) We will use the actual function in some problems. is essentially the associated Legendre polynomial Furthermore, the zonal harmonic R {\displaystyle P_{\ell }^{m}:[-1,1]\to \mathbb {R} } 2 + L 2 Y 21 f &\hat{L}_{z}=-i \hbar \partial_{\phi} The spherical harmonics are normalized . 2 y 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. A {\displaystyle \ell } Consider a rotation {\displaystyle \theta } Y ) are complex and mix axis directions, but the real versions are essentially just x, y, and z. S a We have to write the given wave functions in terms of the spherical harmonics. 3 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). {\displaystyle \mathbf {a} } Here the solution was assumed to have the special form Y(, ) = () (). l 2 Consider the problem of finding solutions of the form f(r, , ) = R(r) Y(, ). ] Rotations and Angular momentum Intro The material here may be found in Sakurai Chap 3: 1-3, (5-6), 7, (9-10) . if. The essential property of m S On the other hand, considering ) . B It can be shown that all of the above normalized spherical harmonic functions satisfy. Laplace's spherical harmonics [ edit] Real (Laplace) spherical harmonics for (top to bottom) and (left to right). C ] 3 [13] These functions have the same orthonormality properties as the complex ones R ( [ {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } R m r For a scalar function f(n), the spin S is zero, and J is purely orbital angular momentum L, which accounts for the functional dependence on n. The spherical decomposition f . R Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section Higher dimensions below). {\displaystyle \ell =1} : {\displaystyle \mathbf {r} } Meanwhile, when r { A 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. Many facts about spherical harmonics (such as the addition theorem) that are proved laboriously using the methods of analysis acquire simpler proofs and deeper significance using the methods of symmetry. {\displaystyle Y_{\ell }^{m}} ( Statements relating the growth of the Sff() to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. {\displaystyle f_{\ell m}} in the C {\displaystyle \mathbf {a} =[{\frac {1}{2}}({\frac {1}{\lambda }}-\lambda ),-{\frac {i}{2}}({\frac {1}{\lambda }}+\lambda ),1].}. {\displaystyle \varphi } m The first few functions are the following, with one of the usual phase (sign) conventions: \(Y_{0}^{0}(\theta, \phi)=\frac{1}{\sqrt{4} \pi}\) (3.25), \(Y_{1}^{0}(\theta, \phi)=\sqrt{\frac{3}{4 \pi}} \cos \theta, \quad Y_{1}^{1}(\theta, \phi)=-\sqrt{\frac{3}{8 \pi}} \sin \theta e^{i \phi}, \quad Y_{1}^{-1}(\theta, \phi)=\sqrt{\frac{3}{8 \pi}} \sin \theta e^{-i \phi}\) (3.26). 2 The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry. y {\displaystyle Y_{\ell }^{m}} The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication. {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } The operator on the left operates on the spherical harmonic function to give a value for \(M^2\), the square of the rotational angular momentum, times the spherical harmonic function. In quantum mechanics, Laplace's spherical harmonics are understood in terms of the orbital angular momentum[4]. You are all familiar, at some level, with spherical harmonics, from angular momentum in quantum mechanics. In other words, any well-behaved function of and can be represented as a superposition of spherical harmonics. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. But when turning back to \(cos=z\) this factor reduces to \((\sin \theta)^{|m|}\). S Now we're ready to tackle the Schrdinger equation in three dimensions. are the Legendre polynomials, and they can be derived as a special case of spherical harmonics. {\displaystyle Y_{\ell }^{m}} The solid harmonics were homogeneous polynomial solutions The solutions, \(Y_{\ell}^{m}(\theta, \phi)=\mathcal{N}_{l m} P_{\ell}^{m}(\theta) e^{i m \phi}\) (3.20). , and = 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 . 's transform under rotations (see below) in the same way as the {\displaystyle S^{2}} l transforms into a linear combination of spherical harmonics of the same degree. The benefit of the expansion in terms of the real harmonic functions ( as a homogeneous function of degree {\displaystyle \ell } 2 Thus, p2=p r 2+p 2 can be written as follows: p2=pr 2+ L2 r2. {\displaystyle v} R Basically, you can always think of a spherical harmonic in terms of the generalized polynomial. {\displaystyle A_{m}(x,y)} but may be expressed more abstractly in the complete, orthonormal spherical ket basis. 