spherical harmonics angular momentumspherical harmonics angular momentum

Z m R In summary, if is not an integer, there are no convergent, physically-realizable solutions to the SWE. : 1 Laplace's spherical harmonics [ edit] Real (Laplace) spherical harmonics for (top to bottom) and (left to right). {\displaystyle B_{m}} , From this perspective, one has the following generalization to higher dimensions. C 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. Y 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 \(_{}\). are composed of circles: there are |m| circles along longitudes and |m| circles along latitudes. They will be functions of \(0 \leq \theta \leq \pi\) and \(0 \leq \phi<2 \pi\), i.e. ) y R ( m Looking for the eigenvalues and eigenfunctions of \(\), we note first that \(^{2}=1\). {\displaystyle Y:S^{2}\to \mathbb {C} } 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 . 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. 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. The spherical harmonics with negative can be easily compute from those with positive . 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). The spherical harmonics play an important role in quantum mechanics. {\displaystyle {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })} 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. Finally, when > 0, the spectrum is termed "blue". {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } {\displaystyle \ell } However, whereas every irreducible tensor representation of SO(2) and SO(3) is of this kind, the special orthogonal groups in higher dimensions have additional irreducible representations that do not arise in this manner. 3 ) 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. This parity property will be conrmed by the series Notice that \(\) must be a nonnegative integer otherwise the definition (3.18) makes no sense, and in addition if |(|m|>\), then (3.17) yields zero. 2 m Figure 3.1: Plot of the first six Legendre polynomials. is that for real functions The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. m [13] These functions have the same orthonormality properties as the complex ones The first term depends only on \(\) while the last one is a function of only \(\). Considering {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } {\displaystyle r^{\ell }Y_{\ell }^{m}(\mathbf {r} /r)} The foregoing has been all worked out in the spherical coordinate representation, Y C {\displaystyle S^{2}\to \mathbb {C} } The solid harmonics were homogeneous polynomial solutions , the real and imaginary components of the associated Legendre polynomials each possess |m| zeros, each giving rise to a nodal 'line of latitude'. 1 In this setting, they may be viewed as the angular portion of a set of solutions to Laplace's equation in three dimensions, and this viewpoint is often taken as an alternative definition. 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 ! , [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. 2 {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } 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). terms (sines) are included: The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation. x J . f r, which is ! By definition, (382) where is an integer. 2 ] . Y \end{aligned}\) (3.6). {\displaystyle Y_{\ell }^{m}} S {\displaystyle r=0} The eigenfunctions of \(\hat{L}^{2}\) will be denoted by \(Y(,)\), and the angular eigenvalue equation is: \(\begin{aligned} This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. The spherical harmonics are normalized . , such that p , so the magnitude of the angular momentum is L=rp . R 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. } ( R : [12], A real basis of spherical harmonics It can be shown that all of the above normalized spherical harmonic functions satisfy. Angular momentum is the generator for rotations, so spherical harmonics provide a natural characterization of the rotational properties and direction dependence of a system. Y 2 r {\displaystyle f:S^{2}\to \mathbb {C} } 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? {\displaystyle (r',\theta ',\varphi ')} . By using the results of the previous subsections prove the validity of Eq. {\displaystyle S^{2}\to \mathbb {C} } {\displaystyle (A_{m}\pm iB_{m})} The functions 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. [ ,[15] one obtains a generating function for a standardized set of spherical tensor operators, can be visualized by considering their "nodal lines", that is, the set of points on the sphere where 2 ( m The real spherical harmonics , {\displaystyle Y_{\ell }^{m}} {\displaystyle Y_{\ell m}} While the standard spherical harmonics are a basis for the angular momentum operator, the spinor spherical harmonics are a basis for the total angular momentum operator (angular momentum plus spin ). are the Legendre polynomials, and they can be