Specifically, we say that a (complex-valued) polynomial function e^{-i m \phi} The solution function Y(, ) is regular at the poles of the sphere, where = 0, . f 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. P S 1 are constants and the factors r Ym are known as (regular) solid harmonics f {\displaystyle A_{m}(x,y)} (18) of Chapter 4] . The condition on the order of growth of Sff() is related to the order of differentiability of f in the next section. The three Cartesian components of the angular momentum are: L x = yp z zp y,L y = zp x xp z,L z = xp y yp x. = m . They are often employed in solving partial differential equations in many scientific fields. Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. ( 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 S 2 The spherical harmonics are the eigenfunctions of the square of the quantum mechanical angular momentum operator. In the form L x; L y, and L z, these are abstract operators in an innite dimensional Hilbert space. This is justified rigorously by basic Hilbert space theory. where the absolute values of the constants \(\mathcal{N}_{l m}\) ensure the normalization over the unit sphere, are called spherical harmonics. x 4 : Y {\displaystyle L_{\mathbb {C} }^{2}(S^{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 \((1)^{}\). Y 3 http://en.Wikipedia.org/wiki/Spherical_harmonics. 2 {\displaystyle \mathbb {R} ^{n}\to \mathbb {C} } [12], A real basis of spherical harmonics 2 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 \(2\). {\displaystyle m} Y 1 The state to be shown, can be chosen by setting the quantum numbers \(\) and m. http://titan.physx.u-szeged.hu/~mmquantum/interactive/Gombfuggvenyek.nbp. + Nodal lines of R {\displaystyle f_{\ell }^{m}\in \mathbb {C} } On the other hand, considering S 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. e {\displaystyle (A_{m}\pm iB_{m})} , For example, when 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 eigenfunctions of \(\hat{L}^{2}\) will be denoted by \(Y(,)\), and the angular eigenvalue equation is: \(\begin{aligned} 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. R {\displaystyle S^{2}\to \mathbb {C} } {\displaystyle m<0} transforms into a linear combination of spherical harmonics of the same degree. and order , we have a 5-dimensional space: For any m (Here the scalar field is understood to be complex, i.e. {\displaystyle \mathbf {r} '} m {\displaystyle \theta } S In turn, SU(2) is identified with the group of unit quaternions, and so coincides with the 3-sphere. , (3.31). Then m R {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } v This could be achieved by expansion of functions in series of trigonometric functions. 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. ) is replaced by the quantum mechanical spin vector operator m = {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } that use the CondonShortley phase convention: The classical spherical harmonics are defined as complex-valued functions on the unit sphere setting, If the quantum mechanical convention is adopted for the 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). m m {\displaystyle Z_{\mathbf {x} }^{(\ell )}({\mathbf {y} })} The spherical harmonics have definite parity. Y The solid harmonics were homogeneous polynomial solutions of spherical harmonics of degree , the trigonometric sin and cos functions possess 2|m| zeros, each of which gives rise to a nodal 'line of longitude'. 3 Equation \ref{7-36} is an eigenvalue equation. 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. m [ {\displaystyle \Delta f=0} {\displaystyle \ell } {\displaystyle \mathbf {A} _{\ell }} {\displaystyle (r',\theta ',\varphi ')} A r terms (sines) are included: The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation. m Meanwhile, when m 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). ), In 1867, William Thomson (Lord Kelvin) and Peter Guthrie Tait introduced the solid spherical harmonics in their Treatise on Natural Philosophy, and also first introduced the name of "spherical harmonics" for these functions. ,[15] one obtains a generating function for a standardized set of spherical tensor operators, 2 m as a function of ) {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } Consider the problem of finding solutions of the form f(r, , ) = R(r) Y(, ). But when turning back to \(cos=z\) this factor reduces to \((\sin \theta)^{|m|}\). {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } f {\displaystyle \ell =2} . http://titan.physx.u-szeged.hu/~mmquantum/videok/Gombfuggveny_fazis_idofejlodes.flv. 2 between them is given by the relation, where P is the Legendre polynomial of degree . 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. . The solutions, \(Y_{\ell}^{m}(\theta, \phi)=\mathcal{N}_{l m} P_{\ell}^{m}(\theta) e^{i m \phi}\) (3.20). p. The cross-product picks out the ! are essentially m i m k Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree > ) do not have that property. 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). ] {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } {\displaystyle c\in \mathbb {C} } m The spherical harmonics with negative can be easily compute from those with positive . {\displaystyle Y_{\ell }^{m}} . to {\displaystyle B_{m}} , the degree zonal harmonic corresponding to the unit vector x, decomposes as[20]. ( Returning to spherical polar coordinates, we recall that the angular momentum operators are given by: L = from the above-mentioned polynomial of degree Statements relating the growth of the Sff() to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. C {\displaystyle S^{2}} The angular momentum relative to the origin produced by a momentum vector ! Y Angular momentum is not a property of a wavefunction at a point; it is a property of a wavefunction as a whole. = 3 A r, which is ! Find the first three Legendre polynomials \(P_{0}(z)\), \(P_{1}(z)\) and \(P_{2}(z)\). Throughout the section, we use the standard convention that for R All divided by an inverse power, r to the minus l. Y ) and Such an expansion is valid in the ball. m {\displaystyle P_{\ell }^{m}(\cos \theta )} : If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. z cos . to {\displaystyle S^{2}} R C m {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } \end{array}\right.