Abstract. In summary, if is not an integer, there are no convergent, physically-realizable solutions to the SWE. {\displaystyle \mathbf {a} } With respect to this group, the sphere is equivalent to the usual Riemann sphere. . 3 [13] These functions have the same orthonormality properties as the complex ones to correspond to a (smooth) function Y {\displaystyle (r,\theta ,\varphi )} = , {\displaystyle \Re [Y_{\ell }^{m}]=0} {\displaystyle \mathbf {r} } Y 4 , The angular momentum relative to the origin produced by a momentum vector ! Details of the calculation: ( r) = (x + y - 3z)f (r) = (rsincos + rsinsin - 3rcos)f (r) [1] These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above. {\displaystyle f_{\ell m}} . Then The solutions, \(Y_{\ell}^{m}(\theta, \phi)=\mathcal{N}_{l m} P_{\ell}^{m}(\theta) e^{i m \phi}\) (3.20). , which can be seen to be consistent with the output of the equations above. } The angular components of . {\displaystyle v} m R 2 ( of Laplace's equation. > Spherical coordinates, elements of vector analysis. r 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. {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} {\displaystyle \Im [Y_{\ell }^{m}]=0} In quantum mechanics they appear as eigenfunctions of (squared) orbital angular momentum. C Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section Higher dimensions below). / When = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. 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. Spherical Harmonics, and Bessel Functions Physics 212 2010, Electricity and Magnetism Michael Dine Department of Physics . . {\displaystyle \ell } There are two quantum numbers for the rigid rotor in 3D: \(J\) is the total angular momentum quantum number and \(m_J\) is the z-component of the angular momentum. There are several different conventions for the phases of \(\mathcal{N}_{l m}\), so one has to be careful with them. f ) used above, to match the terms and find series expansion coefficients (1) From this denition and the canonical commutation relation between the po-sition and momentum operators, it is easy to verify the commutation relation among the components of the angular momentum . \end{aligned}\) (3.27). 2 {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } We shall now find the eigenfunctions of \(_{}\), that play a very important role in quantum mechanics, and actually in several branches of theoretical physics. 2 Angular momentum and its conservation in classical mechanics. {\displaystyle \mathbb {R} ^{n}\to \mathbb {C} } Figure 3.1: Plot of the first six Legendre polynomials. m 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. C {\displaystyle P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )} 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. . For can be visualized by considering their "nodal lines", that is, the set of points on the sphere where It follows from Equations ( 371) and ( 378) that. Throughout the section, we use the standard convention that for m This was a boon for problems possessing spherical symmetry, such as those of celestial mechanics originally studied by Laplace and Legendre. < 0 p is ! m and another of S Any function of and can be expanded in the spherical harmonics . where the superscript * denotes complex conjugation. 0 {\displaystyle (r',\theta ',\varphi ')} m Z Laplace equation. \(\hat{L}^{2}=-\hbar^{2}\left(\partial_{\theta \theta}^{2}+\cot \theta \partial_{\theta}+\frac{1}{\sin ^{2} \theta} \partial_{\phi \phi}^{2}\right)=-\hbar^{2} \Delta_{\theta \phi}\) (3.7). : ) For a given value of , there are 2 + 1 independent solutions of this form, one for each integer m with m . 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. 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 . | {\displaystyle r>R} m 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. {\displaystyle \mathbf {A} _{\ell }} Y {\displaystyle \{\pi -\theta ,\pi +\varphi \}} (the irregular solid harmonics but may be expressed more abstractly in the complete, orthonormal spherical ket basis. , and 1 = Looking for the eigenvalues and eigenfunctions of \(\), we note first that \(^{2}=1\). For example, when Find the first three Legendre polynomials \(P_{0}(z)\), \(P_{1}(z)\) and \(P_{2}(z)\). ) Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree are constants and the factors r Ym are known as (regular) solid harmonics The real spherical harmonics ( P Nodal lines of 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. 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. : ) {\displaystyle Y_{\ell }^{m}} only, or equivalently of the orientational unit vector m , i.e. In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum[4]. {\displaystyle y} x 2 1 m 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. , then, a We will use the actual function in some problems. m ) S One concludes that the spherical harmonics in the solution for the electron wavefunction in the hydrogen atom identify the angular momentum of the electron. 1-62. P . \end {aligned} V (r) = V (r). R ) . &\Pi_{\psi_{+}}(\mathbf{r})=\quad \psi_{+}(-\mathbf{r})=\psi_{+}(\mathbf{r}) \\ is just the space of restrictions to the sphere {\displaystyle r=\infty } ) specified by these angles. ] : S The spherical harmonics, more generally, are important in problems with spherical symmetry. The spherical harmonics have definite parity. q S Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with P As to what's "really" going on, it's exactly the same thing that you have in the quantum mechanical addition of angular momenta. {\displaystyle \mathbf {r} } S In turn, SU(2) is identified with the group of unit quaternions, and so coincides with the 3-sphere. 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