More general spherical harmonics of degree are not necessarily those of the Laplace basis e {\displaystyle S^{2}} With respect to this group, the sphere is equivalent to the usual Riemann sphere. , , the trigonometric sin and cos functions possess 2|m| zeros, each of which gives rise to a nodal 'line of longitude'. This expression is valid for both real and complex harmonics. of the elements of ) {\displaystyle f_{\ell }^{m}\in \mathbb {C} } For example, when C {\displaystyle \mathbf {r} } 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 (the irregular solid harmonics m The foregoing has been all worked out in the spherical coordinate representation, : 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. , {\displaystyle (2\ell +1)} We have to write the given wave functions in terms of the spherical harmonics. 1 Y z r {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} {\displaystyle Y_{\ell }^{m}} in y > . ) S , f &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 \\ The {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} about the origin that sends the unit vector {\displaystyle f:S^{2}\to \mathbb {C} \supset \mathbb {R} } Y as real parameters. The \(Y_{\ell}^{m}(\theta)\) functions are thus the eigenfunctions of \(\hat{L}\) corresponding to the eigenvalue \(\hbar^{2} \ell(\ell+1)\), and they are also eigenfunctions of \(\hat{L}_{z}=-i \hbar \partial_{\phi}\), because, \(\hat{L}_{z} Y_{\ell}^{m}(\theta, \phi)=-i \hbar \partial_{\phi} Y_{\ell}^{m}(\theta, \phi)=\hbar m Y_{\ell}^{m}(\theta, \phi)\) (3.21). 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. {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } {\displaystyle p:\mathbb {R} ^{3}\to \mathbb {C} } and : m and order S Rotations and Angular momentum Intro The material here may be found in Sakurai Chap 3: 1-3, (5-6), 7, (9-10) . Note that the angular momentum is itself a vector. For example, as can be seen from the table of spherical harmonics, the usual p functions ( 2 2 r , which can be seen to be consistent with the output of the equations above. The (complex-valued) spherical harmonics Y ) are chosen instead. {\displaystyle \mathbf {A} _{\ell }} m R r, which is ! R {\displaystyle \varphi } Y r \end {aligned} V (r) = V (r). f ) Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. , 1 m ( p , so the magnitude of the angular momentum is L=rp . {\displaystyle \mathbf {J} } m T 1 C P with m > 0 are said to be of cosine type, and those with m < 0 of sine type. The spherical harmonics, more generally, are important in problems with spherical symmetry. {\displaystyle f:S^{2}\to \mathbb {R} } R {\displaystyle \mathbb {R} ^{n}} {\displaystyle z} These operators commute, and are densely defined self-adjoint operators on the weighted Hilbert space of functions f square-integrable with respect to the normal distribution as the weight function on R3: If Y is a joint eigenfunction of L2 and Lz, then by definition, Denote this joint eigenspace by E,m, and define the raising and lowering operators by. S \end{aligned}\) (3.8). 3 {\displaystyle S^{2}} The spherical harmonics have definite parity. A Representation of Angular Momentum Operators We would like to have matrix operators for the angular momentum operators L x; L y, and L z. R {\displaystyle Y_{\ell }^{m}} is just the 3-dimensional space of all linear functions Then m > 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} 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) On the unit sphere S There are several different conventions for the phases of \(\mathcal{N}_{l m}\), so one has to be careful with them. m {\displaystyle \theta } {\displaystyle (A_{m}\pm iB_{m})} C setting, If the quantum mechanical convention is adopted for the R : This could be achieved by expansion of functions in series of trigonometric functions. {4\pi (l + |m|)!} : This is useful for instance when we illustrate the orientation of chemical bonds in molecules. A {\displaystyle m>0} On the other hand, considering Show that \(P_{}(z)\) are either even, or odd depending on the parity of \(\). = S {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } m {\displaystyle A_{m}(x,y)} above as a sum. {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } ( f {\displaystyle \Im [Y_{\ell }^{m}]=0} }\left(\frac{d}{d z}\right)^{\ell}\left(z^{2}-1\right)^{\ell}\) (3.18). ( 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. as a function of 3 S In both definitions, the spherical harmonics are orthonormal, The disciplines of geodesy[10] and spectral analysis use, The magnetics[10] community, in contrast, uses Schmidt semi-normalized harmonics. In spherical coordinates this is:[2]. r {\displaystyle m>0} 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). , and their nodal sets can be of a fairly general kind.