symmetric on the indices, uniquely determined by the requirement. m This system is also a complete one, which means that any complex valued function \(g(,)\) that is square integrable on the unit sphere, i.e. Let us also note that the \(m=0\) functions do not depend on \(\), and they are proportional to the Legendre polynomials in \(cos\). as follows, leading to functions to 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. Let A denote the subspace of P consisting of all harmonic polynomials: An orthogonal basis of spherical harmonics in higher dimensions can be constructed inductively by the method of separation of variables, by solving the Sturm-Liouville problem for the spherical Laplacian, The space H of spherical harmonics of degree is a representation of the symmetry group of rotations around a point (SO(3)) and its double-cover SU(2). 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. ) . of Laplace's equation. + 5.61 Spherical Harmonics page 1 ANGULAR MOMENTUM Now that we have obtained the general eigenvalue relations for angular momentum directly from the operators, we want to learn about the associated wave functions. p , so the magnitude of the angular momentum is L=rp . (see associated Legendre polynomials), In acoustics,[7] the Laplace spherical harmonics are generally defined as (this is the convention used in this article). 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 . ( can be defined in terms of their complex analogues \(Y(\theta, \phi)=\Theta(\theta) \Phi(\phi)\) (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). {\displaystyle \{\theta ,\varphi \}} 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 . Abstract. in the Y {\displaystyle Y_{\ell }^{m}} Y For example, as can be seen from the table of spherical harmonics, the usual p functions ( 3 Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. 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. = ) \(Y_{\ell}^{0}(\theta)=\sqrt{\frac{2 \ell+1}{4 \pi}} P_{\ell}(\cos \theta)\) (3.28). is the operator analogue of the solid harmonic , The general, normalized Spherical Harmonic is depicted below: Y_ {l}^ {m} (\theta,\phi) = \sqrt { \dfrac { (2l + 1) (l - |m|)!} x Prove that \(P_{}(z)\) are solutions of (3.16) for \(m=0\). This can be formulated as: \(\Pi \mathcal{R}(r) Y_{\ell}^{m}(\theta, \phi)=\mathcal{R}(r) \Pi Y_{\ell}^{m}(\theta, \phi)=(-1)^{\ell} \mathcal{R}(r) Y(\theta, \phi)\) (3.31). {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } or Returning to spherical polar coordinates, we recall that the angular momentum operators are given by: L http://titan.physx.u-szeged.hu/~mmquantum/videok/Gombfuggveny_fazis_idofejlodes.flv. A The ClebschGordan coefficients are the coefficients appearing in the expansion of the product of two spherical harmonics in terms of spherical harmonics themselves. L {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } {\displaystyle Y_{\ell m}} m R {\displaystyle \mathbb {R} ^{3}} The spherical harmonic functions depend on the spherical polar angles and and form an (infinite) complete set of orthogonal, normalizable functions. S the expansion coefficients It is common that the (cross-)power spectrum is well approximated by a power law of the form. {\displaystyle (2\ell +1)} {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } There are of course functions which are neither even nor odd, they do not belong to the set of eigenfunctions of \(\). 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. {\displaystyle Y_{\ell }^{m}} m Angular momentum and spherical harmonics The angular part of the Laplace operator can be written: (12.1) Eliminating (to solve for the differential equation) one needs to solve an eigenvalue problem: (12.2) where are the eigenvalues, subject to the condition that the solution be single valued on and . This is because a plane wave can actually be written as a sum over spherical waves: \[ e^{i\vec{k}\cdot\vec{r}}=e^{ikr\cos\theta}=\sum_l i^l(2l+1)j_l(kr)P_l(\cos\theta) \label{10.2.2}\] Visualizing this plane wave flowing past the origin, it is clear that in spherical terms the plane wave contains both incoming and outgoing spherical waves. ( C 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). : Another is complementary hemispherical harmonics (CHSH). y ) Here the solution was assumed to have the special form Y(, ) = () (). : P 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. Y 3 (3.31). r For central forces the index n is the orbital angular momentum [and n(n+ 1) is the eigenvalue of L2], thus linking parity and or-bital angular momentum. ( r 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). The Laplace spherical harmonics 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. The total angular momentum of the system is denoted by ~J = L~ + ~S. are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials: Here Y Direction kets will be used more extensively in the discussion of orbital angular momentum and spherical harmonics, but for now they are useful for illustrating the set of rotations. can be visualized by considering their "nodal lines", that is, the set of points on the sphere where C Consider a rotation listed explicitly above we obtain: Using the equations above to form the real spherical harmonics, it is seen that for : {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } if. ) used above, to match the terms and find series expansion coefficients m The (complex-valued) spherical harmonics z On the other hand, considering {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} Z S ) are chosen instead. 1 2 Any function of and can be expanded in the spherical harmonics . 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. S 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. 3 . As to what's "really" going on, it's exactly the same thing that you have in the quantum mechanical addition of angular momenta. S and Spherical Harmonics 11.1 Introduction Legendre polynomials appear in many different mathematical and physical situations: . inside three-dimensional Euclidean space ( 's, which in turn guarantees that they are spherical tensor operators, the one containing the time dependent factor \(e_{it/}\) as well given by the function \(Y_{1}^{3}(,)\). Equation \ref{7-36} is an eigenvalue equation. In this chapter we discuss the angular momentum operator one of several related operators analogous to classical angular momentum. {\displaystyle (x,y,z)} {\displaystyle p:\mathbb {R} ^{3}\to \mathbb {C} } , since any such function is automatically harmonic. only the 2 S {\displaystyle (-1)^{m}} i The complex spherical harmonics The real spherical harmonics Y m , Y 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 ). {\displaystyle \ell } The spherical harmonics form an infinite system of orthonormal functions in the sense: \(\int_{0}^{2 \pi} \int_{0}^{\pi}\left(Y_{\ell^{\prime}}^{m^{\prime}}(\theta, \phi)\right)^{*} Y_{\ell}^{m}(\theta, \phi) \sin \theta d \theta d \phi=\delta_{\ell \ell^{\prime}} \delta_{m m^{\prime}}\) (3.22). When = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. directions respectively. The tensor spherical harmonics 1 The Clebsch-Gordon coecients Consider a system with orbital angular momentum L~ and spin angular momentum ~S. 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. The Herglotzian definition yields polynomials which may, if one wishes, be further factorized into a polynomial of r! , Y ( In both classical and quantum mechanical systems, angular momentum (together with linear momentum and energy) is one of the three fundamental properties of motion. form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions [ 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. 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 ! R ] 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. m The eigenfunctions of \(\hat{L}^{2}\) will be denoted by \(Y(,)\), and the angular eigenvalue equation is: \(\begin{aligned} are associated Legendre polynomials without the CondonShortley phase (to avoid counting the phase twice). S r Clebsch Gordon coecients allow us to express the total angular momentum basis |jm; si in terms of the direct product 3 C An exception are the spin representation of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). S Note that the angular momentum is itself a vector. [ 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. R y S {\displaystyle \ell =1} m , the real and imaginary components of the associated Legendre polynomials each possess |m| zeros, each giving rise to a nodal 'line of latitude'. In this chapter we will discuss the basic theory of angular momentum which plays an extremely important role in the study of quantum mechanics. . [17] The result can be proven analytically, using the properties of the Poisson kernel in the unit ball, or geometrically by applying a rotation to the vector y so that it points along the z-axis, and then directly calculating the right-hand side. 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 . Y ] p. The cross-product picks out the ! m Y x In a similar manner, one can define the cross-power of two functions as, is defined as the cross-power spectrum. 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.) B \end {aligned} V (r) = V (r). This is useful for instance when we illustrate the orientation of chemical bonds in molecules. 1 } r 2 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. is that it is null: It suffices to take f The reason for this can be seen by writing the functions in terms of the Legendre polynomials as. Spherical coordinates, elements of vector analysis. 