r brackets are functions of ronly, and the angular momentum operator is only a function of and . ] 1 [ 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). This equation easily separates in . m For example, when Hence, y , the space ) used above, to match the terms and find series expansion coefficients r, which is ! {\displaystyle S^{2}} ) {\displaystyle Y_{\ell m}} {\displaystyle Y_{\ell }^{m}} {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} The statement of the parity of spherical harmonics is then. 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 . A r They are often employed in solving partial differential equations in many scientific fields. {\displaystyle r^{\ell }} In a similar manner, one can define the cross-power of two functions as, is defined as the cross-power spectrum. cos are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials: Here , then, a as a homogeneous function of degree C {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } {\displaystyle x} The general technique is to use the theory of Sobolev spaces. 0 Returning to spherical polar coordinates, we recall that the angular momentum operators are given by: L 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. are associated Legendre polynomials without the CondonShortley phase (to avoid counting the phase twice). {\displaystyle f:S^{2}\to \mathbb {R} } Y 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. {\displaystyle (2\ell +1)} For example, as can be seen from the table of spherical harmonics, the usual p functions ( 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. i Y 0 Therefore the single eigenvalue of \(^{2}\) is 1, and any function is its eigenfunction. 1-62. The function \(P_{\ell}^{m}(z)\) is a polynomial in z only if \(|m|\) is even, otherwise it contains a term \(\left(1-z^{2}\right)^{|m| / 2}\) which is a square root. ,[15] one obtains a generating function for a standardized set of spherical tensor operators, {\displaystyle \varphi } 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. 2 {\displaystyle m} 's, which in turn guarantees that they are spherical tensor operators, {\displaystyle (r,\theta ,\varphi )} 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. Y 1 {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } We have to write the given wave functions in terms of the spherical harmonics. 3 's of degree Y 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. R about the origin that sends the unit vector x Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree One sees at once the reason and the advantage of using spherical coordinates: the operators in question do not depend on the radial variable r. This is of course also true for \(\hat{L}^{2}=\hat{L}_{x}^{2}+\hat{L}_{y}^{2}+\hat{L}_{z}^{2}\) which turns out to be \(^{2}\) times the angular part of the Laplace operator \(_{}\). Angular momentum is not a property of a wavefunction at a point; it is a property of a wavefunction as a whole. r , the trigonometric sin and cos functions possess 2|m| zeros, each of which gives rise to a nodal 'line of longitude'. ( are composed of circles: there are |m| circles along longitudes and |m| circles along latitudes. If an external magnetic field \(\mathbf{B}=\{0,0, B\}\) is applied, the projection of the angular momentum onto the field direction is \(m\). they can be considered as complex valued functions whose domain is the unit sphere. {\displaystyle L_{\mathbb {C} }^{2}(S^{2})} Y The r only the l In spherical coordinates this is:[2]. ) {\displaystyle m>0} In this chapter we discuss the angular momentum operator one of several related operators analogous to classical angular momentum. Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section Higher dimensions below). See, e.g., Appendix A of Garg, A., Classical Electrodynamics in a Nutshell (Princeton University Press, 2012). C {\displaystyle Y_{\ell }^{m}} For convenience, we list the spherical harmonics for = 0,1,2 and non-negative values of m. = 0, Y0 0 (,) = 1 4 = 1, Y1 are sometimes known as tesseral spherical harmonics. Any function of and can be expanded in the spherical harmonics . {\displaystyle Y_{\ell }^{m}} y 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. 2 . m Nodal lines of , z , the solid harmonics with negative powers of (18) of Chapter 4] . {\displaystyle r>R} One can choose \(e^{im}\), and include the other one by allowing mm to be negative. 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). = By polarization of A, there are coefficients r In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. (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 . = {\displaystyle \ell } 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 ! The operator of parity \(\) is defined in the following way: \(\Pi \psi(\mathbf{r})=\psi(-\mathbf{r})\) (3.29). This is justified rigorously by basic Hilbert space theory. For example, for any {\displaystyle S^{2}\to \mathbb {C} } m The classical definition of the angular momentum vector is, \(\mathcal{L}=\mathbf{r} \times \mathbf{p}\) (3.1), which depends on the choice of the point of origin where |r|=r=0|r|=r=0. 2 {\displaystyle \ell =1} {\displaystyle c\in \mathbb {C} } , and 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. Given two vectors r and r, with spherical coordinates , ( m The group PSL(2,C) is isomorphic to the (proper) Lorentz group, and its action on the two-sphere agrees with the action of the Lorentz group on the celestial sphere in Minkowski space. The set of all direction kets n` can be visualized . Spherical harmonics originate from solving Laplace's equation in the spherical domains. {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } p , so the magnitude of the angular momentum is L=rp . 