A 1 The spherical harmonic functions depend on the spherical polar angles and and form an (infinite) complete set of orthogonal, normalizable functions. r m but may be expressed more abstractly in the complete, orthonormal spherical ket basis. { by \(\mathcal{R}(r)\). m The quantum number \(\) is called angular momentum quantum number, or sometimes for a historical reason as azimuthal quantum number, while m is the magnetic quantum number. The solid harmonics were homogeneous polynomial solutions 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). from the above-mentioned polynomial of degree only, or equivalently of the orientational unit vector , Then \(e^{im(+2)}=e^{im}\), and \(e^{im2}=1\) must hold. The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. {\displaystyle S^{2}} (8.2) 8.2 Angular momentum operator For a quantum system the angular momentum is an observable, we can measure the angular momentum of a particle in a given quantum state. m {\displaystyle \lambda } 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. The (complex-valued) spherical harmonics are eigenfunctions of the square of the orbital angular momentum operator and therefore they represent the different quantized configurations of atomic orbitals . R {\displaystyle f:S^{2}\to \mathbb {C} \supset \mathbb {R} } R {\displaystyle S^{n-1}\to \mathbb {C} } Y Y &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 ClebschGordan coefficients are the coefficients appearing in the expansion of the product of two spherical harmonics in terms of spherical harmonics themselves. We will use the actual function in some problems. m Y The solutions, \(Y_{\ell}^{m}(\theta, \phi)=\mathcal{N}_{l m} P_{\ell}^{m}(\theta) e^{i m \phi}\) (3.20). S 2 A They occur in . : {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } r The spherical harmonics are orthogonal functions, and are properly normalized with respect to integration over the entire solid angle: (381) The spherical harmonics also form a complete set for representing general functions of and . Functions that are solutions to Laplace's equation are called harmonics. = C This operator thus must be the operator for the square of the angular momentum. In quantum mechanics they appear as eigenfunctions of (squared) orbital angular momentum. 2 The spherical harmonics called \(J_J^{m_J}\) are functions whose probability \(|Y_J^{m_J}|^2\) has the well known shapes of the s, p and d orbitals etc learned in general chemistry. {\displaystyle \Re [Y_{\ell }^{m}]=0} q Meanwhile, when , {\displaystyle \ell } and order ( 3 R are composed of circles: there are |m| circles along longitudes and |m| circles along latitudes. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } ( where the absolute values of the constants \(\mathcal{N}_{l m}\) ensure the normalization over the unit sphere, are called spherical harmonics. But when turning back to \(cos=z\) this factor reduces to \((\sin \theta)^{|m|}\). This equation easily separates in . The set of all direction kets n` can be visualized . ( , the real and imaginary components of the associated Legendre polynomials each possess |m| zeros, each giving rise to a nodal 'line of latitude'. terms (sines) are included: The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation. ( Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with 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. . The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication. [14] An immediate benefit of this definition is that if the vector By polarization of A, there are coefficients {\displaystyle Y_{\ell }^{m}({\mathbf {r} })} One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of , Prove that \(P_{}(z)\) are solutions of (3.16) for \(m=0\). 1 = B and Spherical Harmonics 11.1 Introduction Legendre polynomials appear in many different mathematical and physical situations: . and {\displaystyle r^{\ell }} ) Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. that use the CondonShortley phase convention: The classical spherical harmonics are defined as complex-valued functions on the unit sphere {\displaystyle \ell =2} by setting, The real spherical harmonics ( With the definition of the position and the momentum operators we obtain the angular momentum operator as, \(\hat{\mathbf{L}}=-i \hbar(\mathbf{r} \times \nabla)\) (3.2), The Cartesian components of \(\hat{\mathbf{L}}\) are then, \(\hat{L}_{x}=-i \hbar\left(y \partial_{z}-z \partial_{y}\right), \quad \hat{L}_{y}=-i \hbar\left(z \partial_{x}-x \partial_{z}\right), \quad \hat{L}_{z}=-i \hbar\left(x \partial_{y}-y \partial_{x}\right)\) (3.3), One frequently needs the components of \(\hat{\mathbf{L}}\) in spherical coordinates. Let us also note that the \(m=0\) functions do not depend on \(\), and they are proportional to the Legendre polynomials in \(cos\). C to correspond to a (smooth) function R {\displaystyle {\mathcal {R}}} \(\begin{aligned} http://en.Wikipedia.org/wiki/Spherical_harmonics. The state to be shown, can be chosen by setting the quantum numbers \(\) and m. http://titan.physx.u-szeged.hu/~mmquantum/interactive/Gombfuggvenyek.nbp. {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} That is, they are either even or odd with respect to inversion about the origin. [13] These functions have the same orthonormality properties as the complex ones S m In this chapter we will discuss the basic theory of angular momentum which plays an extremely important role in the study of quantum mechanics. 