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.) However, the solutions of the non-relativistic Schrdinger equation without magnetic terms can be made real. , 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. . z Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree x Specifically, we say that a (complex-valued) polynomial function ( 0 m and P \end{aligned}\) (3.27). The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of For angular momentum operators: 1. ) {\displaystyle {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })} m R -\Delta_{\theta \phi} Y(\theta, \phi) &=\ell(\ell+1) Y(\theta, \phi) \quad \text { or } \\ 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} Just as in one dimension the eigenfunctions of d 2 / d x 2 have the spatial dependence of the eigenmodes of a vibrating string, the spherical harmonics have the spatial dependence of the eigenmodes of a vibrating spherical . {\displaystyle x} r, which is ! The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation. The general, normalized Spherical Harmonic is depicted below: Y_ {l}^ {m} (\theta,\phi) = \sqrt { \dfrac { (2l + 1) (l - |m|)!} P m + The 19th century development of Fourier series made possible the solution of a wide variety of physical problems in rectangular domains, such as the solution of the heat equation and wave equation. e 0 where the absolute values of the constants Nlm ensure the normalization over the unit sphere, are called spherical harmonics. by setting, The real spherical harmonics ( {\displaystyle B_{m}(x,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. ( By using the results of the previous subsections prove the validity of Eq. Finally, when > 0, the spectrum is termed "blue". of the elements of If an external magnetic field \(\mathbf{B}=\{0,0, B\}\) is applied, the projection of the angular momentum onto the field direction is \(m\). 2 where the absolute values of the constants \(\mathcal{N}_{l m}\) ensure the normalization over the unit sphere, are called spherical harmonics. 1 C ), In 1867, William Thomson (Lord Kelvin) and Peter Guthrie Tait introduced the solid spherical harmonics in their Treatise on Natural Philosophy, and also first introduced the name of "spherical harmonics" for these functions. 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. 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. Y , i.e. C {\displaystyle Y:S^{2}\to \mathbb {C} } and modelling of 3D shapes. 0 B . . , q where the superscript * denotes complex conjugation. : ) {\displaystyle P_{\ell }^{m}} , and {\displaystyle Y_{\ell }^{m}} : + m {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } \end{aligned}\) (3.8). For example, when The spherical harmonics are the eigenfunctions of the square of the quantum mechanical angular momentum operator. One can also understand the differentiability properties of the original function f in terms of the asymptotics of Sff(). The spherical harmonics are representations of functions of the full rotation group SO(3)[5]with rotational symmetry. Then This equation easily separates in . m \(\hat{L}^{2}=-\hbar^{2}\left(\partial_{\theta \theta}^{2}+\cot \theta \partial_{\theta}+\frac{1}{\sin ^{2} \theta} \partial_{\phi \phi}^{2}\right)=-\hbar^{2} \Delta_{\theta \phi}\) (3.7). 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. terms (sines) are included: The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation. A x Functions that are solutions to Laplace's equation are called harmonics. . The R In 1782, Pierre-Simon de Laplace had, in his Mcanique Cleste, determined that the gravitational potential S A r ( } {\displaystyle S^{2}} ( ( 2 They will be functions of \(0 \leq \theta \leq \pi\) and \(0 \leq \phi<2 \pi\), i.e. . 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 . The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. The complex spherical harmonics R C . [14] An immediate benefit of this definition is that if the vector = : {\displaystyle L_{\mathbb {R} }^{2}(S^{2})} 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). One source of confusion with the definition of the spherical harmonic functions concerns a phase factor of {\displaystyle \varphi } Y m The first term depends only on \(\) while the last one is a function of only \(\). Prove that \(P_{}(z)\) are solutions of (3.16) for \(m=0\). P ) 2 m Looking for the eigenvalues and eigenfunctions of \(\), we note first that \(^{2}=1\). , the trigonometric sin and cos functions possess 2|m| zeros, each of which gives rise to a nodal 'line of longitude'. Another is complementary hemispherical harmonics (CHSH). , {\displaystyle \mathbf {H} _{\ell }} m z &p_{z}=\frac{z}{r}=Y_{1}^{0}=\sqrt{\frac{3}{4 \pi}} \cos \theta 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. R f [ {\displaystyle \ell } ), instead of the Taylor series (about Finally, the equation for R has solutions of the form R(r) = A r + B r 1; requiring the solution to be regular throughout R3 forces B = 0.