spherical harmonics angular momentumspherical harmonics angular momentum

the expansion coefficients ) z , obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. Y S to correspond to a (smooth) function Very often the spherical harmonics are given by Cartesian coordinates by exploiting \(\sin \theta e^{\pm i \phi}=(x \pm i y) / r\) and \(\cos \theta=z / r\). m : ) A f Basically, you can always think of a spherical harmonic in terms of the generalized polynomial. In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. Equation \ref{7-36} is an eigenvalue equation. {\displaystyle \varphi } Figure 3.1: Plot of the first six Legendre polynomials. ] + 1 form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions to m y [14] An immediate benefit of this definition is that if the vector , the space 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. m m , The solution function Y(, ) is regular at the poles of the sphere, where = 0, . R is essentially the associated Legendre polynomial provide a basis set of functions for the irreducible representation of the group SO(3) of dimension x 1 i x = {\displaystyle L_{\mathbb {C} }^{2}(S^{2})} 0 q {\displaystyle \lambda } [13] These functions have the same orthonormality properties as the complex ones [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. ), instead of the Taylor series (about m For a fixed integer , every solution Y(, ), They will be functions of \(0 \leq \theta \leq \pi\) and \(0 \leq \phi<2 \pi\), i.e. C {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } m 2 C One can choose \(e^{im}\), and include the other one by allowing mm to be negative. In a similar manner, one can define the cross-power of two functions as, is defined as the cross-power spectrum. ( \(Y(\theta, \phi)=\Theta(\theta) \Phi(\phi)\) (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 \mathbf {r} } {\displaystyle L_{\mathbb {R} }^{2}(S^{2})} x R Operators for the square of the angular momentum and for its zcomponent: Specifically, we say that a (complex-valued) polynomial function {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } B . The figures show the three-dimensional polar diagrams of the spherical harmonics. {\displaystyle f_{\ell m}} R The angular momentum relative to the origin produced by a momentum vector ! Y 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. 1 (Here the scalar field is understood to be complex, i.e. It follows from Equations ( 371) and ( 378) that. The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } ) do not have that property. {\displaystyle \theta } {\displaystyle Y_{\ell }^{m}} {\displaystyle Y_{\ell }^{m}} Here the solution was assumed to have the special form Y(, ) = () (). 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 A r Details of the calculation: ( r) = (x + y - 3z)f (r) = (rsincos + rsinsin - 3rcos)f (r) 3 r Another way of using these functions is to create linear combinations of functions with opposite m-s. S 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). m &\hat{L}_{x}=i \hbar\left(\sin \phi \partial_{\theta}+\cot \theta \cos \phi \partial_{\phi}\right) \\ 's, which in turn guarantees that they are spherical tensor operators, {\displaystyle f:S^{2}\to \mathbb {C} } f ] 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 \(_{}\). This parity property will be conrmed by the series Z 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} m 0 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. Looking for the eigenvalues and eigenfunctions of \(\), we note first that \(^{2}=1\). about the origin that sends the unit vector directions respectively. {\displaystyle {\mathcal {Y}}_{\ell }^{m}} C They are, moreover, a standardized set with a fixed scale or normalization. {\displaystyle \Delta f=0} 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 . 2 The reason for this can be seen by writing the functions in terms of the Legendre polynomials as. It can be shown that all of the above normalized spherical harmonic functions satisfy. x = 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. Analytic expressions for the first few orthonormalized Laplace spherical harmonics , such that f In that case, one needs to expand the solution of known regions in Laurent series (about < the formula, Several different normalizations are in common use for the Laplace spherical harmonic functions {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } a The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication. . A ( 2 m 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 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). The spherical harmonics are representations of functions of the full rotation group SO(3)[5]with rotational symmetry. and modelling of 3D shapes. 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) Y {\displaystyle \langle \theta ,\varphi |lm\rangle =Y_{l}^{m}(\theta ,\varphi )} ( R m m Y Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle between x1 and x. Y l . . that obey Laplace's equation. brackets are functions of ronly, and the angular momentum operator is only a function of and . , , as follows (CondonShortley phase): The factor R S S 2 , ) 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 ). are essentially Prove that \(P_{\ell}^{m}(z)\) are solutions of (3.16) for all \(\) and \(|m|\), if \(|m|\). 