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} The eigenvalues of \(\) itself are then \(1\), and we have the following two possibilities: \(\begin{aligned} {\displaystyle m>0} C m {\displaystyle \ell } r {\displaystyle f:S^{2}\to \mathbb {R} } For convenience, we list the spherical harmonics for = 0,1,2 and non-negative values of m. = 0, Y0 0 (,) = 1 4 = 1, Y1 ) {\displaystyle \theta } P m Y 's, which in turn guarantees that they are spherical tensor operators, C Y ( \(\sin \theta \frac{d}{d \theta}\left(\sin \theta \frac{d \Theta}{d \theta}\right)+\left[\ell(\ell+1) \sin ^{2} \theta-m^{2}\right] \Theta=0\) (3.16), is more complicated. {\displaystyle Y_{\ell m}} m Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis, each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. Looking for the eigenvalues and eigenfunctions of \(\), we note first that \(^{2}=1\). 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. We demonstrate this with the example of the p functions. A {\displaystyle r=\infty } {\displaystyle \ell =1} Spherical Harmonics, and Bessel Functions Physics 212 2010, Electricity and Magnetism Michael Dine Department of Physics . T Spherical harmonics originate from solving Laplace's equation in the spherical domains. ) The convergence of the series holds again in the same sense, namely the real spherical harmonics From this perspective, one has the following generalization to higher dimensions. Spherical harmonics are ubiquitous in atomic and molecular physics. {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } 3 R Y } {\displaystyle Z_{\mathbf {x} }^{(\ell )}({\mathbf {y} })} Rotations and Angular momentum Intro The material here may be found in Sakurai Chap 3: 1-3, (5-6), 7, (9-10) . There are of course functions which are neither even nor odd, they do not belong to the set of eigenfunctions of \(\). are guaranteed to be real, whereas their coefficients {\displaystyle Y_{\ell }^{m}} {\displaystyle (r,\theta ,\varphi )} {\displaystyle (A_{m}\pm iB_{m})} A In particular, if Sff() decays faster than any rational function of as , then f is infinitely differentiable. C of spherical harmonics of degree .) By using the results of the previous subsections prove the validity of Eq. The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of S One sees at once the reason and the advantage of using spherical coordinates: the operators in question do not depend on the radial variable r. This is of course also true for \(\hat{L}^{2}=\hat{L}_{x}^{2}+\hat{L}_{y}^{2}+\hat{L}_{z}^{2}\) which turns out to be \(^{2}\) times the angular part of the Laplace operator \(_{}\). = + The essential property of ) 0 > Therefore the single eigenvalue of \(^{2}\) is 1, and any function is its eigenfunction. The 2 m e ] [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). 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 Now we're ready to tackle the Schrdinger equation in three dimensions. {\displaystyle Y_{\ell }^{m}} ) When = 0, the spectrum is "white" as each degree possesses equal power. ( Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. ] C being a unit vector, In terms of the spherical angles, parity transforms a point with coordinates are sometimes known as tesseral spherical harmonics. (See Applications of Legendre polynomials in physics for a more detailed analysis. 1 , which can be seen to be consistent with the output of the equations above. If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. . where \(P_{}(z)\) is the \(\)-th Legendre polynomial, defined by the following formula, (called the Rodrigues formula): \(P_{\ell}(z):=\frac{1}{2^{\ell} \ell ! \end{array}\right.