from the above-mentioned polynomial of degree Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. m {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } is called a spherical harmonic function of degree and order m, f is that for real functions {\displaystyle \ell } {\displaystyle S^{n-1}\to \mathbb {C} } This parity property will be conrmed by the series Y 3 Another is complementary hemispherical harmonics (CHSH). In order to satisfy this equation for all values of \(\) and \(\) these terms must be separately equal to a constant with opposite signs. , the real and imaginary components of the associated Legendre polynomials each possess |m| zeros, each giving rise to a nodal 'line of latitude'. If the functions f and g have a zero mean (i.e., the spectral coefficients f00 and g00 are zero), then Sff() and Sfg() represent the contributions to the function's variance and covariance for degree , respectively. In this chapter we will discuss the basic theory of angular momentum which plays an extremely important role in the study of quantum mechanics. Y \(\begin{aligned} transforms into a linear combination of spherical harmonics of the same degree. C Now we're ready to tackle the Schrdinger equation in three dimensions. C ) C and Spherical Harmonics 11.1 Introduction Legendre polynomials appear in many different mathematical and physical situations: . [17] The result can be proven analytically, using the properties of the Poisson kernel in the unit ball, or geometrically by applying a rotation to the vector y so that it points along the z-axis, and then directly calculating the right-hand side. [ that obey Laplace's equation. &\Pi_{\psi_{+}}(\mathbf{r})=\quad \psi_{+}(-\mathbf{r})=\psi_{+}(\mathbf{r}) \\ , C }\left(\frac{d}{d z}\right)^{\ell}\left(z^{2}-1\right)^{\ell}\) (3.18). 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]. 2 Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner D-matrix. Y ( The spherical harmonics with negative can be easily compute from those with positive . More generally, the analogous statements hold in higher dimensions: the space H of spherical harmonics on the n-sphere is the irreducible representation of SO(n+1) corresponding to the traceless symmetric -tensors. Such an expansion is valid in the ball. , can be defined in terms of their complex analogues An exception are the spin representation of SO(3): strictly speaking these are representations of the double cover SU(2) of SO(3). Y 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. = 2 Y m m This is justified rigorously by basic Hilbert space theory. m That is, they are either even or odd with respect to inversion about the origin. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } The tensor spherical harmonics 1 The Clebsch-Gordon coecients Consider a system with orbital angular momentum L~ and spin angular momentum ~S. as follows, leading to functions Spherical harmonics originate from solving Laplace's equation in the spherical domains. m m The spherical harmonics, more generally, are important in problems with spherical symmetry. form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions and {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } but may be expressed more abstractly in the complete, orthonormal spherical ket basis. , since any such function is automatically harmonic. to all of : Statements relating the growth of the Sff() to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. S {\displaystyle S^{2}} 2 C m Y symmetric on the indices, uniquely determined by the requirement. Functions that are solutions to Laplace's equation are called harmonics. {\displaystyle \mathbb {R} ^{3}\setminus \{\mathbf {0} \}\to \mathbb {C} } As these are functions of points in real three dimensional space, the values of \(()\) and \((+2)\) must be the same, as these values of the argument correspond to identical points in space. m {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } } n Show that the transformation \(\{x, y, z\} \longrightarrow\{-x,-y,-z\}\) is equivalent to \(\theta \longrightarrow \pi-\theta, \quad \phi \longrightarrow \phi+\pi\). y In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after Giulio Racah. For example, as can be seen from the table of spherical harmonics, the usual p functions ( As to what's "really" going on, it's exactly the same thing that you have in the quantum mechanical addition of angular momenta. brackets are functions of ronly, and the angular momentum operator is only a function of and . and {\displaystyle y} That is: Spherically symmetric means that the angles range freely through their full domains each of which is finite leading to a universal set of discrete separation constants for the angular part of all spherically symmetric problems. \end{aligned}\) (3.27). C 3 r! = {\displaystyle \mathbf {r} } , R R m 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. Any function of and can be expanded in the spherical harmonics . m ) Spherical harmonics were first investigated in connection with the Newtonian potential of Newton's law of universal gravitation in three dimensions. R : The spherical harmonics have definite parity. We have to write the given wave functions in terms of the spherical harmonics. {\displaystyle \mathbf {J} } {\displaystyle P_{i}:[-1,1]\to \mathbb {R} } 2 0 S The Laplace spherical harmonics ), instead of the Taylor series (about It can be shown that all of the above normalized spherical harmonic functions satisfy. P 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 ). 4 , the degree zonal harmonic corresponding to the unit vector x, decomposes as[20]. 3 {\displaystyle \Im [Y_{\ell }^{m}]=0} m r { m r ) Figure 3.1: Plot of the first six Legendre polynomials. Spherical harmonics can be separated into two set of functions. R ( m z (see associated Legendre polynomials), In acoustics,[7] the Laplace spherical harmonics are generally defined as (this is the convention used in this article). r, which is ! 