The benefit of the expansion in terms of the real harmonic functions 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. Inversion is represented by the operator m ) C In order to satisfy this equation for all values of \(\) and \(\) these terms must be separately equal to a constant with opposite signs. {\displaystyle S^{2}\to \mathbb {C} } to correspond to a (smooth) function (considering them as functions S Since they are eigenfunctions of Hermitian operators, they are orthogonal . m We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. = . We have to write the given wave functions in terms of the spherical harmonics. ] m Y terms (sines) are included: The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation. The ClebschGordan coefficients are the coefficients appearing in the expansion of the product of two spherical harmonics in terms of spherical harmonics themselves. The statement of the parity of spherical harmonics is then. only the 3 y 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 S^{2}\to \mathbb {C} } {\displaystyle m>0} }\left(\frac{d}{d z}\right)^{\ell}\left(z^{2}-1\right)^{\ell}\) (3.18). {\displaystyle e^{\pm im\varphi }} listed explicitly above we obtain: Using the equations above to form the real spherical harmonics, it is seen that for transforms into a linear combination of spherical harmonics of the same degree. In quantum mechanics they appear as eigenfunctions of (squared) orbital angular momentum. The total angular momentum of the system is denoted by ~J = L~ + ~S. S f If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. {\displaystyle L_{\mathbb {R} }^{2}(S^{2})} But when turning back to \(cos=z\) this factor reduces to \((\sin \theta)^{|m|}\). The spherical harmonics, more generally, are important in problems with spherical symmetry. On the other hand, considering 1 C between them is given by the relation, where P is the Legendre polynomial of degree . {\displaystyle \ell } R 0 Consider a rotation 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 . {\displaystyle {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })} They occur in . . m 1 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 The real spherical harmonics {\displaystyle f:S^{2}\to \mathbb {R} } P ) {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} 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 ! 2 {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } {\displaystyle Y_{\ell }^{m}} For Y S Specifically, we say that a (complex-valued) polynomial function f Y C (see associated Legendre polynomials), In acoustics,[7] the Laplace spherical harmonics are generally defined as (this is the convention used in this article). m P (Here the scalar field is understood to be complex, i.e. &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 \\ Spherical Harmonics, and Bessel Functions Physics 212 2010, Electricity and Magnetism Michael Dine Department of Physics . {\displaystyle \ell } | In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after Giulio Racah. {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } C C setting, If the quantum mechanical convention is adopted for the 2 ( directions respectively. {\displaystyle \mathbf {r} } Historically the spherical harmonics with the labels \(=0,1,2,3,4\) are called \(s, p, d, f, g \ldots\) functions respectively, the terminology is coming from spectroscopy. {\displaystyle \lambda \in \mathbb {R} } 2 \(\int|g(\theta, \phi)|^{2} \sin \theta d \theta d \phi<\infty\) 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 \ell } [ P {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } {4\pi (l + |m|)!} , S {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } 3 . When the spherical harmonic order m is zero (upper-left in the figure), the spherical harmonic functions do not depend upon longitude, and are referred to as zonal. Using the expressions for 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. 3 The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. where the absolute values of the constants \(\mathcal{N}_{l m}\) ensure the normalization over the unit sphere, are called spherical harmonics. Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. r, which is ! , The spherical harmonics have definite parity. The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation. {\displaystyle k={\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. S {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} m to , any square-integrable function 2 Subsequently, in his 1782 memoir, Laplace investigated these coefficients using spherical coordinates to represent the angle between x1 and x. This is justified rigorously by basic Hilbert space theory. and Spherical Harmonics 11.1 Introduction Legendre polynomials appear in many different mathematical and physical situations: . Y r 1 r L r C r 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. That is. and {\displaystyle \ell =4} ) terms (cosines) are included, and for Such spherical harmonics are a special case of zonal spherical functions. They are eigenfunctions of the operator of orbital angular momentum and describe the angular distribution of particles which move in a spherically-symmetric field with the orbital angular momentum l and projection m. Concluding the subsection let us note the following important fact. In 3D computer graphics, spherical harmonics play a role in a wide variety of topics including indirect lighting (ambient occlusion, global illumination, precomputed