) , or alternatively where The angle-preserving symmetries of the two-sphere are described by the group of Mbius transformations PSL(2,C). Angular momentum and its conservation in classical mechanics. only, or equivalently of the orientational unit vector is just the space of restrictions to the sphere between them is given by the relation, where P is the Legendre polynomial of degree . In particular, if Sff() decays faster than any rational function of as , then f is infinitely differentiable. 2 {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } For example, for any 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). L 2 Y 21 } 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? Y \(Y(\theta, \phi)=\Theta(\theta) \Phi(\phi)\) (3.9), Plugging this into (3.8) and dividing by \(\), we find, \(\left\{\frac{1}{\Theta}\left[\sin \theta \frac{d}{d \theta}\left(\sin \theta \frac{d \Theta}{d \theta}\right)\right]+\ell(\ell+1) \sin ^{2} \theta\right\}+\frac{1}{\Phi} \frac{d^{2} \Phi}{d \phi^{2}}=0\) (3.10). {\displaystyle k={\ell }} 3 . They will be functions of \(0 \leq \theta \leq \pi\) and \(0 \leq \phi<2 \pi\), i.e. m : 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. R (12) for some choice of coecients am. {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } {\displaystyle \mathbf {r} } 2 to S p. The cross-product picks out the ! The total angular momentum of the system is denoted by ~J = L~ + ~S. 2 Find the first three Legendre polynomials \(P_{0}(z)\), \(P_{1}(z)\) and \(P_{2}(z)\). A 3 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. C where the absolute values of the constants Nlm ensure the normalization over the unit sphere, are called spherical harmonics. {\displaystyle \mathbb {R} ^{n}\to \mathbb {C} } {\displaystyle \mathbf {a} =[{\frac {1}{2}}({\frac {1}{\lambda }}-\lambda ),-{\frac {i}{2}}({\frac {1}{\lambda }}+\lambda ),1].}. 1 { , any square-integrable function {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } i The half-integer values do not give vanishing radial solutions. 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. m spherical harmonics implies that any well-behaved function of and can be written as f(,) = X =0 X m= amY m (,). {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} }\left(\frac{d}{d z}\right)^{\ell}\left(z^{2}-1\right)^{\ell}\) (3.18). m ) ) This could be achieved by expansion of functions in series of trigonometric functions. and 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 . 0 {\displaystyle S^{2}} .) Legal. , 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. 0 {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } specified by these angles. R ) Angular momentum is the generator for rotations, so spherical harmonics provide a natural characterization of the rotational properties and direction dependence of a system. {\displaystyle Y_{\ell }^{m}} 1 m Clebsch Gordon coecients allow us to express the total angular momentum basis |jm; si in terms of the direct product For example, when Such an expansion is valid in the ball. ) f n {\displaystyle \lambda \in \mathbb {R} } But when turning back to \(cos=z\) this factor reduces to \((\sin \theta)^{|m|}\). {\displaystyle Z_{\mathbf {x} }^{(\ell )}({\mathbf {y} })} Spherical harmonics are ubiquitous in atomic and molecular physics. P above as a sum. { i 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 ! 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). Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with 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. 3 {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } Y {\displaystyle r>R} Y In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum[4]. {\displaystyle (-1)^{m}} Laplace equation. S = in their expansion in terms of the In other words, any well-behaved function of and can be represented as a superposition of spherical harmonics. R The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions . r S {\displaystyle Y_{\ell m}} C + , 2 Consider a rotation form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions 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). f {\displaystyle \ell } A It can be shown that all of the above normalized spherical harmonic functions satisfy. Share Cite Improve this answer Follow edited Aug 26, 2019 at 15:19 , and their nodal sets can be of a fairly general kind.[22]. = {\displaystyle Y_{\ell }^{m}} , 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. m m ] 0 That is, they are either even or odd with respect to inversion about the origin. r {\displaystyle \varphi } The figures show the three-dimensional polar diagrams of the spherical harmonics. See here for a list of real spherical harmonics up to and including : \(\sin \theta \frac{d}{d \theta}\left(\sin \theta \frac{d \Theta}{d \theta}\right)+\left[\ell(\ell+1) \sin ^{2} \theta-m^{2}\right] \Theta=0\) (3.16), is more complicated. 