: | , the space of the elements of {\displaystyle \ell } \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) The spherical harmonics, more generally, are important in problems with spherical symmetry. cos \(Y(\theta, \phi)=\Theta(\theta) \Phi(\phi)\) (3.9), Plugging this into (3.8) and dividing by \(\), we find, \(\left\{\frac{1}{\Theta}\left[\sin \theta \frac{d}{d \theta}\left(\sin \theta \frac{d \Theta}{d \theta}\right)\right]+\ell(\ell+1) \sin ^{2} \theta\right\}+\frac{1}{\Phi} \frac{d^{2} \Phi}{d \phi^{2}}=0\) (3.10). The 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. R = m C ( 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). ( is called a spherical harmonic function of degree and order m, ) {\displaystyle Y_{\ell m}} {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } Here, it is important to note that the real functions span the same space as the complex ones would. S 2 Y By analogy with classical mechanics, the operator L 2, that represents the magnitude squared of the angular momentum vector, is defined (7.1.2) L 2 = L x 2 + L y 2 + L z 2. In that case, one needs to expand the solution of known regions in Laurent series (about Then \(e^{im(+2)}=e^{im}\), and \(e^{im2}=1\) must hold. 2 Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3). to that use the CondonShortley phase convention: The classical spherical harmonics are defined as complex-valued functions on the unit sphere ) {\displaystyle \ell =1} . e^{-i m \phi} They are, moreover, a standardized set with a fixed scale or normalization. For example, for any {\displaystyle \gamma } S i m The set of all direction kets n` can be visualized . Y m y ), instead of the Taylor series (about The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. {\displaystyle m<0} and Consider the problem of finding solutions of the form f(r, , ) = R(r) Y(, ). The angular momentum relative to the origin produced by a momentum vector ! 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. 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. L 2 Y 21 and order the angular momentum and the energy of the particle are measured simultane-ously at time t= 0, what values can be obtained for each observable and with what probabilities? The spinor spherical harmonics are the natural spinor analog of the vector spherical harmonics. One can determine the number of nodal lines of each type by counting the number of zeros of i ( ,[15] one obtains a generating function for a standardized set of spherical tensor operators, {\displaystyle L_{\mathbb {R} }^{2}(S^{2})} m , and B C T You are all familiar, at some level, with spherical harmonics, from angular momentum in quantum mechanics. 2 . More general spherical harmonics of degree are not necessarily those of the Laplace basis The spherical harmonics are representations of functions of the full rotation group SO(3)[5]with rotational symmetry. . The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication. That is, they are either even or odd with respect to inversion about the origin. ) 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} ( With respect to this group, the sphere is equivalent to the usual Riemann sphere. {\displaystyle 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} {\displaystyle r^{\ell }} Y Chapters 1 and 2. Specifically, if, A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. = S {\displaystyle \ell } On the other hand, considering Y Given two vectors r and r, with spherical coordinates from the above-mentioned polynomial of degree m r! : {\displaystyle \theta } R , obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. 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 . [5] As suggested in the introduction, this perspective is presumably the origin of the term spherical harmonic (i.e., the restriction to the sphere of a harmonic function). \(\int|g(\theta, \phi)|^{2} \sin \theta d \theta d \phi<\infty\) 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: {\displaystyle \mathbb {R} ^{n}} This equation easily separates in . {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } The Laplace spherical harmonics 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. 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 \(\). B {\displaystyle \mathbf {r} } {\displaystyle x} {\displaystyle \ell } 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). R are composed of circles: there are |m| circles along longitudes and |m| circles along latitudes. The angle-preserving symmetries of the two-sphere are described by the group of Mbius transformations PSL(2,C). S m m . Meanwhile, when Statements relating the growth of the Sff() to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. L The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation. m R 2 r Y This can be formulated as: \(\Pi \mathcal{R}(r) Y_{\ell}^{m}(\theta, \phi)=\mathcal{R}(r) \Pi Y_{\ell}^{m}(\theta, \phi)=(-1)^{\ell} \mathcal{R}(r) Y(\theta, \phi)\) (3.31). 1 and another of , since any such function is automatically harmonic. 1 . The general, normalized Spherical Harmonic is depicted below: Y_ {l}^ {m} (\theta,\phi) = \sqrt { \dfrac { (2l + 1) (l - |m|)!} as follows, leading to functions {\displaystyle \mathbf {H} _{\ell }} Such spherical harmonics are a special case of zonal spherical functions. Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. f [ (12) for some choice of coecients am. 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 , f Essentially all the properties of the spherical harmonics can be derived from this generating function. , i.e. : {\displaystyle Y_{\ell }^{m}} To make full use of rotational symmetry and angular momentum, we will restrict our attention to spherically symmetric potentials, \begin {aligned} V (\vec {r}) = V (r). , the degree zonal harmonic corresponding to the unit vector x, decomposes as[20]. {\displaystyle v} 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. C {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } = ) do not have that property. y Y They occur in . In summary, if is not an integer, there are no convergent, physically-realizable solutions to the SWE. ( {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } by \(\mathcal{R}(r)\). For example, when Angular momentum and its conservation in classical mechanics. Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle between x1 and x. 2 { {\displaystyle (r,\theta ,\varphi )} e^{i m \phi} \\ 2 r , 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 ). R x Y {\displaystyle \varphi } (Here the scalar field is understood to be complex, i.e. 2 m The foregoing has been all worked out in the spherical coordinate representation, C to directions respectively. {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } Note that the angular momentum is itself a vector. ] P f 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. 2 are associated Legendre polynomials without the CondonShortley phase (to avoid counting the phase twice). One concludes that the spherical harmonics in the solution for the electron wavefunction in the hydrogen atom identify the angular momentum of the electron. 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 ! : ) The functions R When = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. ) are complex and mix axis directions, but the real versions are essentially just x, y, and z. , Y 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). [ S Find \(P_{2}^{0}(\theta)\), \(P_{2}^{1}(\theta)\), \(P_{2}^{2}(\theta)\). Inversion is represented by the operator Now we're ready to tackle the Schrdinger equation in three dimensions. m {\displaystyle (2\ell +1)} ( C Y = {\displaystyle m>0} ( {\displaystyle Y_{\ell }^{m}} where Show that \(P_{}(z)\) are either even, or odd depending on the parity of \(\). More generally, hypergeometric series can be generalized to describe the symmetries of any symmetric space; in particular, hypergeometric series can be developed for any Lie group. {\displaystyle Y_{\ell }^{m}} Y = ] i Specifically, we say that a (complex-valued) polynomial function This could be achieved by expansion of functions in series of trigonometric functions. ( 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). Clebsch Gordon coecients allow us to express the total angular momentum basis |jm; si in terms of the direct product The spherical harmonics are the eigenfunctions of the square of the quantum mechanical angular momentum operator. The 3-D wave equation; spherical harmonics. One can choose \(e^{im}\), and include the other one by allowing mm to be negative. in their expansion in terms of the with m > 0 are said to be of cosine type, and those with m < 0 of sine type. {\displaystyle \psi _{i_{1}\dots i_{\ell }}} Thus, the wavefunction can be written in a form that lends to separation of variables. {\displaystyle Y_{\ell }^{m}} &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 \\ In particular, if Sff() decays faster than any rational function of as , then f is infinitely differentiable. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } Therefore the single eigenvalue of \(^{2}\) is 1, and any function is its eigenfunction. R S &\Pi_{\psi_{+}}(\mathbf{r})=\quad \psi_{+}(-\mathbf{r})=\psi_{+}(\mathbf{r}) \\ See, e.g., Appendix A of Garg, A., Classical Electrodynamics in a Nutshell (Princeton University Press, 2012). {\displaystyle \mathbb {R} ^{3}} z (the irregular solid harmonics Abstract. {\displaystyle \ell } From this perspective, one has the following generalization to higher dimensions. m Y but may be expressed more abstractly in the complete, orthonormal spherical ket basis. For R In quantum mechanics the constants \(\ell\) and \(m\) are called the azimuthal quantum number and magnetic quantum number due to their association with rotation and how the energy of an . Y As to what's "really" going on, it's exactly the same thing that you have in the quantum mechanical addition of angular momenta. \end{aligned}\) (3.27). and modelling of 3D shapes. 1 {\displaystyle \Delta f=0} spherical harmonics implies that any well-behaved function of and can be written as f(,) = X =0 X m= amY m (,). r terms (cosines) are included, and for form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable 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. 1 only the 3 = \end{aligned}\) (3.6). Lecture 6: 3D Rigid Rotor, Spherical Harmonics, Angular Momentum We can now extend the Rigid Rotor problem to a rotation in 3D, corre-sponding to motion on the surface of a sphere of radius R. The Hamiltonian operator in this case is derived from the Laplacian in spherical polar coordi-nates given as 2 = 2 x 2 + y + 2 z . Let us also note that the \(m=0\) functions do not depend on \(\), and they are proportional to the Legendre polynomials in \(cos\). The solutions, \(Y_{\ell}^{m}(\theta, \phi)=\mathcal{N}_{l m} P_{\ell}^{m}(\theta) e^{i m \phi}\) (3.20). are guaranteed to be real, whereas their coefficients , the real and imaginary components of the associated Legendre polynomials each possess |m| zeros, each giving rise to a nodal 'line of latitude'. , one has. in m is an associated Legendre polynomial, N is a normalization constant, and and represent colatitude and longitude, respectively. {\displaystyle e^{\pm im\varphi }} , any square-integrable function S {\displaystyle S^{2}\to \mathbb {C} } m {\displaystyle r=\infty } 2 ( One might wonder what is the reason for writing the eigenvalue in the form \((+1)\), but as it will turn out soon, there is no loss of generality in this notation. Y r m R {\displaystyle r^{\ell }Y_{\ell }^{m}(\mathbf {r} /r)} B m The statement of the parity of spherical harmonics is then. {\displaystyle \mathbf {r} } S R That is, a polynomial p is in P provided that for any real 2 The spherical harmonics form an infinite system of orthonormal functions in the sense: \(\int_{0}^{2 \pi} \int_{0}^{\pi}\left(Y_{\ell^{\prime}}^{m^{\prime}}(\theta, \phi)\right)^{*} Y_{\ell}^{m}(\theta, \phi) \sin \theta d \theta d \phi=\delta_{\ell \ell^{\prime}} \delta_{m m^{\prime}}\) (3.22). q , [ {\displaystyle \mathbf {r} } = {\displaystyle (x,y,z)} ) ( {\displaystyle \ell } Spherical harmonics can be separated into two set of functions. : Any function of and can be expanded in the spherical harmonics . : 1 The spherical harmonics with negative can be easily compute from those with positive . m The eigenvalues of \(\) itself are then \(1\), and we have the following two possibilities: \(\begin{aligned} 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. 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