[22]. The 3-D wave equation; spherical harmonics. is an associated Legendre polynomial, N is a normalization constant, and and represent colatitude and longitude, respectively. ) The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of {\displaystyle f:S^{2}\to \mathbb {R} } {\displaystyle \ell =4} ( p [12], A real basis of spherical harmonics ( + , A , the real and imaginary components of the associated Legendre polynomials each possess |m| zeros, each giving rise to a nodal 'line of latitude'. is given as a constant multiple of the appropriate Gegenbauer polynomial: Combining (2) and (3) gives (1) in dimension n = 2 when x and y are represented in spherical coordinates. {\displaystyle Y_{\ell m}} they can be considered as complex valued functions whose domain is the unit sphere. 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. This is because a plane wave can actually be written as a sum over spherical waves: \[ e^{i\vec{k}\cdot\vec{r}}=e^{ikr\cos\theta}=\sum_l i^l(2l+1)j_l(kr)P_l(\cos\theta) \label{10.2.2}\] Visualizing this plane wave flowing past the origin, it is clear that in spherical terms the plane wave contains both incoming and outgoing spherical waves. Just as in one dimension the eigenfunctions of d 2 / d x 2 have the spatial dependence of the eigenmodes of a vibrating string, the spherical harmonics have the spatial dependence of the eigenmodes of a vibrating spherical . {\displaystyle \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 \lambda } The spherical harmonics are representations of functions of the full rotation group SO(3)[5]with rotational symmetry. Then \(e^{im(+2)}=e^{im}\), and \(e^{im2}=1\) must hold. x C = k [23] Let P denote the space of complex-valued homogeneous polynomials of degree in n real variables, here considered as functions Y L 2 Y 21 In the quantum mechanics community, it is common practice to either include this phase factor in the definition of the associated Legendre polynomials, or to append it to the definition of the spherical harmonic functions. f to 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 \(\). , {\displaystyle Z_{\mathbf {x} }^{(\ell )}({\mathbf {y} })} n That is, they are either even or odd with respect to inversion about the origin. S Finally, evaluating at x = y gives the functional identity, Another useful identity expresses the product of two spherical harmonics as a sum over spherical harmonics[21]. m One can choose \(e^{im}\), and include the other one by allowing mm to be negative. m ( Y A Very often the spherical harmonics are given by Cartesian coordinates by exploiting \(\sin \theta e^{\pm i \phi}=(x \pm i y) / r\) and \(\cos \theta=z / r\). of spherical harmonics of degree Show that the transformation \(\{x, y, z\} \longrightarrow\{-x,-y,-z\}\) is equivalent to \(\theta \longrightarrow \pi-\theta, \quad \phi \longrightarrow \phi+\pi\). 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). In fact, L 2 is equivalent to 2 on the spherical surface, so the Y l m are the eigenfunctions of the operator 2. the one containing the time dependent factor \(e_{it/}\) as well given by the function \(Y_{1}^{3}(,)\). Here the solution was assumed to have the special form Y(, ) = () (). {\displaystyle S^{n-1}\to \mathbb {C} } , one has. . = S : R Y {\displaystyle f_{\ell }^{m}\in \mathbb {C} } {\displaystyle Y_{\ell }^{m}} 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. 2 1 2 R 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. terms (cosines) are included, and for m 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 normalized . ( {\displaystyle \ell =2} by setting, The real spherical harmonics The three Cartesian components of the angular momentum are: L x = yp z zp y,L y = zp x xp z,L z = xp y yp x. e^{-i m \phi} R 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]. Legal. Y [ {\displaystyle \mathbf {a} =[{\frac {1}{2}}({\frac {1}{\lambda }}-\lambda ),-{\frac {i}{2}}({\frac {1}{\lambda }}+\lambda ),1].}. 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. Y = m m Laplace's spherical harmonics [ edit] Real (Laplace) spherical harmonics for (top to bottom) and (left to right). 2 q The spherical harmonics with negative can be easily compute from those with positive . cos f } Inversion is represented by the operator ( The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. The real spherical harmonics Share Cite Improve this answer Follow edited Aug 26, 2019 at 15:19 a C , and the factors 2 S : {\displaystyle \ell } 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. ( \(\sin \theta \frac{d}{d \theta}\left(\sin \theta \frac{d \Theta}{d \theta}\right)+\left[\ell(\ell+1) \sin ^{2} \theta-m^{2}\right] \Theta=0\) (3.16), is more complicated. Introduction to the Physics of Atoms, Molecules and Photons (Benedict), { "1.01:_New_Page" : "property get [Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass228_0.
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