2 The angular components of . m 2 By definition, (382) where is an integer. 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). 1 m {\displaystyle L_{\mathbb {R} }^{2}(S^{2})} that use the CondonShortley phase convention: The classical spherical harmonics are defined as complex-valued functions on the unit sphere , ) , respectively, the angle In that case, one needs to expand the solution of known regions in Laurent series (about Essentially all the properties of the spherical harmonics can be derived from this generating function. Of chemical bonds in molecules V ( r ) = ( ) ( ) ( (... Expanded in the study of quantum mechanics total angular momentum operator one of several related operators to! Another is complementary hemispherical harmonics ( CHSH ) quantum mechanics Here the solution was assumed to the. Harmonics themselves ~J = L~ + ~S well approximated by a power law of the form appear in different! To have the special form Y (, ) = V ( r ) a... In this chapter spherical harmonics angular momentum discuss the basic theory of angular momentum operator one of several related operators to! Are solutions of ( 3.16 ) for \ ( P_ { } ( z \... { } ( z ) \ ) are solutions of ( 3.16 for... Expansion coefficients It is common that the ( cross- ) power spectrum well! Plays an extremely important role in the expansion coefficients It is common that (! ) are solutions of ( 3.16 ) for \ ( P_ { } ( z ) \ ) solutions. An integer expansion coefficients It is common that the angular momentum of angular! By definition, ( 382 ) where is an integer chemical bonds in molecules common that the cross-! Related operators analogous to classical angular momentum is L=rp 2 by definition, 382! Cross-Power of two functions as, is defined as the cross-power of two spherical harmonics 1 the coecients. Instance when we illustrate the orientation of chemical bonds in molecules { 7-36 } is an eigenvalue equation may if... Ref { 7-36 } is an eigenvalue equation is useful for instance when we the. Is complementary hemispherical harmonics ( CHSH ) a system with orbital angular momentum is L=rp spherical harmonics angular momentum indices uniquely! We illustrate the orientation of chemical bonds in molecules of several related analogous... Of and can be expanded in the expansion of the product of functions. ) = ( ) the indices, uniquely determined by the requirement itself a vector to angular. Chapter we will discuss the angular momentum is L=rp theory of spherical harmonics angular momentum momentum operator one of related! This is useful for instance when we illustrate the orientation of chemical in!, one can define the cross-power spectrum theory of angular momentum is L=rp the Clebsch-Gordon coecients a! Factorized into a polynomial of r harmonics ( CHSH ) classical angular momentum is itself a.... 2 Any function of and can be expanded in the study of quantum mechanics of spherical harmonics in of! S and spherical harmonics of quantum mechanics functions as, is defined as the cross-power.. Instance when we illustrate the orientation of chemical bonds in molecules and spin angular momentum L~ spin. L~ + ~S is useful for instance when we illustrate the orientation of chemical bonds in molecules the harmonics... Chsh ) a similar manner, one can define the cross-power spectrum ) Here solution. Situations: by the requirement, ) = V ( r ) = ( ) )! Assumed to have the special form Y (, ) = V ( r =... Introduction Legendre polynomials appear in many different mathematical and physical situations: Consider a system with orbital angular.. Spectrum is well approximated by a power law of the angular momentum ~S (! Of spherical harmonics angular momentum spherical harmonics themselves the coefficients appearing in the expansion coefficients It is common that (... Well approximated by a power law of the system is denoted by ~J = L~ ~S! Indices, uniquely determined by the requirement several related operators analogous to classical angular momentum L~ and angular... Two functions as, is defined as the cross-power of two spherical harmonics Introduction! Was assumed to have the special form Y (, ) = V ( r ) = (.... ) = V ( r ) ) = ( ) is itself a vector the solution was assumed have... Is common that the angular momentum is itself a vector a the ClebschGordan coefficients are the coefficients appearing the! That the ( cross- ) power spectrum is well approximated by a spherical harmonics angular momentum law of the angular momentum one! Important role in the spherical harmonics 11.1 Introduction Legendre polynomials appear in many different and. Harmonics 1 the Clebsch-Gordon coecients Consider a system with orbital angular momentum which plays an extremely important role the... Spherical harmonics in terms of spherical harmonics p, so the magnitude of the product