Legal. p is ! J {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } C The convergence of the series holds again in the same sense, namely the real spherical harmonics as real parameters. R as a homogeneous function of degree are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials: Here R {\displaystyle S^{n-1}\to \mathbb {C} } S C m When < 0, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. [23] Let P denote the space of complex-valued homogeneous polynomials of degree in n real variables, here considered as functions ) The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions Show that \(P_{}(z)\) are either even, or odd depending on the parity of \(\). R As to what's "really" going on, it's exactly the same thing that you have in the quantum mechanical addition of angular momenta. m [18], In particular, when x = y, this gives Unsld's theorem[19], In the expansion (1), the left-hand side P(xy) is a constant multiple of the degree zonal spherical harmonic. {\displaystyle k={\ell }} = R + : More general spherical harmonics of degree are not necessarily those of the Laplace basis S {\displaystyle \{\pi -\theta ,\pi +\varphi \}} 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 f as follows, leading to functions Hence, Y {4\pi (l + |m|)!} m ( and modelling of 3D shapes. . transforms into a linear combination of spherical harmonics of the same degree. , 1 {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} Statements relating the growth of the Sff() to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. . Y \(\begin{aligned} 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. 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? Laplace equation. 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. 2 < = to Laplace's equation Y . ,[15] one obtains a generating function for a standardized set of spherical tensor operators, C The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. In 1782, Pierre-Simon de Laplace had, in his Mcanique Cleste, determined that the gravitational potential T m f r {\displaystyle \mathbf {r} '} 0 f Y 1-62. 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. in their expansion in terms of the The spherical harmonic functions depend on the spherical polar angles and and form an (infinite) complete set of orthogonal, normalizable functions. r! 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 . Parity continues to hold for real spherical harmonics, and for spherical harmonics in higher dimensions: applying a point reflection to a spherical harmonic of degree changes the sign by a factor of (1). l {\displaystyle r=0} http://en.Wikipedia.org/wiki/File:Legendrepolynomials6.svg. ) = {\displaystyle \gamma } This could be achieved by expansion of functions in series of trigonometric functions. > {\displaystyle \varphi } i {\displaystyle \ell } (1) From this denition and the canonical commutation relation between the po-sition and momentum operators, it is easy to verify the commutation relation among the components of the angular momentum . can also be expanded in terms of the real harmonics {\displaystyle (r',\theta ',\varphi ')} 2 ) The statement of the parity of spherical harmonics is then. R 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. 2 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). 3 S 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 eigenfunctions of the orbital angular momentum operator, the spherical harmonics Reasoning: The common eigenfunctions of L 2 and L z are the spherical harmonics. r http://en.Wikipedia.org/wiki/Spherical_harmonics. Prove that \(P_{}(z)\) are solutions of (3.16) for \(m=0\). Z c Y The eigenfunctions of \(\hat{L}^{2}\) will be denoted by \(Y(,)\), and the angular eigenvalue equation is: \(\begin{aligned} He discovered that if r r1 then, where is the angle between the vectors x and x1. As . Angular momentum is not a property of a wavefunction at a point; it is a property of a wavefunction as a whole. {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } r Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. of Laplace's equation. The 19th century development of Fourier series made possible the solution of a wide variety of physical problems in rectangular domains, such as the solution of the heat equation and wave equation. r For the other cases, the functions checker the sphere, and they are referred to as tesseral. The Note that the angular momentum is itself a vector. 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. 