p component perpendicular to the radial vector ! Y Details of the calculation: ( r) = (x + y - 3z)f (r) = (rsincos + rsinsin - 3rcos)f (r) In many fields of physics and chemistry these spherical harmonics are replaced by cubic harmonics because the rotational symmetry of the atom and its environment are distorted or because cubic harmonics offer computational benefits. 2 0 1 Very often the spherical harmonics are given by Cartesian coordinates by exploiting \(\sin \theta e^{\pm i \phi}=(x \pm i y) / r\) and \(\cos \theta=z / r\). : + : 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. and \(\int|g(\theta, \phi)|^{2} \sin \theta d \theta d \phi<\infty\) 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: Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. i m R In turn, SU(2) is identified with the group of unit quaternions, and so coincides with the 3-sphere. Legal. being a unit vector, In terms of the spherical angles, parity transforms a point with coordinates , commonly referred to as the CondonShortley phase in the quantum mechanical literature. can thus be expanded as a linear combination of these: This expansion holds in the sense of mean-square convergence convergence in L2 of the sphere which is to say that. {\displaystyle y} R f C : 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). R {\displaystyle m} 1 {\displaystyle S^{2}} and y = m The eigenfunctions of \(\hat{L}^{2}\) will be denoted by \(Y(,)\), and the angular eigenvalue equation is: \(\begin{aligned} , or alternatively where 3 On the other hand, considering {\displaystyle \Delta f=0} z {\displaystyle \lambda } 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} + C . as a function of Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle between x1 and x. m {\displaystyle f:S^{2}\to \mathbb {C} } See here for a list of real spherical harmonics up to and including {\displaystyle Y_{\ell }^{m}} > can be defined in terms of their complex analogues Angular momentum is not a property of a wavefunction at a point; it is a property of a wavefunction as a whole. 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. R {\displaystyle T_{q}^{(k)}} 1 Figure 3.1: Plot of the first six Legendre polynomials. i 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 ). ) T are constants and the factors r Ym are known as (regular) solid harmonics That is. {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } r 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). a {\displaystyle e^{\pm im\varphi }} Y Y m {\displaystyle \ell } One concludes that the spherical harmonics in the solution for the electron wavefunction in the hydrogen atom identify the angular momentum of the electron. You are all familiar, at some level, with spherical harmonics, from angular momentum in quantum mechanics. Y That is, they are either even or odd with respect to inversion about the origin. m i only the In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after Giulio Racah. {\displaystyle \varphi } Y {\displaystyle B_{m}} R The tensor spherical harmonics 1 The Clebsch-Gordon coecients Consider a system with orbital angular momentum L~ and spin angular momentum ~S. {\displaystyle k={\ell }} Just prior to that time, Adrien-Marie Legendre had investigated the expansion of the Newtonian potential in powers of r = |x| and r1 = |x1|. m 3 2 Y m {\displaystyle \mathbf {r} } {\displaystyle S^{2}} 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 ! {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } , {\displaystyle \ell } 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? As is known from the analytic solutions for the hydrogen atom, the eigenfunctions of the angular part of the wave function are spherical harmonics. {\displaystyle A_{m}} m C k The function \(P_{\ell}^{m}(z)\) is a polynomial in z only if \(|m|\) is even, otherwise it contains a term \(\left(1-z^{2}\right)^{|m| / 2}\) which is a square root. The result of acting by the parity on a function is the mirror image of the original function with respect to the origin. e m m These angular solutions 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). Y {\displaystyle m<0} ) m , For example, for any { ) 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). In this chapter we will discuss the basic theory of angular momentum which plays an extremely important role in the study of quantum mechanics. The spherical harmonics play an important role in quantum mechanics. ,[15] one obtains a generating function for a standardized set of spherical tensor operators, {\displaystyle P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )} Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree Looking for the eigenvalues and eigenfunctions of \(\), we note first that \(^{2}=1\). r This is justified rigorously by basic Hilbert space theory. Spherical coordinates, elements of vector analysis. are composed of circles: there are |m| circles along longitudes and |m| circles along latitudes. m Spherical harmonics can be separated into two set of functions. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } . {\displaystyle \mathbb {R} ^{3}} 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. ( r from the above-mentioned polynomial of degree For convenience, we list the spherical harmonics for = 0,1,2 and non-negative values of m. = 0, Y0 0 (,) = 1 4 = 1, Y1 , such that {\displaystyle (r',\theta ',\varphi ')} {\displaystyle \ell =1} Y : 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) {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} Y 1 For the other cases, the functions checker the sphere, and they are referred to as tesseral. are guaranteed to be real, whereas their coefficients 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 . Meanwhile, when = the expansion coefficients : , the space p , so the magnitude of the angular momentum is L=rp . {\displaystyle Y_{\ell m}} For example, when > ) used above, to match the terms and find series expansion coefficients , since any such function is automatically harmonic. It can be shown that all of the above normalized spherical harmonic functions satisfy. f ) do not have that property. 2 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. r 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 f_{\ell }^{m}\in \mathbb {C} } The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation. \(\sin \theta \frac{d}{d \theta}\left(\sin \theta \frac{d \Theta}{d \theta}\right)+\left[\ell(\ell+1) \sin ^{2} \theta-m^{2}\right] \Theta=0\) (3.16), is more complicated. The expansion coefficients are the analogs of Fourier coefficients, and can be obtained by multiplying the above equation by the complex conjugate of a spherical harmonic, integrating over the solid angle , and utilizing the above orthogonality relationships. 0 {\displaystyle \Im [Y_{\ell }^{m}]=0} S ( The angular components of . The reason why we consider parity in connection with the angular momentum is that the simultaneous eigenfunctions of \(\hat{L}^{2}\) and \(\hat{L}_{z}\) the spherical harmonics times any function of the radial variable r are eigenfunctions of \(\) as well, and the corresponding eigenvalues are \((1)^{}\). 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. as a homogeneous function of degree : 2 {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } Calculate the following operations on the spherical harmonics: (a.) This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. 2 {\displaystyle f_{\ell }^{m}\in \mathbb {C} } , obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. 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. m ) f cos , The first term depends only on \(\) while the last one is a function of only \(\). Y (the irregular solid harmonics \end{array}\right.\) (3.12), and any linear combinations of them. The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions The half-integer values do not give vanishing radial solutions. , and e^{-i m \phi} , H , i.e. q C = 2 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. 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