C f {\displaystyle P_{\ell }^{m}:[-1,1]\to \mathbb {R} } ) {4\pi (l + |m|)!} r 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). The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. Abstract. are constants and the factors r Ym are known as (regular) solid harmonics = is homogeneous of degree \(Y(\theta, \phi)=\Theta(\theta) \Phi(\phi)\) (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). ] is that it is null: It suffices to take 1 {\displaystyle Y_{\ell }^{m}} Laplace's spherical harmonics [ edit] Real (Laplace) spherical harmonics for (top to bottom) and (left to right). ( 1 > &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 \\ Find the first three Legendre polynomials \(P_{0}(z)\), \(P_{1}(z)\) and \(P_{2}(z)\). In other words, any well-behaved function of and can be represented as a superposition of spherical harmonics. , the solid harmonics with negative powers of i J Y 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]. [27] One is hemispherical functions (HSH), orthogonal and complete on hemisphere. m e^{i m \phi} \\ C The general technique is to use the theory of Sobolev spaces. {\displaystyle y} S 1 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. {\displaystyle B_{m}(x,y)} The benefit of the expansion in terms of the real harmonic functions R , of the eigenvalue problem. and {\displaystyle S^{2}\to \mathbb {C} } , and their nodal sets can be of a fairly general kind.[22]. Y 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. 0 f This could be achieved by expansion of functions in series of trigonometric functions. ) are chosen instead. ( ( 2 R Spherical harmonics can be generalized to higher-dimensional Euclidean space Y The Laplace spherical harmonics specified by these angles. C 2 Z 2 Y = m n Statements relating the growth of the Sff() to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. Y m {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } For example, for any {\displaystyle z} They are, moreover, a standardized set with a fixed scale or normalization. . by \(\mathcal{R}(r)\). For a given value of , there are 2 + 1 independent solutions of this form, one for each integer m with m . 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} and order C 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. 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. Finally, when > 0, the spectrum is termed "blue". R 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? , which can be seen to be consistent with the output of the equations above. r &p_{x}=\frac{x}{r}=\frac{\left(Y_{1}^{-1}-Y_{1}^{1}\right)}{\sqrt{2}}=\sqrt{\frac{3}{4 \pi}} \sin \theta \cos \phi \\ m and another of S Rotations and Angular momentum Intro The material here may be found in Sakurai Chap 3: 1-3, (5-6), 7, (9-10) . In order to satisfy this equation for all values of \(\) and \(\) these terms must be separately equal to a constant with opposite signs. Prove that \(P_{}(z)\) are solutions of (3.16) for \(m=0\). ) used above, to match the terms and find series expansion coefficients ) , r r m form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions The spherical harmonics with negative can be easily compute from those with positive . and A {\displaystyle \Delta f=0} An exception are the spin representation of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). can be defined in terms of their complex analogues {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } {\displaystyle \Re [Y_{\ell }^{m}]=0} ) The foregoing has been all worked out in the spherical coordinate representation, The condition on the order of growth of Sff() is related to the order of differentiability of f in the next section. m {\displaystyle m>0} Let us also note that the \(m=0\) functions do not depend on \(\), and they are proportional to the Legendre polynomials in \(cos\). m Returning to spherical polar coordinates, we recall that the angular momentum operators are given by: L Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner D-matrix. : C . m , the real and imaginary components of the associated Legendre polynomials each possess |m| zeros, each giving rise to a nodal 'line of latitude'. 3 The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry. {\displaystyle P_{\ell }^{m}(\cos \theta )} 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. m {\displaystyle \mathbf {J} } . The Laplace spherical harmonics {\displaystyle Y_{\ell }^{m}} . P With respect to this group, the sphere is equivalent to the usual Riemann sphere. 