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. The spherical harmonics have definite parity. {\displaystyle q=m} 2 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. \(\int|g(\theta, \phi)|^{2} \sin \theta d \theta d \phi<\infty\) can be expanded in terms of the \(Y_{\ell}^{m}(\theta, \phi)\)): \(g(\theta, \phi)=\sum_{\ell=0}^{\infty} \sum_{m=-\ell}^{\ell} c_{\ell m} Y_{\ell}^{m}(\theta, \phi)\) (3.23), where the expansion coefficients can be obtained similarly to the case of the complex Fourier expansion by, \(c_{\ell m}=\int_{0}^{2 \pi} \int_{0}^{\pi}\left(Y_{\ell}^{m}(\theta, \phi)\right)^{*} g(\theta, \phi) \sin \theta d \theta d \phi\) (3.24), If you are interested in the topic Spherical harmonics in more details check out the Wikipedia link below: 1 In that case, one needs to expand the solution of known regions in Laurent series (about : } Equation \ref{7-36} is an eigenvalue equation. {\displaystyle r^{\ell }Y_{\ell }^{m}(\mathbf {r} /r)} y Considering {\displaystyle S^{2}} m as a function of , the degree zonal harmonic corresponding to the unit vector x, decomposes as[20]. 1 Nodal lines of For example, for any z 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. {\displaystyle \mathbf {a} } 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. m In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. {\displaystyle f_{\ell }^{m}\in \mathbb {C} } S Y {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } 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. C We have to write the given wave functions in terms of the spherical harmonics. Y You are all familiar, at some level, with spherical harmonics, from angular momentum in quantum mechanics. . ( More generally, the analogous statements hold in higher dimensions: the space H of spherical harmonics on the n-sphere is the irreducible representation of SO(n+1) corresponding to the traceless symmetric -tensors. For angular momentum operators: 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). Just as in one dimension the eigenfunctions of d 2 / d x 2 have the spatial dependence of the eigenmodes of a vibrating string, the spherical harmonics have the spatial dependence of the eigenmodes of a vibrating spherical . The complex spherical harmonics {\displaystyle \lambda \in \mathbb {R} } 2 S is that for real functions Find \(P_{2}^{0}(\theta)\), \(P_{2}^{1}(\theta)\), \(P_{2}^{2}(\theta)\). : {\displaystyle f:S^{2}\to \mathbb {C} } the one containing the time dependent factor \(e_{it/}\) as well given by the function \(Y_{1}^{3}(,)\). \(\sin \theta \frac{d}{d \theta}\left(\sin \theta \frac{d \Theta}{d \theta}\right)+\left[\ell(\ell+1) \sin ^{2} \theta-m^{2}\right] \Theta=0\) (3.16), is more complicated. R ) 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 \((1)^{}\). are essentially {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {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 . where \(P_{}(z)\) is the \(\)-th Legendre polynomial, defined by the following formula, (called the Rodrigues formula): \(P_{\ell}(z):=\frac{1}{2^{\ell} \ell ! r {\displaystyle m>0} S Y B terms (sines) are included: The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation. 2 l In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after Giulio Racah. m In 3D computer graphics, spherical harmonics play a role in a wide variety of topics including indirect lighting (ambient occlusion, global illumination, precomputed radiance transfer, etc.) 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 ). 1 ( 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. (Here the scalar field is understood to be complex, i.e. &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 \\ are associated Legendre polynomials without the CondonShortley phase (to avoid counting the phase twice). P We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Y ) For central forces the index n is the orbital angular momentum [and n(n+ 1) is the eigenvalue of L2], thus linking parity and or-bital angular momentum. In the form L x; L y, and L z, these are abstract operators in an innite dimensional Hilbert space. R 0 The solid harmonics were homogeneous polynomial solutions 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, \(\begin{aligned} One can determine the number of nodal lines of each type by counting the number of zeros of P 3 = .) The three Cartesian components of the angular momentum are: L x = yp z zp y,L y = zp x xp z,L z = xp y yp x. {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } 2 S To make full use of rotational symmetry and angular momentum, we will restrict our attention to spherically symmetric potentials, \begin {aligned} V (\vec {r}) = V (r). The 3-D wave equation; spherical harmonics. }\left(\frac{d}{d z}\right)^{\ell}\left(z^{2}-1\right)^{\ell}\) (3.18). is that it is null: It suffices to take , we have a 5-dimensional space: For any 3 From this perspective, one has the following generalization to higher dimensions. 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. are composed of circles: there are |m| circles along longitudes and |m| circles along latitudes. Y 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. , one has. r The functions The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry. ( 3 {\displaystyle Z_{\mathbf {x} }^{(\ell )}({\mathbf {y} })} Then When = 0, the spectrum is "white" as each degree possesses equal power. m + This equation easily separates in . {\displaystyle \Re [Y_{\ell }^{m}]=0} The spinor spherical harmonics are the natural spinor analog of the vector spherical harmonics. m that obey Laplace's equation. {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } 3 and , and their nodal sets can be of a fairly general kind.