m|)!} Legal. : Y r a 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. 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 . m = form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials: Here Looking for the eigenvalues and eigenfunctions of \(\), we note first that \(^{2}=1\). r m The reason for this can be seen by writing the functions in terms of the Legendre polynomials as. can also be expanded in terms of the real harmonics from the above-mentioned polynomial of degree or ) are chosen instead. 2 The operator on the left operates on the spherical harmonic function to give a value for \(M^2\), the square of the rotational angular momentum, times the spherical harmonic function. {\displaystyle \mathbf {a} =[{\frac {1}{2}}({\frac {1}{\lambda }}-\lambda ),-{\frac {i}{2}}({\frac {1}{\lambda }}+\lambda ),1].}. 's of degree ) ( 0 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 \(\). Abstract. {\displaystyle (r,\theta ,\varphi )} and {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } Since they are eigenfunctions of Hermitian operators, they are orthogonal . 0 f For example, for any Z 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. Y 's transform under rotations (see below) in the same way as the {\displaystyle Y_{\ell }^{m}} ) brackets are functions of ronly, and the angular momentum operator is only a function of and . {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {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. In a similar manner, one can define the cross-power of two functions as, is defined as the cross-power spectrum. Therefore the single eigenvalue of \(^{2}\) is 1, and any function is its eigenfunction. {\displaystyle {\mathcal {R}}} When = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. R , . {\displaystyle \Im [Y_{\ell }^{m}]=0} S Nodal lines of {\displaystyle f:S^{2}\to \mathbb {R} } {\displaystyle \{\theta ,\varphi \}} He discovered that if r r1 then, where is the angle between the vectors x and x1. This is useful for instance when we illustrate the orientation of chemical bonds in molecules. {\displaystyle f_{\ell }^{m}\in \mathbb {C} } {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } {\displaystyle Y_{\ell }^{m}} the formula, Several different normalizations are in common use for the Laplace spherical harmonic functions [13] These functions have the same orthonormality properties as the complex ones The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. 3 The spherical harmonic functions depend on the spherical polar angles and and form an (infinite) complete set of orthogonal, normalizable functions. ) = {\displaystyle S^{2}} Chapters 1 and 2. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. 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 . S m {\displaystyle (r',\theta ',\varphi ')} R in the + m S Consider the problem of finding solutions of the form f(r, , ) = R(r) Y(, ). Operators for the square of the angular momentum and for its zcomponent: \end{array}\right.\) (3.12), and any linear combinations of them. {\displaystyle B_{m}} by setting, The real spherical harmonics For example, as can be seen from the table of spherical harmonics, the usual p functions ( The Laplace spherical harmonics The spherical harmonics are orthonormal: that is, Y l, m Yl, md = ll mm, and also form a complete set. In spherical coordinates this is:[2]. : J In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum[4]. P m R i y , which can be seen to be consistent with the output of the equations above. , r r 2 ( l We will use the actual function in some problems. f 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. L {\displaystyle \theta } S where the superscript * denotes complex conjugation. The spherical harmonics with negative can be easily compute from those with positive . -\Delta_{\theta \phi} Y(\theta, \phi) &=\ell(\ell+1) Y(\theta, \phi) \quad \text { or } \\ {\displaystyle \{\pi -\theta ,\pi +\varphi \}} {\displaystyle r} {\displaystyle (x,y,z)} m 1 y r, which is ! {4\pi (l + |m|)!} r , z {\displaystyle \mathbb {R} ^{3}} Calculate the following operations on the spherical harmonics: (a.) 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. Y Y ( \(\begin{aligned} 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. 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). [14] An immediate benefit of this definition is that if the vector ) Y is replaced by the quantum mechanical spin vector operator R ) Consider a rotation that use the CondonShortley phase convention: The classical spherical harmonics are defined as complex-valued functions on the unit sphere above. A {\displaystyle \ell } ( + For convenience, we list the spherical harmonics for = 0,1,2 and non-negative values of m. = 0, Y0 0 (,) = 1 4 = 1, Y1 R C 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. 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). m k m m In naming this generating function after Herglotz, we follow Courant & Hilbert 1962, VII.7, who credit unpublished notes by him for its discovery. But when turning back to \(cos=z\) this factor reduces to \((\sin \theta)^{|m|}\). m On the other hand, considering 2 , and the factors 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. &p_{z}=\frac{z}{r}=Y_{1}^{0}=\sqrt{\frac{3}{4 \pi}} \cos \theta On the unit sphere { is essentially the associated Legendre polynomial Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. , such that transforms into a linear combination of spherical harmonics of the same degree. ) used above, to match the terms and find series expansion coefficients [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). Basically, you can always think of a spherical harmonic in terms of the generalized polynomial. r! 1 r 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. L 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. 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. That is, they are either even or odd with respect to inversion about the origin. , {\displaystyle \ell } We will first define the angular momentum operator through the classical relation L = r p and replace p by its operator representation -i [see Eq. , we have a 5-dimensional space: For any 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. The first term depends only on \(\) while the last one is a function of only \(\). give rise to the solid harmonics by extending from Statements relating the growth of the Sff() to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. Just prior to that time, Adrien-Marie Legendre had investigated the expansion of the Newtonian potential in powers of r = |x| and r1 = |x1|. The spherical harmonics have definite parity. can be visualized by considering their "nodal lines", that is, the set of points on the sphere where , . 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. x There are several different conventions for the phases of \(\mathcal{N}_{l m}\), so one has to be careful with them. directions respectively. {\displaystyle (A_{m}\pm iB_{m})} , B 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 . The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence m m ( 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. There are several different conventions for the phases of Nlm, so one has to be careful with them. R > The operator of parity \(\) is defined in the following way: \(\Pi \psi(\mathbf{r})=\psi(-\mathbf{r})\) (3.29). S 2 ( In fact, L 2 is equivalent to 2 on the spherical surface, so the Y l m are the eigenfunctions of the operator 2. 0 : The foregoing has been all worked out in the spherical coordinate representation, H : f Z Meanwhile, when ( 0 {\displaystyle f_{\ell }^{m}} { , The statement of the parity of spherical harmonics is then. p is ! only the : The parallelism of the two definitions ensures that the ) ( {\displaystyle r^{\ell }} { : , 1-62. 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