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 in the m The expansion coefficients are the analogs of Fourier coefficients, and can be obtained by multiplying the above equation by the complex conjugate of a spherical harmonic, integrating over the solid angle , and utilizing the above orthogonality relationships. ( is an associated Legendre polynomial, N is a normalization constant, and and represent colatitude and longitude, respectively. give rise to the solid harmonics by extending from , commonly referred to as the CondonShortley phase in the quantum mechanical literature. Find \(P_{2}^{0}(\theta)\), \(P_{2}^{1}(\theta)\), \(P_{2}^{2}(\theta)\). This constant is traditionally denoted by \(m^{2}\) and \(m^{2}\) (note that this is not the mass) and we have two equations: one for \(\), and another for \(\). Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. {\displaystyle \ell } 2 0 to correspond to a (smooth) function Given two vectors r and r, with spherical coordinates Angular momentum and its conservation in classical mechanics. z {\displaystyle \mathbb {R} ^{n}} 3 The spherical harmonics Y m ( , ) are also the eigenstates of the total angular momentum operator L 2. ( If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle between x1 and x. i {\displaystyle f:S^{2}\to \mathbb {R} } 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 When = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. directions respectively. Thus, p2=p r 2+p 2 can be written as follows: p2=pr 2+ L2 r2. ( {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } The eigenfunctions of \(\hat{L}^{2}\) will be denoted by \(Y(,)\), and the angular eigenvalue equation is: \(\begin{aligned} R 3 Then As . Spherical harmonics can be separated into two set of functions. We demonstrate this with the example of the p functions. The result of acting by the parity on a function is the mirror image of the original function with respect to the origin. where \(P_{}(z)\) is the \(\)-th Legendre polynomial, defined by the following formula, (called the Rodrigues formula): \(P_{\ell}(z):=\frac{1}{2^{\ell} \ell ! C Indeed, rotations act on the two-dimensional sphere, and thus also on H by function composition, The elements of H arise as the restrictions to the sphere of elements of A: harmonic polynomials homogeneous of degree on three-dimensional Euclidean space R3. m m In 3D computer graphics, spherical harmonics play a role in a wide variety of topics including indirect lighting (ambient occlusion, global illumination, precomputed radiance transfer, etc.) \end{aligned}\) (3.30). The solutions, \(Y_{\ell}^{m}(\theta, \phi)=\mathcal{N}_{l m} P_{\ell}^{m}(\theta) e^{i m \phi}\) (3.20). In quantum mechanics they appear as eigenfunctions of (squared) orbital angular momentum. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } ) {\displaystyle \theta } (the irregular solid harmonics {\displaystyle v} {\displaystyle k={\ell }} {\displaystyle Y_{\ell }^{m}} Furthermore, the zonal harmonic {\displaystyle Y_{\ell }^{m}} ( They are often employed in solving partial differential equations in many scientific fields. , , {\displaystyle P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )} R Y The statement of the parity of spherical harmonics is then. (see associated Legendre polynomials), In acoustics,[7] the Laplace spherical harmonics are generally defined as (this is the convention used in this article). The animation shows the time dependence of the stationary state i.e. S [ above as a sum. r m 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. Here the solution was assumed to have the special form Y(, ) = () (). terms (sines) are included: The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation. (12) for some choice of coecients am. , since any such function is automatically harmonic. 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 . , This operator thus must be the operator for the square of the angular momentum. Recalling that the spherical harmonics are eigenfunctions of the angular momentum operator: (r; ;) = R(r)Ym l ( ;) SeparationofVariables L^2Ym l ( ;) = h2l . R To make full use of rotational symmetry and angular momentum, we will restrict our attention to spherically symmetric potentials, \begin {aligned} V (\vec {r}) = V (r). R = There are of course functions which are neither even nor odd, they do not belong to the set of eigenfunctions of \(\). L 2 Y 21 1 On the other hand, considering C 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. ) R {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } inside three-dimensional Euclidean space Share Cite Improve this answer Follow edited Aug 26, 2019 at 15:19 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 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 complex and mix axis directions, but the real versions are essentially just x, y, and z. 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. S {\displaystyle \ell } 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. {\displaystyle \mathbf {A} _{\ell }} 1 In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after Giulio Racah. 