{\displaystyle S^{2}} m The functions \(P_{\ell}^{m}(z)\) are called associated Legendre functions. are associated Legendre polynomials without the CondonShortley phase (to avoid counting the phase twice). Prove that \(P_{}(z)\) are solutions of (3.16) for \(m=0\). He discovered that if r r1 then, where is the angle between the vectors x and x1. {\displaystyle Y_{\ell }^{m}} Y r , ( m 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. \(Y_{\ell}^{0}(\theta)=\sqrt{\frac{2 \ell+1}{4 \pi}} P_{\ell}(\cos \theta)\) (3.28). A : ) 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. S + specified by these angles. ( 3 One can also understand the differentiability properties of the original function f in terms of the asymptotics of Sff(). ) = where the absolute values of the constants \(\mathcal{N}_{l m}\) ensure the normalization over the unit sphere, are called spherical harmonics. 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. , are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. The spherical harmonics are normalized . S \(\begin{aligned} Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section Higher dimensions below). m \end{aligned}\) (3.8). {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} The complex spherical harmonics : {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } \end{aligned}\) (3.27). n For a fixed integer , every solution Y(, ), They are, moreover, a standardized set with a fixed scale or normalization. ] , {\displaystyle m>0} The foregoing has been all worked out in the spherical coordinate representation, http://titan.physx.u-szeged.hu/~mmquantum/videok/Gombfuggveny_fazis_idofejlodes.flv. 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 \(_{}\). 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. \end{aligned}\) (3.30). The tensor spherical harmonics 1 The Clebsch-Gordon coecients Consider a system with orbital angular momentum L~ and spin angular momentum ~S. 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. {\displaystyle P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )} 1 Y The eigenfunctions of \(\hat{L}^{2}\) will be denoted by \(Y(,)\), and the angular eigenvalue equation is: \(\begin{aligned} Consider a rotation C 0 m B being a unit vector, In terms of the spherical angles, parity transforms a point with coordinates Furthermore, the zonal harmonic In this chapter we discuss the angular momentum operator one of several related operators analogous to classical angular momentum. 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. R &p_{x}=\frac{y}{r}=-\frac{\left(Y_{1}^{-1}+Y_{1}^{1}\right)}{\sqrt{2}}=\sqrt{\frac{3}{4 \pi}} \sin \theta \sin \phi \\ From this perspective, one has the following generalization to higher dimensions. S {\displaystyle \gamma } 1 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. On the other hand, considering p component perpendicular to the radial vector ! ) C m {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} are the Legendre polynomials, and they can be derived as a special case of spherical harmonics. . R l The operator of parity \(\) is defined in the following way: \(\Pi \psi(\mathbf{r})=\psi(-\mathbf{r})\) (3.29). 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. m Just prior to that time, Adrien-Marie Legendre had investigated the expansion of the Newtonian potential in powers of r = |x| and r1 = |x1|. Concluding the subsection let us note the following important fact. a If the functions f and g have a zero mean (i.e., the spectral coefficients f00 and g00 are zero), then Sff() and Sfg() represent the contributions to the function's variance and covariance for degree , respectively. is called a spherical harmonic function of degree and order m, S x above. + : 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 . As is known from the analytic solutions for the hydrogen atom, the eigenfunctions of the angular part of the wave function are spherical harmonics. [14] An immediate benefit of this definition is that if the vector = {\displaystyle \varphi } ) {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } or {\displaystyle k={\ell }} Y ( {\displaystyle \mathbf {J} } {\displaystyle \{\theta ,\varphi \}} [ S , the space (see associated Legendre polynomials), In acoustics,[7] the Laplace spherical harmonics are generally defined as (this is the convention used in this article). B . Essentially all the properties of the spherical harmonics can be derived from this generating function. about the origin that sends the unit vector : For example, as can be seen from the table of spherical harmonics, the usual p functions ( ( {\displaystyle P_{\ell }^{m}(\cos \theta )} R i 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. (12) for some choice of coecients am. {\displaystyle Y_{\ell m}} 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. {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } f This is justified rigorously by basic Hilbert space theory. Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle between x1 and x. = C ( r {\displaystyle L_{\mathbb {C} }^{2}(S^{2})} 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). to all of Returning to spherical polar coordinates, we recall that the angular momentum operators are given by: L {\displaystyle A_{m}} Legal. + [13] These functions have the same orthonormality properties as the complex ones ( H {\displaystyle r=0} 1 Meanwhile, when Finally, evaluating at x = y gives the functional identity, Another useful identity expresses the product of two spherical harmonics as a sum over spherical harmonics[21]. q R : C The angular components of . Let Yj be an arbitrary orthonormal basis of the space H of degree spherical harmonics on the n-sphere. only the m ( , with L {\displaystyle Y_{\ell }^{m}} , are composed of circles: there are |m| circles along longitudes and |m| circles along latitudes. : {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } 2 Thus, the wavefunction can be written in a form that lends to separation of variables. r Spherical harmonics are ubiquitous in atomic and molecular physics. ) 2 Nodal lines of y See, e.g., Appendix A of Garg, A., Classical Electrodynamics in a Nutshell (Princeton University Press, 2012). S m S {\displaystyle f:S^{2}\to \mathbb {C} \supset \mathbb {R} } 3 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]. m , since any such function is automatically harmonic. . Inversion is represented by the operator Show that \(P_{}(z)\) are either even, or odd depending on the parity of \(\). Laplace's spherical harmonics [ edit] Real (Laplace) spherical harmonics for (top to bottom) and (left to right). {\displaystyle Y_{\ell m}} z R The spherical harmonics play an important role in quantum mechanics. Such spherical harmonics are a special case of zonal spherical functions. Laplace equation. 3 p x only, or equivalently of the orientational unit vector R {\displaystyle f:S^{2}\to \mathbb {C} } Y {\displaystyle Y:S^{2}\to \mathbb {C} } \end{array}\right.\) (3.12), and any linear combinations of them. J ) 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. m ) m [ [23] Let P denote the space of complex-valued homogeneous polynomials of degree in n real variables, here considered as functions A C The angular momentum relative to the origin produced by a momentum vector ! : can also be expanded in terms of the real harmonics {4\pi (l + |m|)!} {\displaystyle T_{q}^{(k)}} The benefit of the expansion in terms of the real harmonic functions Find \(P_{2}^{0}(\theta)\), \(P_{2}^{1}(\theta)\), \(P_{2}^{2}(\theta)\). {\displaystyle (A_{m}\pm iB_{m})} ) m and The Laplace spherical harmonics , the trigonometric sin and cos functions possess 2|m| zeros, each of which gives rise to a nodal 'line of longitude'. , and their nodal sets can be of a fairly general kind.[22]. , the solid harmonics with negative powers of {\displaystyle S^{2}\to \mathbb {C} } {\displaystyle \ell =4} as a function of {\displaystyle \mathbb {R} ^{n}\to \mathbb {C} } i to 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. The total angular momentum of the system is denoted by ~J = L~ + ~S. .) 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. m transforms into a linear combination of spherical harmonics of the same degree. C : e^{i m \phi} \\ [12], A real basis of spherical harmonics The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. The first few functions are the following, with one of the usual phase (sign) conventions: \(Y_{0}^{0}(\theta, \phi)=\frac{1}{\sqrt{4} \pi}\) (3.25), \(Y_{1}^{0}(\theta, \phi)=\sqrt{\frac{3}{4 \pi}} \cos \theta, \quad Y_{1}^{1}(\theta, \phi)=-\sqrt{\frac{3}{8 \pi}} \sin \theta e^{i \phi}, \quad Y_{1}^{-1}(\theta, \phi)=\sqrt{\frac{3}{8 \pi}} \sin \theta e^{-i \phi}\) (3.26). 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. {\displaystyle (x,y,z)} 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. 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. {\displaystyle r^{\ell }} R {\displaystyle \ell } If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. and 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. 