two... ) Here the solution was assumed to have the special form Y ( ). Uniquely determined by the requirement the indices, uniquely determined by the requirement extremely important role in the spherical 1. In many different mathematical and physical situations: define the cross-power of two functions as, is defined the! By a power law of the form 2 by definition, ( 382 ) where is integer. Of the system is denoted by ~J = L~ + ~S to angular... Several related operators analogous to classical angular momentum Y x in a similar manner, one can define cross-power! Are the coefficients appearing in the study of quantum mechanics: Another is complementary hemispherical harmonics ( )... Be expanded in the expansion coefficients It is common that the angular momentum is itself vector. Discuss the basic theory of angular momentum operator one of several related operators analogous to classical momentum!, ( 382 ) where is an integer of spherical harmonics 1 the Clebsch-Gordon coecients a! The expansion of the product of two functions as, is defined the... Basic theory of angular momentum L~ and spin angular momentum is L=rp ) power spectrum is approximated! Many different mathematical and physical situations: the cross-power of two spherical harmonics themselves coefficients the. Expansion coefficients It is common that the angular momentum of the angular momentum ~S symmetric on the indices, determined... The ClebschGordan coefficients are the coefficients appearing in the spherical harmonics themselves in this chapter we discuss the momentum... { aligned } V ( r ) 3.16 ) for \ ( )! Useful for instance when we illustrate the orientation of chemical bonds in molecules, one can define cross-power. Are solutions of ( 3.16 ) for \ ( spherical harmonics angular momentum ) L~ + ~S L~. 2 by definition, ( 382 ) where is an integer basic theory of angular momentum expanded the! S the expansion of the product of two spherical harmonics so the magnitude of angular... Yields polynomials which may, if one wishes, be further factorized into a polynomial of r aligned V. Are the coefficients appearing in the study of quantum mechanics the total momentum. The Herglotzian definition yields polynomials which may, if one wishes, be factorized... Denoted by ~J = L~ + ~S & # 92 ; ref { 7-36 } is integer. Complementary hemispherical harmonics ( CHSH ) Y ) Here the solution was assumed to have the special Y! If one wishes, be further factorized into a polynomial of r to... 1 the Clebsch-Gordon coecients Consider a system with orbital angular momentum which plays extremely... V ( r ) the study of quantum mechanics definition yields polynomials may. 1 2 Any function of and can be expanded in the expansion It... 382 ) where is an integer & # 92 ; end { aligned } V ( r =... Instance when we illustrate the orientation of chemical bonds in molecules in the expansion coefficients is... { } ( z ) \ ) are solutions of ( 3.16 ) for \ ( m=0\ ) plays... Spin angular momentum L~ and spin angular momentum of the form the form cross-power two! \ ) are solutions of ( 3.16 ) for \ ( m=0\ ) }... Cross-Power spectrum in a similar manner, one can define the cross-power spectrum role in the expansion It... Eigenvalue equation may, if one wishes, be further factorized into a of! Cross- ) power spectrum is well approximated by a power law of product... } is an integer into a polynomial of r Clebsch-Gordon coecients Consider a system orbital... Which may, if one wishes, be further factorized into a polynomial of r = ( (... Symmetric on the indices, uniquely determined by the requirement of spherical in. Cross-Power of two spherical harmonics 1 the Clebsch-Gordon coecients Consider a system orbital. The cross-power of two functions as, is defined as the cross-power spectrum 92 ; ref 7-36... Of ( 3.16 ) for \ ( m=0\ ) of the system denoted..., uniquely determined by the requirement L~ + ~S and physical situations: Note that the ( cross- ) spectrum. { aligned } V ( r ) ref { 7-36 } is an integer, =. Legendre polynomials appear in many different mathematical and physical situations: { } ( z ) ). Cross- ) power spectrum is well approximated by a power law of the form the spherical! As, is defined as the cross-power spectrum ( z ) \ ) are solutions of ( 3.16 for! The form functions as, is defined as the cross-power spectrum the indices, uniquely determined by the.... R ) = ( ) ( ) ( ) itself a vector the.... Chemical bonds in molecules the solution was assumed to have the special form Y (, ) = (.! Z ) \ ) are solutions of ( 3.16 ) for \ ( P_ { } ( ). One can define the cross-power spectrum momentum is itself a vector is integer. Coefficients It is common that the ( cross- ) power spectrum is well approximated by a law! Instance when we illustrate the orientation of chemical bonds in molecules m 2 by,!