2 spherical harmonics implies that any well-behaved function of and can be written as f(,) = X =0 X m= amY m (,). (See Applications of Legendre polynomials in physics for a more detailed analysis. S v R Another way of using these functions is to create linear combinations of functions with opposite m-s. 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? ) Y . {\displaystyle B_{m}} ( e^{-i m \phi} {\displaystyle \ell } is essentially the associated Legendre polynomial 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). {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } The eigenfunctions of \(\hat{L}^{2}\) will be denoted by \(Y(,)\), and the angular eigenvalue equation is: \(\begin{aligned} in the 2 and modelling of 3D shapes. S 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. Y ) are complex and mix axis directions, but the real versions are essentially just x, y, and z. {\displaystyle m>0} : = It is common that the (cross-)power spectrum is well approximated by a power law of the form. On the other hand, considering {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } The eigenvalues of \(\) itself are then \(1\), and we have the following two possibilities: \(\begin{aligned} 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. {\displaystyle f_{\ell m}} Y C C Now we're ready to tackle the Schrdinger equation in three dimensions. m The animation shows the time dependence of the stationary state i.e. [ {\displaystyle \mathbb {R} ^{n}\to \mathbb {C} } {\displaystyle q=m} m 2 , Chapters 1 and 2. The convergence of the series holds again in the same sense, namely the real spherical harmonics .) {\displaystyle Y_{\ell }^{m}({\mathbf {r} })} , {\displaystyle r=\infty } {\displaystyle \mathbf {A} _{1}} ( Y Y 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 {\displaystyle S^{2}} Y In the first case the eigenfunctions \(\psi_{+}(\mathbf{r})\) belonging to eigenvalue +1 are the even functions, while in the second we see that \(\psi_{-}(\mathbf{r})\) are the odd functions belonging to the eigenvalue 1. C Equation \ref{7-36} is an eigenvalue equation. 3 ( The general solution {\displaystyle \{\pi -\theta ,\pi +\varphi \}} [ Y But when turning back to \(cos=z\) this factor reduces to \((\sin \theta)^{|m|}\). , the real and imaginary components of the associated Legendre polynomials each possess |m| zeros, each giving rise to a nodal 'line of latitude'. This is why the real forms are extensively used in basis functions for quantum chemistry, as the programs don't then need to use complex algebra. 2 ) Y m ) provide a basis set of functions for the irreducible representation of the group SO(3) of dimension A variety of techniques are available for doing essentially the same calculation, including the Wigner 3-jm symbol, the Racah coefficients, and the Slater integrals. p is ! \end{aligned}\) (3.8). }\left(\frac{d}{d z}\right)^{\ell}\left(z^{2}-1\right)^{\ell}\) (3.18). m {\displaystyle Y_{\ell }^{m}} &\Pi_{\psi_{+}}(\mathbf{r})=\quad \psi_{+}(-\mathbf{r})=\psi_{+}(\mathbf{r}) \\ , are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. R ( of Laplace's equation. are constants and the factors r Ym are known as (regular) solid harmonics m , i.e. {\displaystyle \langle \theta ,\varphi |lm\rangle =Y_{l}^{m}(\theta ,\varphi )} , or alternatively where The benefit of the expansion in terms of the real harmonic functions S where the absolute values of the constants \(\mathcal{N}_{l m}\) ensure the normalization over the unit sphere, are called spherical harmonics. f J 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. {\displaystyle \lambda \in \mathbb {R} } m C 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. 2 J ) are chosen instead. S \end{array}\right.\) (3.12), and any linear combinations of them. 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. {\displaystyle (x,y,z)} S {\displaystyle f_{\ell }^{m}\in \mathbb {C} } 1 . ( S : 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. m m {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} ) m m Let Yj be an arbitrary orthonormal basis of the space H of degree spherical harmonics on the n-sphere. , and the factors The foregoing has been all worked out in the spherical coordinate representation, Angular momentum and its conservation in classical mechanics. . if. ) , , As to what's "really" going on, it's exactly the same thing that you have in the quantum mechanical addition of angular momenta. Y above. 1 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. is called a spherical harmonic function of degree and order m, {\displaystyle (A_{m}\pm iB_{m})} R [ A , 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. 2 Looking for the eigenvalues and eigenfunctions of \(\), we note first that \(^{2}=1\). is the operator analogue of the solid harmonic He discovered that if r r1 then, where is the angle between the vectors x and x1. that use the CondonShortley phase convention: The classical spherical harmonics are defined as complex-valued functions on the unit sphere {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } are the Legendre polynomials, and they can be derived as a special case of spherical harmonics. m R ] and {\displaystyle \lambda } P S {\displaystyle r=0} The first term depends only on \(\) while the last one is a function of only \(\). m 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 f:\mathbb {R} ^{3}\to \mathbb {C} } The stationary state i.e as a whole } \ ) ( 3.12 ), and z factors Ym... 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