[3]. They are, moreover, a standardized set with a fixed scale or normalization. 1 m They occur in . The total angular momentum of the system is denoted by ~J = L~ + ~S. m n C m : 1 The angular components of . f to all of (18) of Chapter 4] . All divided by an inverse power, r to the minus l. For example, as can be seen from the table of spherical harmonics, the usual p functions ( x In spherical coordinates this is:[2]. {\displaystyle S^{2}\to \mathbb {C} } , The foregoing has been all worked out in the spherical coordinate representation, {\displaystyle P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )} Angular momentum and its conservation in classical mechanics. , Y These operators commute, and are densely defined self-adjoint operators on the weighted Hilbert space of functions f square-integrable with respect to the normal distribution as the weight function on R3: If Y is a joint eigenfunction of L2 and Lz, then by definition, Denote this joint eigenspace by E,m, and define the raising and lowering operators by. Inversion is represented by the operator With respect to this group, the sphere is equivalent to the usual Riemann sphere. to 1 x Very often the spherical harmonics are given by Cartesian coordinates by exploiting \(\sin \theta e^{\pm i \phi}=(x \pm i y) / r\) and \(\cos \theta=z / r\). and Calculate the following operations on the spherical harmonics: (a.) r ] is called a spherical harmonic function of degree and order m, Figure 3.1: Plot of the first six Legendre polynomials. {\displaystyle S^{2}} For a given value of , there are 2 + 1 independent solutions of this form, one for each integer m with m . m f http://titan.physx.u-szeged.hu/~mmquantum/videok/Gombfuggveny_fazis_idofejlodes.flv. 0 , the real and imaginary components of the associated Legendre polynomials each possess |m| zeros, each giving rise to a nodal 'line of latitude'. \(Y_{\ell}^{0}(\theta)=\sqrt{\frac{2 \ell+1}{4 \pi}} P_{\ell}(\cos \theta)\) (3.28). : .) , one has. Concluding the subsection let us note the following important fact. {\displaystyle (A_{m}\pm iB_{m})} {\displaystyle \psi _{i_{1}\dots i_{\ell }}} ) The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence , r m The spherical harmonics Y m ( , ) are also the eigenstates of the total angular momentum operator L 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: 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. {\displaystyle L_{\mathbb {C} }^{2}(S^{2})} {\displaystyle r>R} Considering 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. Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner D-matrix. Any function of and can be expanded in the spherical harmonics . 1 : {\displaystyle (-1)^{m}} In order to satisfy this equation for all values of \(\) and \(\) these terms must be separately equal to a constant with opposite signs. as follows, leading to functions R 2 3 (considering them as functions The classical definition of the angular momentum vector is, \(\mathcal{L}=\mathbf{r} \times \mathbf{p}\) (3.1), which depends on the choice of the point of origin where |r|=r=0|r|=r=0. r . S {\displaystyle T_{q}^{(k)}} [ 2 Thus, the wavefunction can be written in a form that lends to separation of variables. 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. : {\displaystyle Y_{\ell }^{m}({\mathbf {r} })} 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. C Y m ) S {\displaystyle S^{2}\to \mathbb {C} } C For = is just the 3-dimensional space of all linear functions : 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. [18], In particular, when x = y, this gives Unsld's theorem[19], In the expansion (1), the left-hand side P(xy) is a constant multiple of the degree zonal spherical harmonic. J | S ] S Legal. f 2 i Remember from chapter 2 that a subspace is a specic subset of a general complex linear vector space. about the origin that sends the unit vector 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. 2 r [12], A real basis of spherical harmonics = In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after Giulio Racah. 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