m The real spherical harmonics Notice, however, that spherical harmonics are not functions on the sphere which are harmonic with respect to the Laplace-Beltrami operator for the standard round metric on the sphere: the only harmonic functions in this sense on the sphere are the constants, since harmonic functions satisfy the Maximum principle. r The animation shows the time dependence of the stationary state i.e. z [ C {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } Spherical coordinates, elements of vector analysis. 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. : ( We have to write the given wave functions in terms of the spherical harmonics. : The parallelism of the two definitions ensures that the : {\displaystyle r} m to correspond to a (smooth) function 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). is that for real functions the angular momentum and the energy of the particle are measured simultane-ously at time t= 0, what values can be obtained for each observable and with what probabilities? Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. Z 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} r Another way of using these functions is to create linear combinations of functions with opposite m-s. 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. \(\hat{L}^{2}=-\hbar^{2}\left(\partial_{\theta \theta}^{2}+\cot \theta \partial_{\theta}+\frac{1}{\sin ^{2} \theta} \partial_{\phi \phi}^{2}\right)=-\hbar^{2} \Delta_{\theta \phi}\) (3.7). ( . The foregoing has been all worked out in the spherical coordinate representation, S : R Spherical Harmonics, and Bessel Functions Physics 212 2010, Electricity and Magnetism Michael Dine Department of Physics . {\displaystyle \mathbf {H} _{\ell }} http://en.Wikipedia.org/wiki/File:Legendrepolynomials6.svg. m See, e.g., Appendix A of Garg, A., Classical Electrodynamics in a Nutshell (Princeton University Press, 2012). f C , the real and imaginary components of the associated Legendre polynomials each possess |m| zeros, each giving rise to a nodal 'line of latitude'. It can be shown that all of the above normalized spherical harmonic functions satisfy. : {\displaystyle y} 1 Let us also note that the \(m=0\) functions do not depend on \(\), and they are proportional to the Legendre polynomials in \(cos\). Specifically, if, A mathematical result of considerable interest and use is called the addition theorem for spherical harmonics. \end{aligned}\) (3.8). 's of degree Y \(\begin{aligned} = a As . 1 {\displaystyle (r,\theta ,\varphi )} r the formula, Several different normalizations are in common use for the Laplace spherical harmonic functions {\displaystyle Y_{\ell }^{m}} {\displaystyle \Im [Y_{\ell }^{m}]=0} {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } terms (sines) are included: The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation. In other words, any well-behaved function of and can be represented as a superposition of spherical harmonics. < 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. The spherical harmonics have definite parity. Thus, p2=p r 2+p 2 can be written as follows: p2=pr 2+ L2 r2. Such an expansion is valid in the ball. {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } Such spherical harmonics are a special case of zonal spherical functions. R (18) of Chapter 4] . The angle-preserving symmetries of the two-sphere are described by the group of Mbius transformations PSL(2,C). {\displaystyle \varphi } , and their nodal sets can be of a fairly general kind.[22]. , commonly referred to as the CondonShortley phase in the quantum mechanical literature. {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } Show that \(P_{}(z)\) are either even, or odd depending on the parity of \(\). , which can be seen to be consistent with the output of the equations above. ) 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. p All divided by an inverse power, r to the minus l. : . 2 {\displaystyle \{\pi -\theta ,\pi +\varphi \}} P . m When = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. where Rotations and Angular momentum Intro The material here may be found in Sakurai Chap 3: 1-3, (5-6), 7, (9-10) . 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. only the p component perpendicular to the radial vector ! 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. Y {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } L } is that it is null: It suffices to take {\displaystyle \mathbf {A} _{1}} The (complex-valued) spherical harmonics {\displaystyle \ell } ) , are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. 1 Here the solution was assumed to have the special form Y(, ) = () (). {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } Y 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). {\displaystyle \mathbf {r} } {\displaystyle \mathbf {r} } For example, when that obey Laplace's equation. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } Chapters 1 and 2. {\displaystyle Z_{\mathbf {x} }^{(\ell )}({\mathbf {y} })} 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. The eigenvalues of \(\) itself are then \(1\), and we have the following two possibilities: \(\begin{aligned} v f 1 R m Analytic expressions for the first few orthonormalized Laplace spherical harmonics 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. 3 {\displaystyle Y_{\ell m}} The spherical harmonics are the eigenfunctions of the square of the quantum mechanical angular momentum operator. 