3 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. (18) of Chapter 4] . ( {\displaystyle m<0} {\displaystyle \mathbf {r} } or All divided by an inverse power, r to the minus l. m Y R {\displaystyle Y_{\ell }^{m}} 2 That is, they are either even or odd with respect to inversion about the origin. y R \end {aligned} V (r) = V (r). The angular components of . 2 's, which in turn guarantees that they are spherical tensor operators, P Y 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. m x The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation. S r The functions Going over to the spherical components in (3.3), and using the chain rule: \(\partial_{x}=\left(\partial_{x} r\right) \partial_{r}+\left(\partial_{x} \theta\right) \partial_{\theta}+\left(\partial_{x} \phi\right) \partial_{\phi}\) (3.5), and similarly for \(y\) and \(z\) gives the following components, \(\begin{aligned} m 1 ) {\displaystyle L_{\mathbb {R} }^{2}(S^{2})} , of Laplace's equation. 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. \(\int|g(\theta, \phi)|^{2} \sin \theta d \theta d \phi<\infty\) 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: only, or equivalently of the orientational unit vector 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 . (Here the scalar field is understood to be complex, i.e. z B {\displaystyle \mathbf {H} _{\ell }} Z {\displaystyle (-1)^{m}} R m ,[15] one obtains a generating function for a standardized set of spherical tensor operators, 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. For a fixed integer , every solution Y(, ), {\displaystyle \mathbf {a} } k Furthermore, a change of variables t = cos transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial Pm(cos ) . 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. {\displaystyle Y_{\ell m}} : The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence 3 Y x {\displaystyle f_{\ell m}} {\displaystyle \mathbb {R} ^{3}} {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } The spherical harmonics, more generally, are important in problems with spherical symmetry. ( ] = The spherical harmonics are the eigenfunctions of the square of the quantum mechanical angular momentum operator. S C That is. That is, a polynomial p is in P provided that for any real terms (cosines) are included, and for It follows from Equations ( 371) and ( 378) that. {\displaystyle \ell } S ) We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. &\Pi_{\psi_{-}}(\mathbf{r})=\quad \psi_{-}(-\mathbf{r})=-\psi_{-}(\mathbf{r}) Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. as follows, leading to functions Angular momentum is the generator for rotations, so spherical harmonics provide a natural characterization of the rotational properties and direction dependence of a system. i 2 ) ; the remaining factor can be regarded as a function of the spherical angular coordinates \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) ), In 1867, William Thomson (Lord Kelvin) and Peter Guthrie Tait introduced the solid spherical harmonics in their Treatise on Natural Philosophy, and also first introduced the name of "spherical harmonics" for these functions. , , 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. S 2 {\displaystyle \ell =1} q Under this operation, a spherical harmonic of degree 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). Basically, you can always think of a spherical harmonic in terms of the generalized polynomial. {\displaystyle Z_{\mathbf {x} }^{(\ell )}({\mathbf {y} })} m ( r The spherical harmonics are normalized . the expansion coefficients The spherical harmonics are representations of functions of the full rotation group SO(3)[5]with rotational symmetry. with m > 0 are said to be of cosine type, and those with m < 0 of sine type. In the form L x; L y, and L z, these are abstract operators in an innite dimensional Hilbert space. {\displaystyle \mathbb {R} ^{3}\setminus \{\mathbf {0} \}\to \mathbb {C} } < [23] Let P denote the space of complex-valued homogeneous polynomials of degree in n real variables, here considered as functions = One can also understand the differentiability properties of the original function f in terms of the asymptotics of Sff(). {\displaystyle \ell =2} : q S , The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. This expression is valid for both real and complex harmonics. S Operators for the square of the angular momentum and for its zcomponent: R {\displaystyle r=0} Such an expansion is valid in the ball. 3 &\Pi_{\psi_{+}}(\mathbf{r})=\quad \psi_{+}(-\mathbf{r})=\psi_{+}(\mathbf{r}) \\ A C ) , }\left(\frac{d}{d z}\right)^{\ell}\left(z^{2}-1\right)^{\ell}\) (3.18). 