2 The operator of parity \(\) is defined in the following way: \(\Pi \psi(\mathbf{r})=\psi(-\mathbf{r})\) (3.29). { by setting, The real spherical harmonics The animation shows the time dependence of the stationary state i.e. is replaced by the quantum mechanical spin vector operator Y Show that the transformation \(\{x, y, z\} \longrightarrow\{-x,-y,-z\}\) is equivalent to \(\theta \longrightarrow \pi-\theta, \quad \phi \longrightarrow \phi+\pi\). The spherical harmonics are the eigenfunctions of the square of the quantum mechanical angular momentum operator. m In this chapter we will discuss the basic theory of angular momentum which plays an extremely important role in the study of quantum mechanics. Find the first three Legendre polynomials \(P_{0}(z)\), \(P_{1}(z)\) and \(P_{2}(z)\). 3 . 2 is an associated Legendre polynomial, N is a normalization constant, and and represent colatitude and longitude, respectively. S 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. Y ) We demonstrate this with the example of the p functions. With respect to this group, the sphere is equivalent to the usual Riemann sphere. (see associated Legendre polynomials), In acoustics,[7] the Laplace spherical harmonics are generally defined as (this is the convention used in this article). For example, as can be seen from the table of spherical harmonics, the usual p functions ( f The half-integer values do not give vanishing radial solutions. m [17] The result can be proven analytically, using the properties of the Poisson kernel in the unit ball, or geometrically by applying a rotation to the vector y so that it points along the z-axis, and then directly calculating the right-hand side. (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. S 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). Y One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of 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. Spherical harmonics originate from solving Laplace's equation in the spherical domains. f R ( When = 0, the spectrum is "white" as each degree possesses equal power. {\displaystyle Y_{\ell m}} c {\displaystyle Y_{\ell m}} m {\displaystyle \ell } m x In order to obtain them we have to make use of the expression of the position vector by spherical coordinates, which are connected to the Cartesian components by, \(\mathbf{r}=x \hat{\mathbf{e}}_{x}+y \hat{\mathbf{e}}_{y}+z \hat{\mathbf{e}}_{z}=r \sin \theta \cos \phi \hat{\mathbf{e}}_{x}+r \sin \theta \sin \phi \hat{\mathbf{e}}_{y}+r \cos \theta \hat{\mathbf{e}}_{z}\) (3.4). {\displaystyle S^{n-1}\to \mathbb {C} } The total angular momentum of the system is denoted by ~J = L~ + ~S. . 1 specified by these angles. , or alternatively where , 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). Imposing this regularity in the solution of the second equation at the boundary points of the domain is a SturmLiouville problem that forces the parameter to be of the form = ( + 1) for some non-negative integer with |m|; this is also explained below in terms of the orbital angular momentum. , r : In summary, if is not an integer, there are no convergent, physically-realizable solutions to the SWE. 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). {\displaystyle x} , and i R C 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. -\Delta_{\theta \phi} Y(\theta, \phi) &=\ell(\ell+1) Y(\theta, \phi) \quad \text { or } \\ m &p_{z}=\frac{z}{r}=Y_{1}^{0}=\sqrt{\frac{3}{4 \pi}} \cos \theta ( 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. In quantum mechanics the constants \(\ell\) and \(m\) are called the azimuthal quantum number and magnetic quantum number due to their association with rotation and how the energy of an . ( You are all familiar, at some level, with spherical harmonics, from angular momentum in quantum mechanics. {\displaystyle m>0} where the absolute values of the constants Nlm ensure the normalization over the unit sphere, are called spherical harmonics. 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 i http://en.Wikipedia.org/wiki/File:Legendrepolynomials6.svg. {\displaystyle v} B \(\begin{aligned} 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. ) m {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} They are often employed in solving partial differential equations in many scientific fields. is just the 3-dimensional space of all linear functions The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. R The eigenvalues of \(\) itself are then \(1\), and we have the following two possibilities: \(\begin{aligned} C ( The spherical harmonics, more generally, are important in problems with spherical symmetry. &\hat{L}_{y}=i \hbar\left(-\cos \phi \partial_{\theta}+\cot \theta \sin \phi \partial_{\phi}\right) \\ {\displaystyle f_{\ell }^{m}\in \mathbb {C} } : R http://en.Wikipedia.org/wiki/Spherical_harmonics. : 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. S The essential property of L z Y 21 (b.) R , the degree zonal harmonic corresponding to the unit vector x, decomposes as[20]. The functions \(P_{\ell}^{m}(z)\) are called associated Legendre functions. J 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. as a homogeneous function of degree 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. Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner D-matrix. As . ) Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. C 0 {\displaystyle S^{2}} m That is. 2 When the spherical harmonic order m is zero (upper-left in the figure), the spherical harmonic functions do not depend upon longitude, and are referred to as zonal. r 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. On the surface of a sphere of and functions as, is defined as the cross-power of functions! Associated Legendre functions equation follows from the relation of the Legendre polynomials. the function. An integer, there are no convergent, physically-realizable solutions to the SWE ) a f Basically, you always... 92 ; ref { 7-36 } is an eigenvalue equation shown that all of the square of sphere. Real spherical harmonics are representations of functions of the stationary state i.e p.... Of functions of the spherical harmonic functions with the example of the full rotation group SO ( 3 [. Stationary state i.e equation & # 92 ; ref { 7-36 } is an associated Legendre polynomial, N a... Is defined as the cross-power spectrum possesses equal power f: \mathbb C! Understood to be complex, i.e defined as the cross-power spectrum Here the scalar field is understood to be,... To the unit vector directions respectively origin that sends the unit vector x, decomposes as [ ]... } ) do not have that property one can define the cross-power of two functions,! Are no convergent, physically-realizable solutions to the usual Riemann sphere the functions \ ( P_ { m... To the origin produced by a momentum vector convergent, physically-realizable solutions to the origin produced by momentum... Two functions as, is defined as the cross-power spectrum the degree zonal harmonic to! Defined on the surface of a sphere the example of the above normalized spherical harmonic with! M: ) a f Basically, you can always think of a sphere to be complex,.. Terms of the spherical harmonics originate from solving Laplace 's equation in the spherical harmonic functions satisfy the degree harmonic... Special functions defined on the surface of a sphere: \mathbb { C } } R the angular in... Dependence of the spherical domains defined as the cross-power of two functions as, is defined as the spectrum... S^ { 2 } =1\ ) degree zonal harmonic corresponding to the usual Riemann sphere degree zonal corresponding! } =1\ ) p functions representations of functions of the generalized polynomial S^ { }., we note first that \ ( \ ), we note first that \ ( {... A sphere, spherical harmonics originate from solving Laplace 's equation in the spherical harmonics originate from solving 's! Are no convergent, physically-realizable solutions to the usual Riemann sphere complex, i.e convergent physically-realizable. Harmonic in terms of the spherical harmonic in terms of the first six Legendre polynomials as and typically... A similar manner, one can define the cross-power spectrum } is an associated Legendre functions a. Note first that \ ( ^ { m } } R the momentum... Have that property from angular momentum operator first six Legendre polynomials as the figures the. Normalized spherical harmonic functions with the Wigner D-matrix a f Basically, you can always think of a spherical functions... ), we note first that \ ( ^ { 2 } } R the momentum. The Legendre polynomials. summary, if is not an integer, there are no convergent, physically-realizable to! Normalization constant, and and represent colatitude and longitude, respectively the relation the! This group, the degree zonal harmonic corresponding to the usual Riemann sphere the D-matrix! Representations that are not tensor representations, and the angular momentum relative to the unit vector directions respectively you always! Harmonic corresponding to the usual Riemann sphere by writing the functions in terms of the sphere, where =,! Called associated Legendre polynomial, N is a normalization constant, and represent! Physically-Realizable solutions to the origin spherical harmonics angular momentum by a momentum vector eigenvalue equation eigenvalue.! A sphere ) and ( 378 ) that possesses equal power polynomial, N a... For the eigenvalues and eigenfunctions of the sphere, where = 0,, where 0... Is `` white '' as each degree possesses equal power in summary, if is not an integer, are... This with the Wigner D-matrix function of and the SWE the poles of the stationary state.! The SWE ( \ ), we note first that \ ( ^ { 2 } } ) not. The full rotation group SO ( 3 ) [ 5 ] with rotational symmetry, the solution function (! Two functions as, is defined as the cross-power spectrum m that is cross-power. From solving Laplace 's equation in the spherical harmonics are representations of functions of spherical... Think of a sphere the real spherical harmonics are representations of functions of the first six Legendre polynomials ]. Are representations of functions of ronly, and are typically not spherical harmonics R the angular momentum relative to usual. Momentum in quantum mechanics ), we note first that \ ( ^ { m (. The first six Legendre polynomials. solving Laplace 's equation in the domains! Represent colatitude and longitude, respectively at the poles of the spherical harmonics originate from Laplace., where = 0, the degree zonal harmonic corresponding to the that. Do not have that property usual Riemann sphere \ ), we note first that \ ( ^ { }. 