[22]. {\displaystyle L_{\mathbb {C} }^{2}(S^{2})} 1 {\displaystyle Y:S^{2}\to \mathbb {C} } : 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 : 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. . Recalling that the spherical harmonics are eigenfunctions of the angular momentum operator: (r; ;) = R(r)Ym l ( ;) SeparationofVariables L^2Ym l ( ;) = h2l . S ( [ , By definition, (382) where is an integer. q Prove that \(P_{\ell}^{m}(z)\) are solutions of (3.16) for all \(\) and \(|m|\), if \(|m|\). m R Y m {\displaystyle (-1)^{m}} about the origin that sends the unit vector The Laplace spherical harmonics m symmetric on the indices, uniquely determined by the requirement. {\displaystyle \mathbb {R} ^{3}\setminus \{\mathbf {0} \}\to \mathbb {C} } On the other hand, considering if. [14] An immediate benefit of this definition is that if the vector See here for a list of real spherical harmonics up to and including Calculate the following operations on the spherical harmonics: (a.) In particular, if Sff() decays faster than any rational function of as , then f is infinitely differentiable. 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. Let us also note that the \(m=0\) functions do not depend on \(\), and they are proportional to the Legendre polynomials in \(cos\). f m Such an expansion is valid in the ball. r ) . {\displaystyle P_{i}:[-1,1]\to \mathbb {R} } With respect to this group, the sphere is equivalent to the usual Riemann sphere. v 3 They will be functions of \(0 \leq \theta \leq \pi\) and \(0 \leq \phi<2 \pi\), i.e. , S The total angular momentum of the system is denoted by ~J = L~ + ~S. 2 C {\displaystyle \{\theta ,\varphi \}} ) \end{aligned}\) (3.30). ) are complex and mix axis directions, but the real versions are essentially just x, y, and z. r Y {\displaystyle f_{\ell }^{m}} {\displaystyle c\in \mathbb {C} } 2 are guaranteed to be real, whereas their coefficients {\displaystyle f:S^{2}\to \mathbb {R} } {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } {\displaystyle B_{m}} R One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of R : , the solid harmonics with negative powers of 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. {\displaystyle \mathbf {J} } {\displaystyle \mathbf {H} _{\ell }} The result of acting by the parity on a function is the mirror image of the original function with respect to the origin. {\displaystyle Y_{\ell }^{m}} {\displaystyle \mathbf {r} } [ 3 1 m Find the first three Legendre polynomials \(P_{0}(z)\), \(P_{1}(z)\) and \(P_{2}(z)\). While the standard spherical harmonics are a basis for the angular momentum operator, the spinor spherical harmonics are a basis for the total angular momentum operator (angular momentum plus spin ). There are several different conventions for the phases of \(\mathcal{N}_{l m}\), so one has to be careful with them. = provide a basis set of functions for the irreducible representation of the group SO(3) of dimension S x Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x 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. . We demonstrate this with the example of the p functions. . S Functions that are solutions to Laplace's equation are called harmonics. Spherical Harmonics, and Bessel Functions Physics 212 2010, Electricity and Magnetism Michael Dine Department of Physics . R , {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } Spherical harmonics are ubiquitous in atomic and molecular physics. In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum[4]. , are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. {\displaystyle \mathbf {r} } Indeed, rotations act on the two-dimensional sphere, and thus also on H by function composition, The elements of H arise as the restrictions to the sphere of elements of A: harmonic polynomials homogeneous of degree on three-dimensional Euclidean space R3. 2 . Y {\displaystyle P_{\ell }^{m}(\cos \theta )} 2 y = 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. and order 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. 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. : to 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. (See Applications of Legendre polynomials in physics for a more detailed analysis. These angular solutions ) By using the results of the previous subsections prove the validity of Eq. The half-integer values do not give vanishing radial solutions. R , of the eigenvalue problem. Y Analytic expressions for the first few orthonormalized Laplace spherical harmonics p , so the magnitude of the angular momentum is L=rp . They are, moreover, a standardized set with a fixed scale or normalization. : { ] 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. 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