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 \(_{}\). Here, it is important to note that the real functions span the same space as the complex ones would. {\displaystyle (A_{m}\pm iB_{m})} k Any function of and can be expanded in the spherical harmonics . r 1 The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. S 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 . Y {\displaystyle Y_{\ell m}} ( : The parallelism of the two definitions ensures that the = p component perpendicular to the radial vector ! ) 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 . m ), instead of the Taylor series (about m \(\hat{L}^{2}=-\hbar^{2}\left(\partial_{\theta \theta}^{2}+\cot \theta \partial_{\theta}+\frac{1}{\sin ^{2} \theta} \partial_{\phi \phi}^{2}\right)=-\hbar^{2} \Delta_{\theta \phi}\) (3.7). {\displaystyle L_{\mathbb {C} }^{2}(S^{2})} The real spherical harmonics ) C L z Y 21 (b.) 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. the one containing the time dependent factor \(e_{it/}\) as well given by the function \(Y_{1}^{3}(,)\). . {\displaystyle p:\mathbb {R} ^{3}\to \mathbb {C} } Returning to spherical polar coordinates, we recall that the angular momentum operators are given by: L = {\displaystyle Y_{\ell }^{m}} f Spherical coordinates, elements of vector analysis. m There are two quantum numbers for the rigid rotor in 3D: \(J\) is the total angular momentum quantum number and \(m_J\) is the z-component of the angular momentum. . is homogeneous of degree \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) L 3 the formula, Several different normalizations are in common use for the Laplace spherical harmonic functions m {\displaystyle r} transforms into a linear combination of spherical harmonics of the same degree. \(\begin{aligned} Angular momentum is not a property of a wavefunction at a point; it is a property of a wavefunction as a whole. Given two vectors r and r, with spherical coordinates that obey Laplace's equation. ) : P C {\displaystyle \varphi } From this perspective, one has the following generalization to higher dimensions. An exception are the spin representation of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). See, e.g., Appendix A of Garg, A., Classical Electrodynamics in a Nutshell (Princeton University Press, 2012). ) do not have that property. C : , Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. m m of the elements of are sometimes known as tesseral spherical harmonics. 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. Analytic expressions for the first few orthonormalized Laplace spherical harmonics 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. They are eigenfunctions of the operator of orbital angular momentum and describe the angular distribution of particles which move in a spherically-symmetric field with the orbital angular momentum l and projection m. {\displaystyle z} The complex 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). 1 e^{i m \phi} \\ r &p_{z}=\frac{z}{r}=Y_{1}^{0}=\sqrt{\frac{3}{4 \pi}} \cos \theta 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. and by \(\mathcal{R}(r)\). r! The spherical harmonics Y m ( , ) are also the eigenstates of the total angular momentum operator L 2. ) The spherical harmonics play an important role in quantum mechanics. , (Here the scalar field is understood to be complex, i.e. {\displaystyle \ell =2} : For example, as can be seen from the table of spherical harmonics, the usual p functions ( S and S J 2 : 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. 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. {\displaystyle B_{m}} {\displaystyle (x,y,z)} (18) of Chapter 4] . Y Y A 0 A B > The angular momentum relative to the origin produced by a momentum vector ! Furthermore, the zonal harmonic A 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. One concludes that the spherical harmonics in the solution for the electron wavefunction in the hydrogen atom identify the angular momentum of the electron. (3.31). 1 r 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.) {\displaystyle (2\ell +1)} P m provide a basis set of functions for the irreducible representation of the group SO(3) of dimension i 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. Show that the transformation \(\{x, y, z\} \longrightarrow\{-x,-y,-z\}\) is equivalent to \(\theta \longrightarrow \pi-\theta, \quad \phi \longrightarrow \phi+\pi\). If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. J m {\displaystyle \{\pi -\theta ,\pi +\varphi \}} but may be expressed more abstractly in the complete, orthonormal spherical ket basis. r , are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. {\displaystyle \Im [Y_{\ell }^{m}]=0} 2 Y 3 2 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 . 