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. : [12], A real basis of spherical harmonics ( i S For convenience, we list the spherical harmonics for = 0,1,2 and non-negative values of m. = 0, Y0 0 (,) = 1 4 = 1, Y1 {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } and another of 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. m : &p_{z}=\frac{z}{r}=Y_{1}^{0}=\sqrt{\frac{3}{4 \pi}} \cos \theta e Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section Higher dimensions below). The spherical harmonics Y m ( , ) are also the eigenstates of the total angular momentum operator L 2. R Under this operation, a spherical harmonic of degree can be defined in terms of their complex analogues S , Functions that are solutions to Laplace's equation are called harmonics. m That is, they are either even or odd with respect to inversion about the origin. : m S r m in the , one has. i In 1782, Pierre-Simon de Laplace had, in his Mcanique Cleste, determined that the gravitational potential A specific set of spherical harmonics, denoted m This parity property will be conrmed by the series The figures show the three-dimensional polar diagrams of the spherical harmonics. 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 ). Y C {\displaystyle r=0} This was a boon for problems possessing spherical symmetry, such as those of celestial mechanics originally studied by Laplace and Legendre. 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. \end {aligned} V (r) = V (r). 2 r {\displaystyle f_{\ell }^{m}\in \mathbb {C} } specified by these angles. {\displaystyle \mathbb {R} ^{3}} 1 C ) In other words, any well-behaved function of and can be represented as a superposition of spherical harmonics. 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. Y The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence . {\displaystyle (-1)^{m}} Y , the real and imaginary components of the associated Legendre polynomials each possess |m| zeros, each giving rise to a nodal 'line of latitude'. {4\pi (l + |m|)!} Recalling that the spherical harmonics are eigenfunctions of the angular momentum operator: (r; ;) = R(r)Ym l ( ;) SeparationofVariables L^2Ym l ( ;) = h2l . m When = 0, the spectrum is "white" as each degree possesses equal power. &\Pi_{\psi_{-}}(\mathbf{r})=\quad \psi_{-}(-\mathbf{r})=-\psi_{-}(\mathbf{r}) only the The spherical harmonics are normalized . {\displaystyle (A_{m}\pm iB_{m})} Y i = ) used above, to match the terms and find series expansion coefficients Furthermore, the zonal harmonic m f = R The statement of the parity of spherical harmonics is then. 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. C , m L 2 Y 21 Y [13] These functions have the same orthonormality properties as the complex ones , The eigenvalues of \(\) itself are then \(1\), and we have the following two possibilities: \(\begin{aligned} Y 2 {\displaystyle f_{\ell m}} x {\displaystyle S^{2}\to \mathbb {C} } \left(\partial_{\theta \theta}^{2}+\cot \theta \partial_{\theta}+\frac{1}{\sin ^{2} \theta} \partial_{\phi \phi}^{2}\right) Y(\theta, \phi) &=-\ell(\ell+1) Y(\theta, \phi) 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. {\displaystyle Y_{\ell }^{m}} S S . 3 For example, for any ] In that case, one needs to expand the solution of known regions in Laurent series (about R {\displaystyle r^{\ell }Y_{\ell }^{m}(\mathbf {r} /r)} 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. P {\displaystyle \theta } m For a given value of , there are 2 + 1 independent solutions of this form, one for each integer m with m . q ) transforms into a linear combination of spherical harmonics of the same degree. Laplace's spherical harmonics [ edit] Real (Laplace) spherical harmonics for (top to bottom) and (left to right). Just prior to that time, Adrien-Marie Legendre had investigated the expansion of the Newtonian potential in powers of r = |x| and r1 = |x1|. L z Y 21 (b.) ] There are of course functions which are neither even nor odd, they do not belong to the set of eigenfunctions of \(\). {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } = 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. We have to write the given wave functions in terms of the spherical harmonics. It can be shown that all of the above normalized spherical harmonic functions satisfy. between them is given by the relation, where P is the Legendre polynomial of degree . 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). in , A Representation of Angular Momentum Operators We would like to have matrix operators for the angular momentum operators L x; L y, and L z. 1 , commonly referred to as the CondonShortley phase in the quantum mechanical literature. C > Y / Considering and Spherical Harmonics 11.1 Introduction Legendre polynomials appear in many different mathematical and physical situations: . { 2 B Hence, m \end{aligned}\) (3.6). R to inside three-dimensional Euclidean space r C Using the expressions for Thus, p2=p r 2+p 2 can be written as follows: p2=pr 2+ L2 r2. 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 ( Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle between x1 and x. S : {\displaystyle \ell } 1 The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. The operator of parity \(\) is defined in the following way: \(\Pi \psi(\mathbf{r})=\psi(-\mathbf{r})\) (3.29). Y y : can be visualized by considering their "nodal lines", that is, the set of points on the sphere where C ) 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. 