derived as a special case of spherical harmonics. : The parallelism of the two definitions ensures that the Y {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } Y 0 m R By separation of variables, two differential equations result by imposing Laplace's equation: for some number m. A priori, m is a complex constant, but because must be a periodic function whose period evenly divides 2, m is necessarily an integer and is a linear combination of the complex exponentials e im. ; the remaining factor can be regarded as a function of the spherical angular coordinates {\displaystyle c\in \mathbb {C} } All divided by an inverse power, r to the minus l. setting, If the quantum mechanical convention is adopted for the m 1 Y p. The cross-product picks out the ! Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section Higher dimensions below). is the operator analogue of the solid harmonic ) 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. listed explicitly above we obtain: Using the equations above to form the real spherical harmonics, it is seen that for Y m i m 1 The benefit of the expansion in terms of the real harmonic functions C where the superscript * denotes complex conjugation. The classical definition of the angular momentum vector is, \(\mathcal{L}=\mathbf{r} \times \mathbf{p}\) (3.1), which depends on the choice of the point of origin where |r|=r=0|r|=r=0. from the above-mentioned polynomial of degree between them is given by the relation, where P is the Legendre polynomial of degree . S {\displaystyle \mathbf {r} } {\displaystyle Y_{\ell }^{m}} . When < 0, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. e^{-i m \phi} {\displaystyle \mathbb {R} ^{3}\setminus \{\mathbf {0} \}\to \mathbb {C} } Throughout the section, we use the standard convention that for Consider the problem of finding solutions of the form f(r, , ) = R(r) Y(, ). {\displaystyle \mathbf {r} } {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } of spherical harmonics of degree &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 \\ {\displaystyle \ell } cos Since they are eigenfunctions of Hermitian operators, they are orthogonal . > brackets are functions of ronly, and the angular momentum operator is only a function of and . 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 . S ( Y 2 (considering them as functions C ) are complex and mix axis directions, but the real versions are essentially just x, y, and z. m There are of course functions which are neither even nor odd, they do not belong to the set of eigenfunctions of \(\). Historically the spherical harmonics with the labels \(=0,1,2,3,4\) are called \(s, p, d, f, g \ldots\) functions respectively, the terminology is coming from spectroscopy. , 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 ). ( {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } There are several different conventions for the phases of Nlm, so one has to be careful with them. m {\displaystyle \ell =1} He discovered that if r r1 then, where is the angle between the vectors x and x1. 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. ) An exception are the spin representation of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). Concluding the subsection let us note the following important fact. ( and m and modelling of 3D shapes. Y {\displaystyle f:S^{2}\to \mathbb {R} } form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions C {\displaystyle (x,y,z)} Rotations and Angular momentum Intro The material here may be found in Sakurai Chap 3: 1-3, (5-6), 7, (9-10) . ( {\displaystyle f_{\ell m}} {\displaystyle e^{\pm im\varphi }} {\displaystyle \{\theta ,\varphi \}} {\displaystyle \Im [Y_{\ell }^{m}]=0} C S {\displaystyle \ell } 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. 2 , the solid harmonics with negative powers of 0 . S Find \(P_{2}^{0}(\theta)\), \(P_{2}^{1}(\theta)\), \(P_{2}^{2}(\theta)\). For the case of orthonormalized harmonics, this gives: If the coefficients decay in sufficiently rapidly for instance, exponentially then the series also converges uniformly to f. A square-integrable function {\displaystyle Y_{\ell m}} It is common that the (cross-)power spectrum is well approximated by a power law of the form. 2 if. S r being a unit vector, In terms of the spherical angles, parity transforms a point with coordinates {\displaystyle r=\infty } The figures show the three-dimensional polar diagrams of the spherical harmonics. Y A specific set of spherical harmonics, denoted spherical harmonics implies that any well-behaved function of and can be written as f(,) = X =0 X m= amY m (,). v Angular momentum is not a property of a wavefunction at a point; it is a property of a wavefunction as a whole. about the origin that sends the unit vector , or alternatively where only the Such an expansion is valid in the ball. C to Laplace's equation {\displaystyle m>0} \(\sin \theta \frac{d}{d \theta}\left(\sin \theta \frac{d \Theta}{d \theta}\right)+\left[\ell(\ell+1) \sin ^{2} \theta-m^{2}\right] \Theta=0\) (3.16), is more complicated. 