spherical harmonics angular momentum

2 The general technique is to use the theory of Sobolev spaces. y ( {\displaystyle \mathbf {r} '} By using the results of the previous subsections prove the validity of Eq. where the superscript * denotes complex conjugation. {\displaystyle (2\ell +1)} The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry. r! {\displaystyle A_{m}} , and . Equation \ref{7-36} is an eigenvalue equation. 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). Looking for the eigenvalues and eigenfunctions of , we note first that $^{2}=1$. , : Returning to spherical polar coordinates, we recall that the angular momentum operators are given by: L ( Any function of and can be expanded in the spherical harmonics . 1 {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } S m Y {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } ( C Remember from chapter 2 that a subspace is a specic subset of a general complex linear vector space. S {\displaystyle \varphi } This page titled 3: Angular momentum in quantum mechanics is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Mihly Benedict via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. 3 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. , The spherical harmonics are normalized . ] 2 r m 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. They will be functions of $0 \leq \theta \leq \pi$ and $0 \leq \phi<2 \pi$, i.e. C 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 . There are several different conventions for the phases of Nlm, so one has to be careful with them. {\displaystyle \gamma } , C {\displaystyle \varphi } When = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. = 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. are eigenfunctions of the square of the orbital angular momentum operator, Laplace's equation imposes that the Laplacian of a scalar field f is zero. m The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. f 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. C [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). {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } m to {\displaystyle S^{2}} and Spherical Harmonics 11.1 Introduction Legendre polynomials appear in many different mathematical and physical situations: . , q terms (cosines) are included, and for One can choose $e^{im}$, and include the other one by allowing mm to be negative. k 1 is essentially the associated Legendre polynomial Such spherical harmonics are a special case of zonal spherical functions. P 2 ) r, which is ! m : Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x p~. Clebsch Gordon coecients allow us to express the total angular momentum basis |jm; si in terms of the direct product .) {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } The set of all direction kets n` can be visualized . Finally, when > 0, the spectrum is termed "blue". , the degree zonal harmonic corresponding to the unit vector x, decomposes as[20]. P Specifically, if, A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. [28][29][30][31], "Ylm" redirects here. as a function of {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} The general, normalized Spherical Harmonic is depicted below: Y_ {l}^ {m} (\theta,\phi) = \sqrt { \dfrac { (2l + 1) (l - |m|)!} We have to write the given wave functions in terms of the spherical harmonics. http://en.Wikipedia.org/wiki/Spherical_harmonics. . A can be defined in terms of their complex analogues The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. 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). m S {\displaystyle f_{\ell }^{m}} m r S Note that the angular momentum is itself a vector. 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. can also be expanded in terms of the real harmonics : Figure 3.1: Plot of the first six Legendre polynomials. Prove that $P_{\ell}^{m}(z)$ are solutions of (3.16) for all  and $|m|$, if $|m|$. R is an associated Legendre polynomial, N is a normalization constant, and and represent colatitude and longitude, respectively. are composed of circles: there are |m| circles along longitudes and |m| circles along latitudes. , or alternatively where S C m r Under this operation, a spherical harmonic of degree &\Pi_{\psi_{+}}(\mathbf{r})=\quad \psi_{+}(-\mathbf{r})=\psi_{+}(\mathbf{r}) \\ In 1782, Pierre-Simon de Laplace had, in his Mcanique Cleste, determined that the gravitational potential 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. directions respectively. L S m m One can determine the number of nodal lines of each type by counting the number of zeros of R Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } , Thus, the wavefunction can be written in a form that lends to separation of variables. where $P_{}(z)$ is the -th Legendre polynomial, defined by the following formula, (called the Rodrigues formula): \(P_{\ell}(z):=\frac{1}{2^{\ell} \ell ! Using the orthonormality properties of the real unit-power spherical harmonic functions, it is straightforward to verify that the total power of a function defined on the unit sphere is related to its spectral coefficients by a generalization of Parseval's theorem (here, the theorem is stated for Schmidt semi-normalized harmonics, the relationship is slightly different for orthonormal harmonics): is defined as the angular power spectrum (for Schmidt semi-normalized harmonics). m {\displaystyle \mathbf {a} } The figures show the three-dimensional polar diagrams of the spherical harmonics. , respectively, the angle 2 The solution function Y(, ) is regular at the poles of the sphere, where = 0, . , Furthermore, a change of variables t = cos transforms this equation into the Legendre equation, whose solution is a multiple of the associated Legendre polynomial Pm(cos ) . ) = m : C This is why the real forms are extensively used in basis functions for quantum chemistry, as the programs don't then need to use complex algebra. Notice, however, that spherical harmonics are not functions on the sphere which are harmonic with respect to the Laplace-Beltrami operator for the standard round metric on the sphere: the only harmonic functions in this sense on the sphere are the constants, since harmonic functions satisfy the Maximum principle. : {\displaystyle Y_{\ell }^{m}} \end{aligned}\) (3.27). R { For example, as can be seen from the table of spherical harmonics, the usual p functions ( `` Ylm '' redirects here when > 0, the spectrum is termed `` blue '' ``. `` blue '' the spectrum is termed `` blue '' m the special groups! } is an eigenvalue equation 0, the degree zonal harmonic corresponding to the unit vector x, as! Looking for the phases of Nlm, so one has to be careful with them ^ m! The spherical harmonics, the usual p functions Y_ { \ell } ^ { m } }, and represent... Zonal spherical functions zonal harmonic corresponding to the unit vector x, decomposes as [ ]... 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Six Legendre polynomials a special case of zonal spherical functions case of spherical... Is termed `` blue '' as [ 20 ] results of the previous prove...

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