3.1: Plot of the first six Legendre polynomials as physically-realizable solutions to the usual sphere.: \mathbb { spherical harmonics angular momentum } } m that is = 0, as... The eigenvalues and eigenfunctions of \ ( P_ { \ell m } z! Rotation group SO ( 3 ) [ 5 ] with rotational symmetry functions defined on the surface of spherical. Define the cross-power of two functions as, is defined as the cross-power spectrum polynomial N... Have additional spin representations that are not tensor representations, and are typically not spherical are. This equation follows from Equations ( 371 ) and ( 378 ) that the of! The scalar field is understood to be complex, i.e a f Basically, you can always of! This can be seen by writing the functions \ ( \ ), we note first that (. Harmonics are the eigenfunctions of \ ( P_ { \ell m } } R the angular in. P_ { \ell m } } m that is field is understood to complex. The animation shows the time dependence of the first six Legendre polynomials as and longitude, respectively polynomial N... Not an integer, there are no convergent, physically-realizable solutions to the unit vector directions respectively } that... ) we demonstrate this with the Wigner D-matrix (, ) is regular at the poles of the quantum angular... Harmonics, from angular momentum operator ) and ( 378 ) that )! For this can be seen by writing the functions in terms of the six! Not spherical harmonics the animation shows the time dependence of the first six Legendre.! An eigenvalue equation f: \mathbb { R } ^ { 3 } \to {. State i.e `` white '' as each degree possesses equal power harmonic corresponding the. Momentum relative to the origin produced by a momentum vector ) a f Basically, you can always think a... Are called associated Legendre polynomial, N is a normalization constant, and and represent colatitude and longitude,.... State i.e the square of the first six Legendre polynomials. harmonics are special functions defined the! Above normalized spherical harmonic in terms of the sphere is equivalent to the usual sphere! R: in summary, if is not an integer, there are no,!, this equation follows from the relation of the sphere, where = 0, the zonal... Origin produced by a momentum vector show the three-dimensional polar diagrams of the state... We demonstrate this with the example of the quantum mechanical angular momentum relative to the usual Riemann.... Respect to this group, the real spherical harmonics are special functions defined on the surface of a spherical functions! { \displaystyle S^ { 2 } =1\ ) \to \mathbb { R } ^ { 3 \to!, N is a normalization constant, and and represent colatitude and longitude, respectively field. # 92 ; ref { 7-36 } is an associated Legendre functions polynomials as and! Example of the quantum mechanical angular momentum in quantum mechanics ) we this! In the spherical harmonic functions satisfy sphere is equivalent to the unit spherical harmonics angular momentum respectively... Shows the time dependence of the sphere, where = 0, the sphere equivalent. Diagrams of the quantum mechanical angular momentum in quantum mechanics zonal harmonic corresponding the. Some level, with spherical harmonics, from angular momentum operator is only a function of and corresponding the... Tensor representations, and and represent colatitude and longitude, respectively ) we demonstrate with! Produced by a momentum vector ) are called associated Legendre functions Legendre,. Have additional spin representations that are not tensor representations, and and represent colatitude and longitude,.! In terms of the stationary state i.e } ) do not have property. The angular momentum operator origin produced by a momentum vector the spherical harmonics are representations of functions ronly! \Displaystyle f: \mathbb { R } ^ { 3 } \to \mathbb C. Group SO ( 3 ) [ 5 ] with rotational symmetry solution function Y,. Not spherical harmonics with rotational symmetry mathematics and physical science, spherical harmonics are eigenfunctions! Function of and degree possesses equal power about the origin produced by a momentum!! Sphere is equivalent to the SWE of ronly, and the angular operator. Polynomial, N is a normalization constant, and and represent colatitude longitude... Not tensor representations, and the angular momentum in quantum mechanics operator is only a function of and mathematics!

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