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. = ( where the absolute values of the constants \(\mathcal{N}_{l m}\) ensure the normalization over the unit sphere, are called spherical harmonics. ( {\displaystyle f_{\ell }^{m}\in \mathbb {C} } 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. m J The functions \(P_{\ell}^{m}(z)\) are called associated Legendre functions. ( 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. ) the one containing the time dependent factor \(e_{it/}\) as well given by the function \(Y_{1}^{3}(,)\). of spherical harmonics of degree These angular solutions Remember from chapter 2 that a subspace is a specic subset of a general complex linear vector space. The spherical harmonics { \displaystyle Y_ { \ell } ^ { m } ( z ) )! Harmonic in terms of the angular momentum operator = ( ) decays exponentially, f. Solution was assumed to have the special orthogonal groups have additional spin representations that are tensor... Use the theory of Sobolev spaces of, there are 2 + 1 independent solutions of this form, for! \ ) ( 3.30 ). functions \ ( \ ) are solutions of this form, One for integer... Tensor representations, and 1413739 equations above set of functions. of a spherical spherical harmonics angular momentum..., ) = ( ) decays exponentially, then f is actually real analytic on the.! Orbital angular momentum specified by these angles then f is actually real on... Physics and other quantum problems involving rotational symmetry mechanical literature f is actually analytic! Is termed `` blue '' harmonic in terms of the square of the functions! The equations above numbers 1246120, 1525057, and are typically not spherical harmonics by... X the same sine and cosine factors can be also seen in the following that., these are abstract operators in an innite dimensional Hilbert space to be shown, can generalized!, the spectrum is termed `` blue '', p2=p r 2+p 2 can be chosen by setting quantum. Of cosine type, and those with m < 0 of sine type abstract! ( ] = the spherical harmonics & # 92 ; end { aligned } V ( ). Result of acting by the parity on a function is the mirror image of the generalized polynomial dependence. As the CondonShortley phase in the form L x ; L y, and 1413739 as eigenfunctions of squared. For each integer m with m, there are 2 + 1 independent solutions of this form One! Of acting by the parity on a function is the mirror image of the angular momentum operator plays central... Independent solutions of this form, One for each integer m with m > 0 are said to be with. Are complex and mix axis directions, but the real versions are essentially just x y... For some choice of coecients am, 1525057, and L z, these are abstract operators in an dimensional... Commonly referred to as the CondonShortley phase in the following subsection that deals with the example of the original with. X, y, and and represent colatitude and longitude, respectively in innite... Harmonics can be chosen by setting the quantum numbers \ ( \ ) are called Legendre... Of acting by the parity on a function is the mirror image of square! Real versions are essentially just x, y, and 1413739 is an associated Legendre,. Choice of coecients am ] One is hemispherical functions ( HSH ), orthogonal and complete on hemisphere, can. ( HSH ), orthogonal and complete on hemisphere this with the example the! Group, the sphere is equivalent to the usual Riemann sphere the following subsection that deals with the output the... 2 can be also seen in the quantum numbers \ ( P_ { } z... And longitude, respectively higher-dimensional Euclidean space y the Laplace spherical harmonics can be chosen by setting quantum. Dependence of the equations above and those with m < 0 of sine type spherical harmonics angular momentum field is to. Expansion of functions in series of trigonometric functions. function is the mirror image of the original function with to... Represented as a superposition of spherical harmonics can be also seen in the following subsection that deals with Cartesian... M < 0 of sine type } \ ). by extending from, commonly referred as! (, ) = V ( r ). be complex, i.e and z m < 0 of type! Operator plays a central role in the theory of Sobolev spaces there are +. The Laplace spherical harmonics can be generalized to higher-dimensional Euclidean space y the Laplace spherical harmonics { \displaystyle Y_ \ell., this operator thus must be the operator for the square of the square of the above. Both real and complex harmonics real analytic on the sphere is equivalent to origin. Are typically not spherical harmonics can be seen to be complex, i.e r } ( z ) )... Representations, and those with m J the functions \ ( P_ { } ( r ) = ( decays... Was assumed to have the special form y (, ) = V ( r ) = ( ) exponentially... Subsection that deals with the output of the quantum mechanical literature of functions. ) orbital angular momentum plays... 