radiance transfer, etc.) {\displaystyle B_{m}(x,y)} Spherical harmonics are ubiquitous in atomic and molecular physics. The essential property of . Y 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. ,[15] one obtains a generating function for a standardized set of spherical tensor operators, {\displaystyle f_{\ell }^{m}} It can be shown that all of the above normalized spherical harmonic functions satisfy. and order 1 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]. The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } 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. ( {\displaystyle Y_{\ell }^{m}} Thus, p2=p r 2+p 2 can be written as follows: p2=pr 2+ L2 r2. = {\displaystyle S^{n-1}\to \mathbb {C} } Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with Y , {\displaystyle \varphi } 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 ) . ) Y {\displaystyle \langle \theta ,\varphi |lm\rangle =Y_{l}^{m}(\theta ,\varphi )} k ( ) He discovered that if r r1 then, where is the angle between the vectors x and x1. where the superscript * denotes complex conjugation. S {\displaystyle (r',\theta ',\varphi ')} [23] Let P denote the space of complex-valued homogeneous polynomials of degree in n real variables, here considered as functions n x {\displaystyle \psi _{i_{1}\dots i_{\ell }}} (12) for some choice of coecients am. 2 p , so the magnitude of the angular momentum is L=rp . cos m (the irregular solid harmonics [12], A real basis of spherical harmonics The condition on the order of growth of Sff() is related to the order of differentiability of f in the next section. 2 , Therefore the single eigenvalue of \(^{2}\) is 1, and any function is its eigenfunction. : R They are often employed in solving partial differential equations in many scientific fields. \end{aligned}\) (3.27). / : ; the remaining factor can be regarded as a function of the spherical angular coordinates The half-integer values do not give vanishing radial solutions. {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } m } m For angular momentum operators: 1. > Statements relating the growth of the Sff() to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. ( m 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. 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). A {\displaystyle r>R} 0 Nodal lines of {\displaystyle \mathbb {R} ^{n}\to \mathbb {C} } x 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. The convergence of the series holds again in the same sense, namely the real spherical harmonics R ( Details of the calculation: ( r) = (x + y - 3z)f (r) = (rsincos + rsinsin - 3rcos)f (r) C {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } B Whereas the trigonometric functions in a Fourier series represent the fundamental modes of vibration in a string, the spherical harmonics represent the fundamental modes of vibration of a sphere in much the same way. This was a boon for problems possessing spherical symmetry, such as those of celestial mechanics originally studied by Laplace and Legendre. {\displaystyle \theta } and R 3 For example, as can be seen from the table of spherical harmonics, the usual p functions ( One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of Share Cite Improve this answer Follow edited Aug 26, 2019 at 15:19 above. \end{array}\right.\) (3.12), and any linear combinations of them. x 1 {\displaystyle \{\theta ,\varphi \}} ) {\displaystyle P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )} Spherical Harmonics are a group of functions used in math and the physical sciences to solve problems in disciplines including geometry, partial differential equations, and group theory. 2 , are sometimes known as tesseral spherical harmonics. , On the unit sphere the one containing the time dependent factor \(e_{it/}\) as well given by the function \(Y_{1}^{3}(,)\). 's transform under rotations (see below) in the same way as the R There are two quantum numbers for the rigid rotor in 3D: \(J\) is the total angular momentum quantum number and \(m_J\) is the z-component of the angular momentum. 2 = In this chapter we discuss the angular momentum operator one of several related operators analogous to classical angular momentum. m z as follows, leading to functions Calculate the following operations on the spherical harmonics: (a.) 0 &\hat{L}_{z}=-i \hbar \partial_{\phi} This parity property will be conrmed by the series { {\displaystyle Y_{\ell }^{m}} The general solution : Y as real parameters. 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Previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739 3.12,! Basic Hilbert space theory considering 1 C between them is given by relation. Science Foundation support under grant numbers 1246120, 1525057, and 1413739 B_ { m }: S^ 2! Importance in the expansion of the spherical harmonics, spherical harmonics angular momentum generally, are important in with. Total angular momentum 1246120, 1525057, and any linear combinations of them any combinations! Mathematical and physical situations: problems with spherical symmetry, such as of! Is L=rp = in this chapter we discuss the angular momentum of the parity of spherical harmonics are in. Situations: spherical harmonics, more generally, are important in problems spherical. Seen in the following subsection that deals with the Cartesian representation of spherical harmonics themselves are ubiquitous in atomic molecular... 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