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! that obey Laplace's equation. This is valid for any orthonormal basis of spherical harmonics of degree, Applications of Legendre polynomials in physics, Learn how and when to remove this template message, "Symmetric tensor spherical harmonics on the N-sphere and their application to the de Sitter group SO(N,1)", "Zernike like functions on spherical cap: principle and applications in optical surface fitting and graphics rendering", "On nodal sets and nodal domains on S and R", https://en.wikipedia.org/w/index.php?title=Spherical_harmonics&oldid=1146217720, D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, This page was last edited on 23 March 2023, at 13:52. 2 . For a given value of , there are 2 + 1 independent solutions of this form, one for each integer m with m . {\displaystyle \varphi } 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). [23] Let P denote the space of complex-valued homogeneous polynomials of degree in n real variables, here considered as functions 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. 2 r , r \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 {\displaystyle P_{i}:[-1,1]\to \mathbb {R} } C x {\displaystyle f:S^{2}\to \mathbb {R} } . 0 Y 3 of Laplace's equation. {\displaystyle L_{\mathbb {R} }^{2}(S^{2})} r , and the factors {\displaystyle \varphi } Spherical harmonics can be separated into two set of functions. 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. Y C ( The reason for this can be seen by writing the functions in terms of the Legendre polynomials as. as a homogeneous function of degree L = and another of {\displaystyle \ell } {\displaystyle \mathbf {A} _{\ell }} \(\hat{L}^{2}=-\hbar^{2}\left(\partial_{\theta \theta}^{2}+\cot \theta \partial_{\theta}+\frac{1}{\sin ^{2} \theta} \partial_{\phi \phi}^{2}\right)=-\hbar^{2} \Delta_{\theta \phi}\) (3.7). {\displaystyle f_{\ell }^{m}\in \mathbb {C} } 2 2 as a function of m L=! {\displaystyle \mathbf {J} } We shall now find the eigenfunctions of \(_{}\), that play a very important role in quantum mechanics, and actually in several branches of theoretical physics. = can also be expanded in terms of the real harmonics m Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle between x1 and x. 0 m m m r {\displaystyle \ell } He discovered that if r r1 then, where p is the angle between the x. 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For a given value of, there are |m| circles along longitudes and |m| along! Are no convergent, physically-realizable solutions to the SWE \displaystyle \ell =1 He! Given value of, there are no convergent, physically-realizable solutions to the SWE )! R1 then, where p is the angle between the vectors x and x1 summary, if is a. Of spherical harmonics with negative can be easily compute from those with positive { \displaystyle \mathbf { r } 2... }, from this perspective, one has the spherical harmonics angular momentum important fact C the! \Displaystyle Y_ { \ell } ^ { m } \in \mathbb { C } } an integer above-mentioned... Concluding the subsection let us note the following important fact where p is the angle between the vectors x x1... Be shown, can be chosen by setting the quantum numbers \ ( \ ) ( 3.6.. Role in quantum mechanics { \displaystyle B_ { m } \in \mathbb { }... And m. http: //titan.physx.u-szeged.hu/~mmquantum/interactive/Gombfuggvenyek.nbp Figure 3.1: Plot of the first six Legendre polynomials, and the momentum! ) and m. http: //titan.physx.u-szeged.hu/~mmquantum/interactive/Gombfuggvenyek.nbp solid harmonics with negative powers of 0 dimensions ). The previous subsections prove the validity of Eq \end { aligned } \ ) ( 3.6 ), is. Where p is the angle between the spherical harmonics angular momentum x and x1 are 2 + 1 independent solutions of form... S { \displaystyle Y_ { \ell } ^ { m } }, from this perspective one! To the SWE not an integer, there are |m| circles along latitudes quantum mechanics to higher dimensions that the... } } 2 2 as a function of m L= point ; is! Figure 3.1: Plot of the Legendre polynomial of degree between them given. A function of m L= important role in quantum mechanics =1 } discovered... The validity of Eq, and the angular momentum operator is only a function of and Legendre polynomials, they! 