\) (3.12), and any linear combinations of them. The spherical harmonics play an important role in quantum mechanics. . For example, as can be seen from the table of spherical harmonics, the usual p functions ( Y We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. and Y : For example, for any Spherical harmonics can be generalized to higher-dimensional Euclidean space to is that for real functions Abstractly, the ClebschGordan coefficients express the tensor product of two irreducible representations of the rotation group as a sum of irreducible representations: suitably normalized, the coefficients are then the multiplicities. Angular momentum is the generator for rotations, so spherical harmonics provide a natural characterization of the rotational properties and direction dependence of a system. {\displaystyle r>R} Y f {\displaystyle x} {\displaystyle L_{\mathbb {C} }^{2}(S^{2})} 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 S^{2}\to \mathbb {C} } Y 0 where the superscript * denotes complex conjugation. 1 : ) Z k {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} Hence, r! The spherical harmonics are normalized . directions respectively. S 2 they can be considered as complex valued functions whose domain is the unit sphere. the one containing the time dependent factor \(e_{it/}\) as well given by the function \(Y_{1}^{3}(,)\). \(\int|g(\theta, \phi)|^{2} \sin \theta d \theta d \phi<\infty\) 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: 2 ( 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]. S m \(Y(\theta, \phi)=\Theta(\theta) \Phi(\phi)\) (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). i The spherical harmonics are orthonormal: that is, Y l, m Yl, md = ll mm, and also form a complete set. The (complex-valued) spherical harmonics This is valid for any orthonormal basis of spherical harmonics of degree, Applications of Legendre polynomials in physics, Learn how and when to remove this template message, "Symmetric tensor spherical harmonics on the N-sphere and their application to the de Sitter group SO(N,1)", "Zernike like functions on spherical cap: principle and applications in optical surface fitting and graphics rendering", "On nodal sets and nodal domains on S and R", https://en.wikipedia.org/w/index.php?title=Spherical_harmonics&oldid=1146217720, D. A. Varshalovich, A. N. Moskalev, V. K. Khersonskii, This page was last edited on 23 March 2023, at 13:52. Now, it is easily demonstrated that if A and B are two general operators then (7.1.3) [ A 2, B] = A [ A, B] + [ A, B] A. There is no requirement to use the CondonShortley phase in the definition of the spherical harmonic functions, but including it can simplify some quantum mechanical operations, especially the application of raising and lowering operators. -\Delta_{\theta \phi} Y(\theta, \phi) &=\ell(\ell+1) Y(\theta, \phi) \quad \text { or } \\ They will be functions of \(0 \leq \theta \leq \pi\) and \(0 \leq \phi<2 \pi\), i.e. {\displaystyle e^{\pm im\varphi }} 2 Y form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions , and their nodal sets can be of a fairly general kind.[22]. This was a boon for problems possessing spherical symmetry, such as those of celestial mechanics originally studied by Laplace and Legendre. C {\displaystyle \ell } about the origin that sends the unit vector {\displaystyle {\mathcal {Y}}_{\ell }^{m}} m m 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. 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). R {\displaystyle y} ( {\displaystyle P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )} , n 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. m In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum[4]. ) R The spherical harmonics are representations of functions of the full rotation group SO(3)[5]with rotational symmetry. m http://titan.physx.u-szeged.hu/~mmquantum/videok/Gombfuggveny_fazis_idofejlodes.flv. R m (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 . y m Spherical harmonics can be separated into two set of functions. m {\displaystyle \mathbf {r} } = only the 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. The spinor spherical harmonics are the natural spinor analog of the vector spherical harmonics. We will first define the angular momentum operator through the classical relation L = r p and replace p by its operator representation -i [see Eq. = S This is well known in quantum mechanics, since [ L 2, L z] = 0, the good quantum numbers are and m. Would it be possible to find another solution analogous to the spherical harmonics Y m ( , ) such that [ L 2, L x or y] = 0? 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Square of the spherical harmonics 11.1 Introduction Legendre polynomials appear in many different mathematical and situations. The orbital angular momentum in many different mathematical and physical situations: analog! ) ( 3.12 ), we note first that \ ( \ ) and spherical spherical harmonics angular momentum. Called harmonics must be the operator for the square of the previous subsections prove the of! A more detailed analysis results of the orbital angular momentum already in set.

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