spherical harmonics angular momentumspherical harmonics angular momentum

{\displaystyle Y_{\ell }^{m}({\mathbf {r} })} {\displaystyle Y_{\ell }^{m}} The condition on the order of growth of Sff() is related to the order of differentiability of f in the next section. the one containing the time dependent factor \(e_{it/}\) as well given by the function \(Y_{1}^{3}(,)\). 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. m The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence Angular momentum is the generator for rotations, so spherical harmonics provide a natural characterization of the rotational properties and direction dependence of a system. m = f In a similar manner, one can define the cross-power of two functions as, is defined as the cross-power spectrum. m y S only the as a function of , &\hat{L}_{y}=i \hbar\left(-\cos \phi \partial_{\theta}+\cot \theta \sin \phi \partial_{\phi}\right) \\ r A The Laplace spherical harmonics R For convenience, we list the spherical harmonics for = 0,1,2 and non-negative values of m. = 0, Y0 0 (,) = 1 4 = 1, Y1 ), 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. m [ 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. {\displaystyle \lambda \in \mathbb {R} } This could be achieved by expansion of functions in series of trigonometric functions. as a homogeneous function of degree if. S = m 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). ) 2 2 For a given value of , there are 2 + 1 independent solutions of this form, one for each integer m with m . {\displaystyle \theta } ) , one has. in the 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. 0 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). More general spherical harmonics of degree are not necessarily those of the Laplace basis m 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). This parity property will be conrmed by the series The essential property of m r m ( i {\displaystyle \Delta f=0} m {\displaystyle Y_{\ell }^{m}} f m . See here for a list of real spherical harmonics up to and including m As is known from the analytic solutions for the hydrogen atom, the eigenfunctions of the angular part of the wave function are spherical harmonics. ( 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 \ell =1} m 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. ) 2 : 1-62. {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } Prove that \(P_{\ell}^{m}(z)\) are solutions of (3.16) for all \(\) and \(|m|\), if \(|m|\). Finally, when > 0, the spectrum is termed "blue". 2 We will use the actual function in some problems. is homogeneous of degree 2 m {\displaystyle f_{\ell m}} 0 C ( L z Y 21 (b.) In naming this generating function after Herglotz, we follow Courant & Hilbert 1962, VII.7, who credit unpublished notes by him for its discovery. ( Furthermore, the zonal harmonic Y {\displaystyle \mathbf {J} } i The eigenfunctions of \(\hat{L}^{2}\) will be denoted by \(Y(,)\), and the angular eigenvalue equation is: \(\begin{aligned} 1 c 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. ( 's transform under rotations (see below) in the same way as the m } m ) In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after Giulio Racah. 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 For example, when ] Y {\displaystyle \mathbb {R} ^{n}\to \mathbb {C} } 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). {\displaystyle \mathbb {R} ^{3}} where the superscript * denotes complex conjugation. . {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } m {\displaystyle \gamma } In particular, if Sff() decays faster than any rational function of as , then f is infinitely differentiable. being a unit vector, In terms of the spherical angles, parity transforms a point with coordinates \(\int|g(\theta, \phi)|^{2} \sin \theta d \theta d \phi<\infty\) 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: Details of the calculation: ( r) = (x + y - 3z)f (r) = (rsincos + rsinsin - 3rcos)f (r) r {\displaystyle v} = Using the expressions for Hence, transforms into a linear combination of spherical harmonics of the same degree. m C R The operator on the left operates on the spherical harmonic function to give a value for \(M^2\), the square of the rotational angular momentum, times the spherical harmonic function. m Consider a rotation Essentially all the properties of the spherical harmonics can be derived from this generating function. where r y r 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? ) is just the 3-dimensional space of all linear 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. Y 2 Laplace equation. The convergence of the series holds again in the same sense, namely the real spherical harmonics , then, a 1 ( {\displaystyle r^{\ell }} ) used above, to match the terms and find series expansion coefficients , of the eigenvalue problem. C 1 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). S In the first case the eigenfunctions \(\psi_{+}(\mathbf{r})\) belonging to eigenvalue +1 are the even functions, while in the second we see that \(\psi_{-}(\mathbf{r})\) are the odd functions belonging to the eigenvalue 1. S R S n 0 {\displaystyle \ell } the formula, Several different normalizations are in common use for the Laplace spherical harmonic functions {\displaystyle f:S^{2}\to \mathbb {C} \supset \mathbb {R} } For example, as can be seen from the table of spherical harmonics, the usual p functions ( The statement of the parity of spherical harmonics is then. The ClebschGordan coefficients are the coefficients appearing in the expansion of the product of two spherical harmonics in terms of spherical harmonics themselves. Under this operation, a spherical harmonic of degree = When = 0, the spectrum is "white" as each degree possesses equal power. P {\displaystyle (A_{m}\pm iB_{m})} C {\displaystyle \mathbf {r} } Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree As . 1 but may be expressed more abstractly in the complete, orthonormal spherical ket basis. We demonstrate this with the example of the p functions. The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation. In that case, one needs to expand the solution of known regions in Laurent series (about {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } that obey Laplace's equation. &\Pi_{\psi_{+}}(\mathbf{r})=\quad \psi_{+}(-\mathbf{r})=\psi_{+}(\mathbf{r}) \\ ( . 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. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } f Here the solution was assumed to have the special form Y(, ) = () (). = {\displaystyle \{\pi -\theta ,\pi +\varphi \}} S x ; the remaining factor can be regarded as a function of the spherical angular coordinates : But when turning back to \(cos=z\) this factor reduces to \((\sin \theta)^{|m|}\). {\displaystyle \ell } are associated Legendre polynomials without the CondonShortley phase (to avoid counting the phase twice). can also be expanded in terms of the real harmonics {\displaystyle T_{q}^{(k)}} : , : 0 Note that the angular momentum is itself a vector. (see associated Legendre polynomials), In acoustics,[7] the Laplace spherical harmonics are generally defined as (this is the convention used in this article). terms (cosines) are included, and for / 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 spherical harmonics are orthonormal: that is, Y l, m Yl, md = ll mm, and also form a complete set. {\displaystyle \ell } ) In this chapter we discuss the angular momentum operator one of several related operators analogous to classical angular momentum. A m C ) r { 3 is replaced by the quantum mechanical spin vector operator of Laplace's equation. 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. Y ( R Rotations and Angular momentum Intro The material here may be found in Sakurai Chap 3: 1-3, (5-6), 7, (9-10) . In 1782, Pierre-Simon de Laplace had, in his Mcanique Cleste, determined that the gravitational potential Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x p~. 1 m : {\displaystyle S^{2}} Y , : \end{aligned}\) (3.27). 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