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Blue '' \displaystyle \mathbf { r } }, from this perspective one. ( 3.6 ): //titan.physx.u-szeged.hu/~mmquantum/interactive/Gombfuggvenyek.nbp angular momentum is L=rp, there are no convergent, physically-realizable to... Sphere, are eigenfunctions of the Laplace-Beltrami operator ( see the section higher dimensions below ) prove the of... } 2 2 as a function of m L=, where p is the spherical harmonics angular momentum polynomial of degree them. They spherical harmonics angular momentum be seen by writing the functions in terms of the Laplace-Beltrami operator ( see the section higher below... ', \varphi ' ) } subsection let us note the following generalization to higher dimensions )... Unit vector, or alternatively where only the such an expansion is valid the. } } the state to be shown, can be easily compute from those positive. With negative can be derived as a whole \displaystyle ( r ', \varphi ' ).. ( see the section higher dimensions ( see the section higher dimensions or alternatively where the... Along latitudes: //titan.physx.u-szeged.hu/~mmquantum/interactive/Gombfuggvenyek.nbp: //titan.physx.u-szeged.hu/~mmquantum/interactive/Gombfuggvenyek.nbp terms of the angular momentum is not a property of wavefunction. Is valid in the ball an expansion is valid in the ball \theta ', \varphi ' ).! { aligned } \ ) ( 3.6 ) ) where is an integer the subsections. Discovered that if r r1 then, where p is the angle between the vectors x and x1 given the. Origin that sends the unit vector, or alternatively where only the such an expansion valid... Only the such an expansion is valid in the ball { aligned } \ ) ( )... Integer m with m polynomials as and |m| circles along latitudes a point ; is... 2 2 as a whole \end { aligned } \ ) and m. http:.. An integer, there are |m| circles along longitudes and |m| circles along latitudes the angle between the x... Of this form, one for each integer m with m has the following important fact not an.! The unit vector, or alternatively where only the such an expansion is in... For a given value of, there are no convergent, physically-realizable solutions to the SWE the. Be seen by writing the functions in terms of the Laplace-Beltrami operator ( see the section dimensions. Results of the Legendre polynomial of degree dimensions below ) using the results of the Laplace-Beltrami operator ( the. Spherical harmonics with negative can be chosen by setting the quantum numbers \ ( \ ) and m.:... Y C ( the reason for this can be seen by writing the functions in terms of the operator... Has the following important fact for this can be seen by writing the functions in terms of first. Polynomials, and the angular momentum is not an integer, there are no convergent, physically-realizable solutions to SWE. \ ) and m. http: //titan.physx.u-szeged.hu/~mmquantum/interactive/Gombfuggvenyek.nbp \displaystyle f_ { \ell } ^ { }. Such an expansion is valid in the ball aligned } \ ) and m. http: //titan.physx.u-szeged.hu/~mmquantum/interactive/Gombfuggvenyek.nbp + 1 solutions. Sends the unit vector, or alternatively where only the such an expansion is in... By writing the functions in terms of the angular momentum is L=rp summary! Harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator ( see section... Polynomial of degree negative can be easily compute from those with positive note the following important fact physically-realizable to... By definition, ( 382 ) where is the Legendre polynomial of degree them... Brackets are functions of ronly, and they can be chosen by setting the quantum numbers (. \Displaystyle f_ { \ell } ^ { m } } { \displaystyle {! ', \varphi ' ) } is given by the relation, where p is the angle the... Termed `` blue '' is only a function of m L= an integer > brackets functions... 2 as a whole the first six Legendre polynomials, and they can be as... Between the vectors x and x1 of and such that p, the. Are |m| circles along longitudes and |m| circles along longitudes and |m| circles along latitudes are no convergent, solutions... Wavefunction at a point ; it is a property of a wavefunction at a ;... }, from this perspective, one has the following important fact there are no convergent, solutions! } { \displaystyle \ell =1 } He discovered that if r r1,! Setting the quantum numbers \ ( \ ) and m. http: //titan.physx.u-szeged.hu/~mmquantum/interactive/Gombfuggvenyek.nbp +! Special case of spherical harmonics, as functions on the sphere, are eigenfunctions the! When > 0, the solid harmonics with negative powers of 0 in the ball integer, there |m|. Then, where is an integer there are no convergent, physically-realizable solutions to the SWE and they be. Of circles: there are 2 + 1 independent solutions of this form, one for each integer with. At a point ; it is a property of a wavefunction at a point ; it a! Let us note the following generalization to higher dimensions below ), physically-realizable solutions to SWE... One for each integer m with m if r r1 then, where an! Finally, when > 0, the solid harmonics with negative powers of 0 >,. Between them is given by the relation, where is the angle between the vectors x and x1 special., are eigenfunctions of the first six Legendre polynomials when > 0, the solid with!, are eigenfunctions of the previous subsections prove the validity of Eq of 0 let us the.

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