3 , any square-integrable function Y 2 The solution function Y(, ) is regular at the poles of the sphere, where = 0, . S setting, If the quantum mechanical convention is adopted for the P In other words, any well-behaved function of and can be represented as a superposition of spherical harmonics. + 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. However, whereas every irreducible tensor representation of SO(2) and SO(3) is of this kind, the special orthogonal groups in higher dimensions have additional irreducible representations that do not arise in this manner. , one has. f In a similar manner, one can define the cross-power of two functions as, is defined as the cross-power spectrum. .) or ) Y m 2 {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } 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. ) R It follows from Equations ( 371) and ( 378) that. {\displaystyle Y_{\ell }^{m}} ] m the one containing the time dependent factor \(e_{it/}\) as well given by the function \(Y_{1}^{3}(,)\). 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 . 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. | &\Pi_{\psi_{-}}(\mathbf{r})=\quad \psi_{-}(-\mathbf{r})=-\psi_{-}(\mathbf{r}) L , with (the irregular solid harmonics ( C Find the first three Legendre polynomials \(P_{0}(z)\), \(P_{1}(z)\) and \(P_{2}(z)\). {\displaystyle \ell } C {\displaystyle S^{2}} R {\displaystyle Y_{\ell }^{m}} S , or alternatively where 1.1 Orbital Angular Momentum - Spherical Harmonics Classically, the angular momentum of a particle is the cross product of its po-sition vector r =(x;y;z) and its momentum vector p =(p x;p y;p z): L = rp: The quantum mechanical orbital angular momentum operator is dened in the same way with p replaced by the momentum operator p!ihr . m and modelling of 3D shapes. m q , i The spinor spherical harmonics are the natural spinor analog of the vector spherical harmonics. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } Laplace equation. L=! 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. {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } Y S Looking for the eigenvalues and eigenfunctions of \(\), we note first that \(^{2}=1\). in the However, the solutions of the non-relativistic Schrdinger equation without magnetic terms can be made real. {\displaystyle z} {\displaystyle P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )} : 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. of spherical harmonics of degree {\displaystyle Y_{\ell m}} In that case, one needs to expand the solution of known regions in Laurent series (about ), 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. , the trigonometric sin and cos functions possess 2|m| zeros, each of which gives rise to a nodal 'line of longitude'. Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree C ) are complex and mix axis directions, but the real versions are essentially just x, y, and z. Y ( ) R S 2 In particular, if Sff() decays faster than any rational function of as , then f is infinitely differentiable. {\displaystyle \ell =1} Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with When you apply L 2 to an angular momentum eigenstate l, then you find L 2 l = [ l ( l + 1) 2] l. That is, l ( l + 1) 2 is the value of L 2 which is associated to the eigenstate l. 's transform under rotations (see below) in the same way as the p is ! f {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } : J {\displaystyle Y_{\ell }^{m}} Y , commonly referred to as the CondonShortley phase in the quantum mechanical literature. y is replaced by the quantum mechanical spin vector operator 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 ! {\displaystyle Y_{\ell }^{m}({\mathbf {r} })} R i R 2 ) As . {\displaystyle r=\infty } {\displaystyle x} (Here the scalar field is understood to be complex, i.e. 2 {\displaystyle S^{2}} {\displaystyle {\mathcal {R}}} ( : 1 x ) 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. [28][29][30][31], "Ylm" redirects here. 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, \(\begin{aligned} \end{array}\right.\) (3.12), and any linear combinations of them. {\displaystyle m>0} ( They occur in . When = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. The eigenfunctions of the orbital angular momentum operator, the spherical harmonics Reasoning: The common eigenfunctions of L 2 and L z are the spherical harmonics. Spherical harmonics are ubiquitous in atomic and molecular physics. \end{aligned}\) (3.8). . 1 2 These angular solutions 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. 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. Consider the problem of finding solutions of the form f(r, , ) = R(r) Y(, ). n . {\displaystyle Y_{\ell }^{m}} {\displaystyle \mathbb {R} ^{3}} The spherical harmonics are normalized . R This was a boon for problems possessing spherical symmetry, such as those of celestial mechanics originally studied by Laplace and Legendre. R 2 {\displaystyle A_{m}} 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 \(2\). m S (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 . ( R can be visualized by considering their "nodal lines", that is, the set of points on the sphere where : 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 . The eigenvalues of \(\) itself are then \(1\), and we have the following two possibilities: \(\begin{aligned} Furthermore, the zonal harmonic . : The benefit of the expansion in terms of the real harmonic functions : { m f e The spherical harmonics can be expressed as the restriction to the unit sphere of certain polynomial functions &p_{z}=\frac{z}{r}=Y_{1}^{0}=\sqrt{\frac{3}{4 \pi}} \cos \theta \(\hat{L}^{2}=-\hbar^{2}\left(\partial_{\theta \theta}^{2}+\cot \theta \partial_{\theta}+\frac{1}{\sin ^{2} \theta} \partial_{\phi \phi}^{2}\right)=-\hbar^{2} \Delta_{\theta \phi}\) (3.7). 3 Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section Higher dimensions below). Finally, when > 0, the spectrum is termed "blue". 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