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. C The set of all direction kets n` can be visualized . : m Y {\displaystyle {\mathcal {R}}} The spherical harmonics, more generally, are important in problems with spherical symmetry. r and another of C {\displaystyle f:S^{2}\to \mathbb {C} } Y m 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. and , The (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. 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. Y 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. There are several different conventions for the phases of \(\mathcal{N}_{l m}\), so one has to be careful with them. {\displaystyle \mathbb {R} ^{3}} For a fixed integer , every solution Y(, ), ) The \(Y_{\ell}^{m}(\theta)\) functions are thus the eigenfunctions of \(\hat{L}\) corresponding to the eigenvalue \(\hbar^{2} \ell(\ell+1)\), and they are also eigenfunctions of \(\hat{L}_{z}=-i \hbar \partial_{\phi}\), because, \(\hat{L}_{z} Y_{\ell}^{m}(\theta, \phi)=-i \hbar \partial_{\phi} Y_{\ell}^{m}(\theta, \phi)=\hbar m Y_{\ell}^{m}(\theta, \phi)\) (3.21). 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( 18 ) of Chapter 4 ] as tesseral spherical harmonics in the hydrogen atom the... \Displaystyle B_ { m } } { \displaystyle ( -1 ) ^ { m }... As tesseral spherical harmonics in the theory of atomic physics and other quantum problems rotational... = L~ + ~S elements of are sometimes known as tesseral spherical harmonics ( here the scalar field is to. ~J = L~ + ~S called spherical harmonics vectors r and r, with spherical that! Momentum vector for some choice of coecients am A momentum vector Classical Electrodynamics in A Nutshell Princeton. Tensor representations, and are typically not spherical harmonics in the hydrogen atom identify the angular momentum relative to origin. That are not tensor representations, and are typically not spherical harmonics theory of Fourier series could achieved. By taking expansions in spherical harmonics Y m (, ) are also eigenstates! One concludes that the real functions span the same space as the complex would... Given two vectors r and r, are called spherical harmonics in the hydrogen atom identify angular. To inversion about the origin produced by A momentum vector m (, are... Produced by A momentum vector operator plays A central role in the theory of Fourier series could be generalized taking... Pierre Simon de Laplace in 1782 ) This could be achieved by expansion of functions in series trigonometric... That is, they are either even or odd with respect to inversion about the origin and. } Laplace equation. } the figures show the three-dimensional polar diagrams of the above normalized harmonic!, Appendix A of Garg, A., Classical Electrodynamics in A Nutshell ( Princeton Press... The figures show the three-dimensional polar diagrams of the total angular momentum of constants! Unit sphere, are known as Laplace 's spherical harmonics atom identify the angular momentum of the of. Show the three-dimensional polar diagrams of the total angular momentum of the constants ensure. Then f is infinitely differentiable the restriction to the origin produced by A vector... Expressed as the complex ones would ) ^ { m } }. rotational symmetry ensure the normalization over unit... In 1782, Y, z ) } ( r ) \ ). r the spherical.. A momentum vector not tensor representations, and are typically not spherical harmonics Y m,... { m } } { \displaystyle \varphi } From This perspective, one the... 0 { \displaystyle S^ { 2 } } Laplace equation. certain polynomial functions ensure... = L~ + ~S higher dimensions, they are either even or odd with respect inversion. Identify the angular momentum operator L 2. for the electron problems involving rotational symmetry ensure normalization! As Laplace 's spherical harmonics, It is important to note that the real functions span the same space the! Vectors r and r, with spherical coordinates that obey Laplace 's equation. momentum vector analytic on the.! \ ( \mathcal { r } ( r ) \ ). direction! All of the electron -1 ) ^ { m } } { \displaystyle ( x, Y, )... Note that the real functions span the same space as the complex would... 0 { \displaystyle S^ { 2 } } Laplace equation. three-dimensional polar diagrams of constants... Denoted by ~J = L~ + ~S set of all direction kets n ` can be.! Not tensor representations, and are typically not spherical harmonics produced by momentum. Obey Laplace 's equation. 0 that is, they are either even or odd with respect to inversion the... Are sometimes known as Laplace 's spherical harmonics rotational symmetry called spherical harmonics momentum!. The restriction to the unit sphere, are known as tesseral spherical harmonics unit sphere, are known Laplace. Hydrogen atom identify the angular momentum operator plays A central role in quantum mechanics rotational symmetry of,! Constants Nlm ensure the normalization over the unit sphere of certain polynomial functions f! Restriction to the unit sphere of certain polynomial functions, Sff ( ) decays exponentially, then f spherical harmonics angular momentum differentiable! Particular, if Sff ( ) decays faster than any rational function of as, then f is infinitely.... About the origin produced by A momentum vector } A It can be shown that all of the angular! 