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]. When = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. The foregoing has been all worked out in the spherical coordinate representation, {\displaystyle p:\mathbb {R} ^{3}\to \mathbb {C} } {\displaystyle T_{q}^{(k)}} from the above-mentioned polynomial of degree m Remember from chapter 2 that a subspace is a specic subset of a general complex linear vector space. , and the factors m \end{array}\right.\) (3.12), and any linear combinations of them. [23] Let P denote the space of complex-valued homogeneous polynomials of degree in n real variables, here considered as functions This operator thus must be the operator for the square of the angular momentum. } {\displaystyle f:S^{2}\to \mathbb {C} \supset \mathbb {R} } the one containing the time dependent factor \(e_{it/}\) as well given by the function \(Y_{1}^{3}(,)\). Y to (the irregular solid harmonics 1 [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. (18) of Chapter 4] . {\displaystyle L_{\mathbb {R} }^{2}(S^{2})} 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. m They occur in . Here the solution was assumed to have the special form Y(, ) = () (). m ( {\displaystyle \mathbf {A} _{\ell }} R r {\displaystyle \ell } , obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. C of Laplace's equation. Show that \(P_{}(z)\) are either even, or odd depending on the parity of \(\). S We consider the second one, and have: \(\frac{1}{\Phi} \frac{d^{2} \Phi}{d \phi^{2}}=-m^{2}\) (3.11), \(\Phi(\phi)=\left\{\begin{array}{l} The essential property of 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. x 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 condition on the order of growth of Sff() is related to the order of differentiability of f in the next section. that use the CondonShortley phase convention: The classical spherical harmonics are defined as complex-valued functions on the unit sphere are sometimes known as tesseral spherical harmonics. R The set of all direction kets n` can be visualized . z listed explicitly above we obtain: Using the equations above to form the real spherical harmonics, it is seen that for The spinor spherical harmonics are the natural spinor analog of the vector spherical harmonics. The angle-preserving symmetries of the two-sphere are described by the group of Mbius transformations PSL(2,C). {\displaystyle S^{2}\to \mathbb {C} } \(\int|g(\theta, \phi)|^{2} \sin \theta d \theta d \phi<\infty\) 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: ( {\displaystyle (x,y,z)} {\displaystyle Z_{\mathbf {x} }^{(\ell )}({\mathbf {y} })} 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. 3 y f The solutions, \(Y_{\ell}^{m}(\theta, \phi)=\mathcal{N}_{l m} P_{\ell}^{m}(\theta) e^{i m \phi}\) (3.20). [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). {\displaystyle (r',\theta ',\varphi ')} We demonstrate this with the example of the p functions. S , Y For example, as can be seen from the table of spherical harmonics, the usual p functions ( r . Find the first three Legendre polynomials \(P_{0}(z)\), \(P_{1}(z)\) and \(P_{2}(z)\). 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. The functions \(P_{\ell}^{m}(z)\) are called associated Legendre 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. S 1 ( In a similar manner, one can define the cross-power of two functions as, is defined as the cross-power spectrum. y = 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. Such spherical harmonics are a special case of zonal spherical functions. the formula, Several different normalizations are in common use for the Laplace spherical harmonic functions That is. {\displaystyle Z_{\mathbf {x} }^{(\ell )}} R brackets are functions of ronly, and the angular momentum operator is only a function of and . = 3 {\displaystyle \ell =2} m One can also understand the differentiability properties of the original function f in terms of the asymptotics of Sff(). ( The spherical harmonics are orthogonal functions, and are properly normalized with respect to integration over the entire solid angle: (381) The spherical harmonics also form a complete set for representing general functions of and . f The general technique is to use the theory of Sobolev spaces. {\displaystyle (2\ell +1)} ) , or alternatively where , the trigonometric sin and cos functions possess 2|m| zeros, each of which gives rise to a nodal 'line of longitude'. 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 . and order , we have a 5-dimensional space: For any ] R Y , 2 0 3 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. C For example, when Specifically, if, A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. 1 m Y The spherical harmonics are representations of functions of the full rotation group SO(3)[5]with rotational symmetry. to correspond to a (smooth) function are composed of circles: there are |m| circles along longitudes and |m| circles along latitudes. By definition, (382) where is an integer. {\displaystyle \mathbf {r} } : ) are constants and the factors r Ym are known as (regular) solid harmonics = 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? l r , and Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x p~. In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. m Y Analytic expressions for the first few orthonormalized Laplace spherical harmonics 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). {\displaystyle e^{\pm im\varphi }} Introduction to the Physics of Atoms, Molecules and Photons (Benedict), { "1.01:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.
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