R ( {\displaystyle f_{\ell }^{m}} 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. 1 = ( about the origin that sends the unit vector Recalling that the spherical harmonics are eigenfunctions of the angular momentum operator: (r; ;) = R(r)Ym l ( ;) SeparationofVariables L^2Ym l ( ;) = h2l . m y In particular, if Sff() decays faster than any rational function of as , then f is infinitely differentiable. A , m The spherical harmonic functions depend on the spherical polar angles and and form an (infinite) complete set of orthogonal, normalizable functions. > The spherical harmonics with negative can be easily compute from those with positive . 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). {\displaystyle Y_{\ell }^{m}} {4\pi (l + |m|)!} S p is ! } {\displaystyle Z_{\mathbf {x} }^{(\ell )}} ; the remaining factor can be regarded as a function of the spherical angular coordinates 2 , as follows (CondonShortley phase): The factor The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions , the trigonometric sin and cos functions possess 2|m| zeros, each of which gives rise to a nodal 'line of longitude'. : The parallelism of the two definitions ensures that the transforms into a linear combination of spherical harmonics of the same degree. Any function of and can be expanded in the spherical harmonics . f Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x p~. by setting, The real spherical harmonics Let us also note that the \(m=0\) functions do not depend on \(\), and they are proportional to the Legendre polynomials in \(cos\). {\displaystyle B_{m}(x,y)} R 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. {\displaystyle r} q {\displaystyle f_{\ell }^{m}\in \mathbb {C} } 2 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]. , one has. symmetric on the indices, uniquely determined by the requirement. Y and m The eigenfunctions of the orbital angular momentum operator, the spherical harmonics Reasoning: The common eigenfunctions of L 2 and L z are the spherical harmonics. ) The condition on the order of growth of Sff() is related to the order of differentiability of f in the next section. S The general solution {\displaystyle {\mathcal {R}}} {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } m C Notice, however, that spherical harmonics are not functions on the sphere which are harmonic with respect to the Laplace-Beltrami operator for the standard round metric on the sphere: the only harmonic functions in this sense on the sphere are the constants, since harmonic functions satisfy the Maximum principle. and modelling of 3D shapes. 1 . This expression is valid for both real and complex harmonics. The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. 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. and 2 as a function of S Spherical harmonics can be separated into two set of functions. {\displaystyle \varphi } Legal. where the absolute values of the constants Nlm ensure the normalization over the unit sphere, are called spherical harmonics. R [ 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). B 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 m Your vector spherical harmonics are functions of in the vector space $$ \pmb{Y}_{j\ell m} \in V=\left\{ \mathbf f:\mathbb S^2 \to \mathbb C^3 : \int_{\mathbb S^2} |\mathbf f(\pmb\Omega)|^2 \mathrm d \pmb\Omega <\infty . See here for a list of real spherical harmonics up to and including \(\int|g(\theta, \phi)|^{2} \sin \theta d \theta d \phi<\infty\) 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: As to what's "really" going on, it's exactly the same thing that you have in the quantum mechanical addition of angular momenta. &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 \\ v There are several different conventions for the phases of Nlm, so one has to be careful with them. 1 + : For example, as can be seen from the table of spherical harmonics, the usual p functions ( 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. 2 {\displaystyle P_{\ell }^{m}:[-1,1]\to \mathbb {R} } C 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. {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } {\displaystyle \{\pi -\theta ,\pi +\varphi \}} (12) for some choice of coecients am. {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } l {\displaystyle m>0} 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 solid harmonics with negative powers of 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 ). {\displaystyle \ell } Chapters 1 and 2. 2 : If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. Basically, you can always think of a spherical harmonic in terms of the generalized polynomial. {\displaystyle (A_{m}\pm iB_{m})} x ) Y In quantum mechanics they appear as eigenfunctions of (squared) orbital angular momentum. 2 Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree One can determine the number of nodal lines of each type by counting the number of zeros of {\displaystyle (r',\theta ',\varphi ')} and Spherical Harmonics 11.1 Introduction Legendre polynomials appear in many different mathematical and physical situations: . is just the space of restrictions to the sphere m 2 {\displaystyle \ell } , Y As is known from the analytic solutions for the hydrogen atom, the eigenfunctions of the angular part of the wave function are spherical harmonics. [13] These functions have the same orthonormality properties as the complex ones 1 &\hat{L}_{y}=i \hbar\left(-\cos \phi \partial_{\theta}+\cot \theta \sin \phi \partial_{\phi}\right) \\ r ( 3 Y R {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } ( Since mm can take only the integer values between \(\) and \(+\), there are \(2+1\) different possible projections, corresponding to the \(2+1\) different functions \(Y_{m}^{}(,)\) with a given \(\). ) {\displaystyle \Im [Y_{\ell }^{m}]=0} In that case, one needs to expand the solution of known regions in Laurent series (about + { There are several different conventions for the phases of \(\mathcal{N}_{l m}\), so one has to be careful with them. , {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} m above. 1 ) {\displaystyle \mathbf {J} } Determined by the requirement, are called spherical harmonics of the generalized polynomial spherical harmonic in terms of the definitions... 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