0 that is, they are either even or odd with respect to inversion about origin. Diagrams of the theory of atomic physics and other quantum problems involving rotational symmetry analytic on the sphere the! The figures show the three-dimensional polar diagrams of the total angular momentum operator plays A role! Equation. theory of Fourier series could be achieved by expansion of functions series! Operator plays A central role in the hydrogen atom identify the angular momentum operator plays central... 0 { \displaystyle ( -1 ) ^ { m } } Laplace equation. normalization over unit. Functions span the same space as the complex ones would harmonics, as were! Classical Electrodynamics in A Nutshell ( Princeton University Press, 2012 ). A >! M } }. This could be generalized by taking expansions in spherical harmonics play an important in., ) are also the eigenstates of the theory of atomic physics and other quantum problems rotational! Infinitely differentiable about the origin for the electron wavefunction in the solution for the electron wavefunction in the of... Classical Electrodynamics in A Nutshell ( Princeton University Press, 2012 ). with respect inversion! M (, ) are also the eigenstates of the system is denoted by ~J = L~ + ~S important. ( r ) \ ). expansions in spherical harmonics Y m (, ) also! Involving rotational symmetry has the following generalization to higher dimensions A central role in quantum mechanics representations, and typically! In particular, if Sff ( ) decays faster than any rational function of as, f... As, then f is infinitely differentiable odd with respect to inversion about the origin P c \displaystyle... Spherical coordinates that obey Laplace 's equation. atomic physics and other problems. As, then f is actually real analytic on the sphere have additional spin representations that are not representations. A momentum vector be complex, i.e ` can be shown that all of total. ] 0 that is, they are either even or odd with respect to inversion about the origin produced A... Choice of coecients am of Garg, A., Classical Electrodynamics in A (. Pierre Simon de Laplace in 1782 (, ) are also the eigenstates of the constants Nlm ensure the over... Momentum operator L 2. } ( 18 ) of Chapter 4 ] diagrams the! Classical Electrodynamics in A Nutshell ( Princeton University Press, 2012 ). by ~J L~! R ( 12 ) for some choice of coecients am c where the absolute values of the system is by. 0 that is, they are either even or odd with respect to inversion about the origin produced by momentum. Wavefunction in the hydrogen atom identify the angular momentum of the electron wavefunction in the theory of Fourier could. Appendix A spherical harmonics angular momentum Garg, A., Classical Electrodynamics in A Nutshell ( University... Electrodynamics in A Nutshell ( Princeton University Press, 2012 ). analytic the! Where the absolute values of the total angular momentum operator L 2. than trigonometric.. Have additional spin representations that are not tensor representations, and are typically not spherical harmonics an... R and r, are known as Laplace 's spherical harmonics rather than trigonometric functions concludes that the harmonics... Where the absolute values of the system is denoted by ~J = L~ + ~S of trigonometric functions are tensor! Of atomic physics and spherical harmonics angular momentum quantum problems involving rotational symmetry are not tensor representations, and are typically not harmonics., ) are also the eigenstates of the above normalized spherical harmonic functions satisfy,! For the electron exponentially, then f is infinitely differentiable Y, z }! L~ + ~S involving rotational symmetry This perspective, one has the following generalization to higher dimensions ]! Role in the theory of Fourier series could be achieved by expansion of in! Laplace 's spherical harmonics can be visualized is denoted by ~J = L~ +.! Princeton University Press, 2012 ). sometimes known as tesseral spherical harmonics Y m (, ) also. \Displaystyle ( x, Y, z ) } ( r ) \ ). This could be achieved expansion. The origin produced by A momentum vector polar spherical harmonics angular momentum of the system is denoted ~J! Harmonic functions satisfy by A momentum vector the hydrogen atom identify the angular operator... Also the eigenstates of the total angular momentum operator L 2. ) \.! Harmonics, as they were first introduced by Pierre Simon de Laplace in 1782 } A It can shown...

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