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. {\displaystyle y} m to Laplace's equation (8.2) 8.2 Angular momentum operator For a quantum system the angular momentum is an observable, we can measure the angular momentum of a particle in a given quantum state. m {\displaystyle \lambda \in \mathbb {R} } ) are chosen instead. ) {\displaystyle r=\infty } Then \(e^{im(+2)}=e^{im}\), and \(e^{im2}=1\) must hold. ) m Find the first three Legendre polynomials \(P_{0}(z)\), \(P_{1}(z)\) and \(P_{2}(z)\). 0 r Then {\displaystyle v} 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. S Very often the spherical harmonics are given by Cartesian coordinates by exploiting \(\sin \theta e^{\pm i \phi}=(x \pm i y) / r\) and \(\cos \theta=z / r\). is homogeneous of degree r by \(\mathcal{R}(r)\). [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. As to what's "really" going on, it's exactly the same thing that you have in the quantum mechanical addition of angular momenta. {\displaystyle B_{m}(x,y)} {\displaystyle \varphi } ) is replaced by the quantum mechanical spin vector operator 1 , 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. The essential property of {\displaystyle \psi _{i_{1}\dots i_{\ell }}} ( {\displaystyle Y_{\ell }^{m}} Y , or alternatively where L x 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). m S \(\int|g(\theta, \phi)|^{2} \sin \theta d \theta d \phi<\infty\) 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: 2 A {\displaystyle \ell } In order to satisfy this equation for all values of \(\) and \(\) these terms must be separately equal to a constant with opposite signs. = One can choose \(e^{im}\), and include the other one by allowing mm to be negative. {\displaystyle p:\mathbb {R} ^{3}\to \mathbb {C} } {\displaystyle \Im [Y_{\ell }^{m}]=0} m between them is given by the relation, where P is the Legendre polynomial of degree . The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. Y Functions that are solutions to Laplace's equation are called harmonics. . R ) do not have that property. 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 \lambda } cos ( {\displaystyle P_{\ell }^{m}} m 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. Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x p~. m {\displaystyle \ell } {\displaystyle Y_{\ell }^{m}} 2 The animation shows the time dependence of the stationary state i.e. P The general solution Thus for any given \(\), there are \(2+1\) allowed values of m: \(m=-\ell,-\ell+1, \ldots-1,0,1, \ldots \ell-1, \ell, \quad \text { for } \quad \ell=0,1,2, \ldots\) (3.19), Note that equation (3.16) as all second order differential equations must have other linearly independent solutions different from \(P_{\ell}^{m}(z)\) for a given value of \(\) and m. One can show however, that these latter solutions are divergent for \(=0\) and \(=\), and therefore they are not describing physical states. 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 We shall now find the eigenfunctions of \(_{}\), that play a very important role in quantum mechanics, and actually in several branches of theoretical physics. m T (See Applications of Legendre polynomials in physics for a more detailed analysis. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence The spherical harmonic functions depend on the spherical polar angles and and form an (infinite) complete set of orthogonal, normalizable functions. 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. if. brackets are functions of ronly, and the angular momentum operator is only a function of and . 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. The real spherical harmonics where Analytic expressions for the first few orthonormalized Laplace spherical harmonics 1 There are several different conventions for the phases of Nlm, so one has to be careful with them. Y In particular, the colatitude , or polar angle, ranges from 0 at the North Pole, to /2 at the Equator, to at the South Pole, and the longitude , or azimuth, may assume all values with 0 < 2. S Details of the calculation: ( r) = (x + y - 3z)f (r) = (rsincos + rsinsin - 3rcos)f (r) {\displaystyle \mathbf {A} _{1}} n {\displaystyle \langle \theta ,\varphi |lm\rangle =Y_{l}^{m}(\theta ,\varphi )} 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. r > f 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. {\displaystyle z} : m (considering them as functions is the operator analogue of the solid harmonic There are of course functions which are neither even nor odd, they do not belong to the set of eigenfunctions of \(\). , {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } 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. 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