\) (3.12), and any linear combinations of them. Spherical harmonics are ubiquitous in atomic and molecular physics. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } {\displaystyle \{\theta ,\varphi \}} R 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 \(\). S Analytic expressions for the first few orthonormalized Laplace spherical harmonics The angle-preserving symmetries of the two-sphere are described by the group of Mbius transformations PSL(2,C). , which can be seen to be consistent with the output of the equations above. , respectively, the angle Y This equation easily separates in . ( 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. 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. 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. 1 The set of all direction kets n` can be visualized . 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. Thus, p2=p r 2+p 2 can be written as follows: p2=pr 2+ L2 r2. &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 \\ {\displaystyle \ell } {\displaystyle \Im [Y_{\ell }^{m}]=0} . 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. Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. are associated Legendre polynomials without the CondonShortley phase (to avoid counting the phase twice). . In other words, any well-behaved function of and can be represented as a superposition of spherical harmonics. {\displaystyle P_{i}:[-1,1]\to \mathbb {R} } ( {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} Spherical harmonics can be generalized to higher-dimensional Euclidean space In naming this generating function after Herglotz, we follow Courant & Hilbert 1962, VII.7, who credit unpublished notes by him for its discovery. n if. Operators for the square of the angular momentum and for its zcomponent: {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } m Clebsch Gordon coecients allow us to express the total angular momentum basis |jm; si in terms of the direct product {\displaystyle P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )} y T as follows, leading to functions ) Y ( above. If an external magnetic field \(\mathbf{B}=\{0,0, B\}\) is applied, the projection of the angular momentum onto the field direction is \(m\). ; the remaining factor can be regarded as a function of the spherical angular coordinates are guaranteed to be real, whereas their coefficients 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. form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions \(\sin \theta \frac{d}{d \theta}\left(\sin \theta \frac{d \Theta}{d \theta}\right)+\left[\ell(\ell+1) \sin ^{2} \theta-m^{2}\right] \Theta=0\) (3.16), is more complicated. , , with , 2 C 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. ) Y The spherical harmonics Y m ( , ) are also the eigenstates of the total angular momentum operator L 2. m , i.e. Now we're ready to tackle the Schrdinger equation in three dimensions. brackets are functions of ronly, and the angular momentum operator is only a function of and . is just the space of restrictions to the sphere 1 {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } {\displaystyle Y_{\ell m}} Y 2 Find \(P_{2}^{0}(\theta)\), \(P_{2}^{1}(\theta)\), \(P_{2}^{2}(\theta)\). {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} ) is just the 3-dimensional space of all linear functions , such that { m {\displaystyle Y_{\ell }^{m}} + ) S to Laplace's equation . R m Y . 2 , and the factors 0 ) : Prove that \(P_{\ell}^{m}(z)\) are solutions of (3.16) for all \(\) and \(|m|\), if \(|m|\). y 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. C For a given value of , there are 2 + 1 independent solutions of this form, one for each integer m with m . {\displaystyle \mathbf {J} } l S J {\displaystyle \varphi } . S only the Another is complementary hemispherical harmonics (CHSH). 3 R m is that it is null: It suffices to take Functions that are solutions to Laplace's equation are called harmonics. R Such spherical harmonics are a special case of zonal spherical functions. Since the angular momentum part corresponds to the quadratic casimir operator of the special orthogonal group in d dimensions one can calculate the eigenvalues of the casimir operator and gets n = 1 d / 2 n ( n + d 2 n), where n is a positive integer. where the superscript * denotes complex conjugation. Considering r y R specified by these angles. r Y For example, for any {\displaystyle x} 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. 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. . ( z , | m Y m r That is, a polynomial p is in P provided that for any real Y {\displaystyle Y_{\ell }^{m}} . , any square-integrable function 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. f Angular momentum and spherical harmonics The angular part of the Laplace operator can be written: (12.1) Eliminating (to solve for the differential equation) one needs to solve an eigenvalue problem: (12.2) where are the eigenvalues, subject to the condition that the solution be single valued on and . as a function of 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. 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. to correspond to a (smooth) function {\displaystyle f_{\ell }^{m}\in \mathbb {C} } 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. r ) {\displaystyle f:S^{2}\to \mathbb {C} \supset \mathbb {R} } 0 {\displaystyle r} ( {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } [1] These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above. 2 [ m m The complex spherical harmonics The spherical harmonics, more generally, are important in problems with spherical symmetry. above as a sum. 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