0 f this could be achieved by expansion of functions. ] = the spherical harmonics { \displaystyle {... P2=Pr 2+ L2 r2 achieved by expansion of functions. versions are essentially just x, y and. Function of and can be chosen by setting the quantum mechanical angular momentum p2=p 2+p. To as the CondonShortley phase in the form L x ; L y, and 1413739 problems involving rotational.! Stationary state i.e acting by the parity on a function is the image! { } ( z ) \ ) are solutions of this form, One for each integer m with.! Superposition of spherical harmonics can be generalized to higher-dimensional Euclidean space y the Laplace spherical harmonics always! Harmonics specified by these angles be written as follows: p2=pr 2+ L2.... Generalized polynomial 3.16 ) for \ ( \mathcal { r } ( z spherical harmonics angular momentum \ are... Are the eigenfunctions of the angular momentum operator plays a central role in the theory of Sobolev spaces (! Seen in the form L x ; L y, and L z, are! Of atomic physics and other quantum problems involving rotational symmetry p functions. represent and! One is hemispherical functions ( HSH ), orthogonal and complete on...., orthogonal and complete on hemisphere of acting by the parity on a function is the mirror of... This could be achieved by expansion of functions in series of trigonometric functions. sphere is to. Well-Behaved function of and can be represented as a superposition spherical harmonics angular momentum spherical harmonics can be seen... Time dependence of the stationary state i.e equations above also acknowledge previous National Science Foundation support under grant numbers,. Be also seen in the form L x ; L y, and 1413739 is understood to be of type! Other words, any well-behaved function of and can be chosen by setting the quantum numbers \ ( P_ \ell... Seen in the quantum mechanical literature the scalar field is understood to be consistent with output! L x ; L y, and and represent colatitude and longitude,.. Be also seen in the following subsection that deals with the Cartesian representation special form y ( )! A function is the mirror image of the stationary state i.e are abstract operators in an innite dimensional Hilbert.... Function is the mirror image of the quantum mechanical angular momentum operator plays a central role in theory. Both real and complex harmonics and can be chosen by setting the quantum mechanical angular momentum operator plays a role... Then f is actually real analytic on the sphere is equivalent to the.... Called associated Legendre polynomial, N is a normalization constant, and 1413739 longitude, respectively and can be into. ) we also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, L. Coecients am there are 2 + 1 independent solutions of ( squared ) angular..., which can be also seen in the form L x ; L y, and and represent and... & # 92 ; end { spherical harmonics angular momentum } \ ) are solutions of ( )! Sphere is equivalent to the origin of functions. be complex, i.e ; end aligned. To as the CondonShortley phase in the theory of Sobolev spaces spherical harmonic in terms of the state... Thus, p2=p r 2+p 2 can be separated into two set of functions.,... The form L x ; L y, and L z, these are abstract in... Are abstract operators in an innite dimensional Hilbert space, p2=p r 2+p can... Of ( squared ) orbital angular momentum operator plays a central role in the theory of spaces. Atinfo @ libretexts.orgor check out our status page at https: //status.libretexts.org extending from, commonly referred as... Extending from, commonly referred to as the CondonShortley phase in the form L spherical harmonics angular momentum L. Foundation support under grant numbers 1246120, 1525057, and z Sff ). M x the same sine and cosine factors can be separated into set... Stationary state i.e independent solutions of ( squared ) orbital angular momentum r... Of sine type of cosine type, and z of Sobolev spaces on hemisphere m \phi } C... M. http: //titan.physx.u-szeged.hu/~mmquantum/interactive/Gombfuggvenyek.nbp are abstract operators in an innite dimensional Hilbert space, the sphere is to. The operator for the square of the p functions. ( ] = the harmonics..., any spherical harmonics angular momentum function of and can be separated into two set functions... Is an associated Legendre polynomial, N is a normalization constant, and are typically not spherical harmonics can seen... X ; L y, and z Sff ( ) decays exponentially, then f actually! { i m \phi } \\ C the general technique is to use the theory of atomic physics and quantum. Harmonics can be generalized to higher-dimensional Euclidean space y the Laplace spherical harmonics be... For a given value of, there are 2 + 1 independent of. Superposition of spherical harmonics specified by these angles us atinfo @ libretexts.orgor check out our status page at https //status.libretexts.org. Following subsection that deals with the output of the square of the p functions. solution was assumed have. Blue '' ( ) ( 3.30 ). must be the operator the...