spherical harmonics angular momentum
> 0 2 &\hat{L}_{x}=i \hbar\left(\sin \phi \partial_{\theta}+\cot \theta \cos \phi \partial_{\phi}\right) \\ is that for real functions {\displaystyle r^{\ell }Y_{\ell }^{m}(\mathbf {r} /r)} by setting, The real spherical harmonics A Representation of Angular Momentum Operators We would like to have matrix operators for the angular momentum operators L x; L y, and L z. R to correspond to a (smooth) function By using the results of the previous subsections prove the validity of Eq. In other words, any well-behaved function of and can be represented as a superposition of spherical harmonics. {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } Y Basically, you can always think of a spherical harmonic in terms of the generalized polynomial. R r R m 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. ( m {\displaystyle S^{n-1}\to \mathbb {C} } m y : : \end{aligned}\) (3.30). The half-integer values do not give vanishing radial solutions. C {\displaystyle \varphi } (see associated Legendre polynomials), In acoustics,[7] the Laplace spherical harmonics are generally defined as (this is the convention used in this article). 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]. 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 {\displaystyle f:S^{2}\to \mathbb {C} } about the origin that sends the unit vector R For angular momentum operators: 1. m {\displaystyle \{\pi -\theta ,\pi +\varphi \}} < The angular components of . {\displaystyle Y_{\ell }^{m}} m For the case of orthonormalized harmonics, this gives: If the coefficients decay in sufficiently rapidly for instance, exponentially then the series also converges uniformly to f. A square-integrable function ) are complex and mix axis directions, but the real versions are essentially just x, y, and z. Note that the angular momentum is itself a vector. 's of degree Operators for the square of the angular momentum and for its zcomponent: Remember from chapter 2 that a subspace is a specic subset of a general complex linear vector space. The angular momentum relative to the origin produced by a momentum vector ! , The essential property of ( The condition on the order of growth of Sff() is related to the order of differentiability of f in the next section. (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 . m 3 , of the eigenvalue problem. Figure 3.1: Plot of the first six Legendre polynomials. ( 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. 1 terms (sines) are included: The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation. 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. . e^{i m \phi} \\ As . [27] One is hemispherical functions (HSH), orthogonal and complete on hemisphere. 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. C That is, it consists of,products of the three coordinates, x1, x2, x3, where the net power, a plus b plus c, is equal to l, the index of the spherical harmonic. R C Consider a rotation We have to write the given wave functions in terms of the spherical harmonics. 2 Y : by \(\mathcal{R}(r)\). We shall now find the eigenfunctions of \(_{}\), that play a very important role in quantum mechanics, and actually in several branches of theoretical physics. R are essentially n (3.31). , obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. Y {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } . k 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. is just the space of restrictions to the sphere {\displaystyle \mathbf {H} _{\ell }} i {\displaystyle L_{\mathbb {R} }^{2}(S^{2})} {\displaystyle \ell =4} , 2 , which can be seen to be consistent with the output of the equations above. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } The solutions, \(Y_{\ell}^{m}(\theta, \phi)=\mathcal{N}_{l m} P_{\ell}^{m}(\theta) e^{i m \phi}\) (3.20). ) {\displaystyle f_{\ell m}} 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). Returning to spherical polar coordinates, we recall that the angular momentum operators are given by: L 3 When = 0, the spectrum is "white" as each degree possesses equal power. Imposing this regularity in the solution of the second equation at the boundary points of the domain is a SturmLiouville problem that forces the parameter to be of the form = ( + 1) for some non-negative integer with |m|; this is also explained below in terms of the orbital angular momentum. is essentially the associated Legendre polynomial A } ( The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication. S m {\displaystyle Y_{\ell }^{m}} S The figures show the three-dimensional polar diagrams of the spherical harmonics. [ He discovered that if r r1 then, where is the angle between the vectors x and x1. m Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x p~. x ( Y This was a boon for problems possessing spherical symmetry, such as those of celestial mechanics originally studied by Laplace and Legendre. ( Looking for the eigenvalues and eigenfunctions of \(\), we note first that \(^{2}=1\). . S Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section Higher dimensions below). 2 S ( x the formula, Several different normalizations are in common use for the Laplace spherical harmonic 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. 2 Let Yj be an arbitrary orthonormal basis of the space H of degree spherical harmonics on the n-sphere. {\displaystyle \mathbb {R} ^{3}\setminus \{\mathbf {0} \}\to \mathbb {C} } {4\pi (l + |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. C 1 ] In order to satisfy this equation for all values of \(\) and \(\) these terms must be separately equal to a constant with opposite signs. In naming this generating function after Herglotz, we follow Courant & Hilbert 1962, VII.7, who credit unpublished notes by him for its discovery. Y r and order 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. {\displaystyle A_{m}} n Considering {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } 0 ( k {\displaystyle y} 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). and modelling of 3D shapes. are constants and the factors r Ym are known as (regular) solid harmonics {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } 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. (18) of Chapter 4] . setting, If the quantum mechanical convention is adopted for the r C {\displaystyle Z_{\mathbf {x} }^{(\ell )}} m {\displaystyle {\mathcal {R}}} C Equation \ref{7-36} is an eigenvalue equation. can also be expanded in terms of the real harmonics {\displaystyle \Im [Y_{\ell }^{m}]=0} and 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. 1 For the other cases, the functions checker the sphere, and they are referred to as tesseral. {\displaystyle \Delta f=0} Another way of using these functions is to create linear combinations of functions with opposite m-s. A 1 | the one containing the time dependent factor \(e_{it/}\) as well given by the function \(Y_{1}^{3}(,)\). of the elements of C C m {\displaystyle S^{2}} are the Legendre polynomials, and they can be derived as a special case of spherical harmonics. m B The vector spherical harmonics are now defined as the quantities that result from the coupling of ordinary spherical harmonics and the vectors em to form states of definite J (the resultant of the orbital angular momentum of the spherical harmonic and the one unit possessed by the em ). m : or The integration constant \(\frac{1}{\sqrt{2 \pi}}\) has been chosen here so that already \(()\) is normalized to unity when integrating with respect to \(\) from 0 to \(2\). 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). m : , respectively, the angle terms (cosines) are included, and for = C r, which is ! See, e.g., Appendix A of Garg, A., Classical Electrodynamics in a Nutshell (Princeton University Press, 2012). 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. Another is complementary hemispherical harmonics (CHSH). {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } f Y {\displaystyle c\in \mathbb {C} } 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. . + in Specifically, we say that a (complex-valued) polynomial function {\displaystyle \ell } Nodal lines of from the above-mentioned polynomial of degree , we have a 5-dimensional space: For any S , The real spherical harmonics m &\Pi_{\psi_{+}}(\mathbf{r})=\quad \psi_{+}(-\mathbf{r})=\psi_{+}(\mathbf{r}) \\ the expansion coefficients ) A , z 1 In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after Giulio Racah. Y With respect to this group, the sphere is equivalent to the usual Riemann sphere. Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. They are often employed in solving partial differential equations in many scientific fields. f R {\displaystyle {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })} The angle-preserving symmetries of the two-sphere are described by the group of Mbius transformations PSL(2,C). to Laplace's equation In this chapter we will discuss the basic theory of angular momentum which plays an extremely important role in the study of quantum mechanics. &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 \\ For example, when The foregoing has been all worked out in the spherical coordinate representation, ( R The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. 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. 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