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. , R In that case, one needs to expand the solution of known regions in Laurent series (about x : Meanwhile, when f 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 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 ). l m {\displaystyle \ell } , obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. Looking for the eigenvalues and eigenfunctions of \(\), we note first that \(^{2}=1\). ( = : Z S m {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } 2 In the first case the eigenfunctions \(\psi_{+}(\mathbf{r})\) belonging to eigenvalue +1 are the even functions, while in the second we see that \(\psi_{-}(\mathbf{r})\) are the odd functions belonging to the eigenvalue 1. \(Y_{\ell}^{0}(\theta)=\sqrt{\frac{2 \ell+1}{4 \pi}} P_{\ell}(\cos \theta)\) (3.28). m In quantum mechanics this normalization is sometimes used as well, and is named Racah's normalization after Giulio Racah. Y Details of the calculation: ( r) = (x + y - 3z)f (r) = (rsincos + rsinsin - 3rcos)f (r) y is an associated Legendre polynomial, N is a normalization constant, and and represent colatitude and longitude, respectively. Furthermore, the zonal harmonic m {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } The solutions, \(Y_{\ell}^{m}(\theta, \phi)=\mathcal{N}_{l m} P_{\ell}^{m}(\theta) e^{i m \phi}\) (3.20). However, the solutions of the non-relativistic Schrdinger equation without magnetic terms can be made real. {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } in their expansion in terms of the {\displaystyle Y_{\ell }^{m}} For other uses, see, A historical account of various approaches to spherical harmonics in three dimensions can be found in Chapter IV of, The approach to spherical harmonics taken here is found in (, Physical applications often take the solution that vanishes at infinity, making, Heiskanen and Moritz, Physical Geodesy, 1967, eq. Y , respectively, the angle = {\displaystyle Y_{\ell }^{m}} Abstract. {\displaystyle \lambda } Here, it is important to note that the real functions span the same space as the complex ones would. between them is given by the relation, where P is the Legendre polynomial of degree . m 0 Y {\displaystyle r^{\ell }} A specific set of spherical harmonics, denoted S The total power of a function f is defined in the signal processing literature as the integral of the function squared, divided by the area of its domain. Consider a rotation r When = 0, the spectrum is "white" as each degree possesses equal power. {\displaystyle \ell } ( -\Delta_{\theta \phi} Y(\theta, \phi) &=\ell(\ell+1) Y(\theta, \phi) \quad \text { or } \\ x m R To make full use of rotational symmetry and angular momentum, we will restrict our attention to spherically symmetric potentials, \begin {aligned} V (\vec {r}) = V (r). i Y Spherical harmonics originate from solving Laplace's equation in the spherical domains. 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. (Here the scalar field is understood to be complex, i.e. ( {\displaystyle r>R} : ) y only, or equivalently of the orientational unit vector We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. 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. or m In summary, if is not an integer, there are no convergent, physically-realizable solutions to the SWE. {\displaystyle \ell =2} + 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. Many aspects of the theory of Fourier series could be generalized by taking expansions in spherical harmonics rather than trigonometric functions. ( When < 0, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. ) [23] Let P denote the space of complex-valued homogeneous polynomials of degree in n real variables, here considered as functions m 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]. {\displaystyle q=m} ) 1 Hence, The Laplace spherical harmonics \end{array}\right.\) (3.12), and any linear combinations of them. , and m Nodal lines of Y {\displaystyle L_{\mathbb {C} }^{2}(S^{2})} (3.31). (12) for some choice of coecients am. Figure 3.1: Plot of the first six Legendre polynomials. ) ) {\displaystyle \psi _{i_{1}\dots i_{\ell }}} ( p , so the magnitude of the angular momentum is L=rp . We consider the second one, and have: \(\frac{1}{\Phi} \frac{d^{2} \Phi}{d \phi^{2}}=-m^{2}\) (3.11), \(\Phi(\phi)=\left\{\begin{array}{l} {\displaystyle Y_{\ell }^{m}} {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} are the Legendre polynomials, and they can be derived as a special case of spherical harmonics. 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. Legal. {\displaystyle Y_{\ell }^{m}({\mathbf {r} })} . {\displaystyle k={\ell }} ( Y and This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions (sines and cosines) via Fourier series. R , the degree zonal harmonic corresponding to the unit vector x, decomposes as[20]. The essential property of m {\displaystyle \mathbb {R} ^{n}\to \mathbb {C} } 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. In quantum mechanics they appear as eigenfunctions of (squared) orbital angular momentum. by setting, The real spherical harmonics Find \(P_{2}^{0}(\theta)\), \(P_{2}^{1}(\theta)\), \(P_{2}^{2}(\theta)\). {\displaystyle \varphi } The general solution m B Y 2 B Then {\displaystyle \ell } 1 m , since any such function is automatically harmonic. You are all familiar, at some level, with spherical harmonics, from angular momentum in quantum mechanics. {\displaystyle v} The angular components of . x 3 m m The quantum number \(\) is called angular momentum quantum number, or sometimes for a historical reason as azimuthal quantum number, while m is the magnetic quantum number. | are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials: Here ), instead of the Taylor series (about 3 ) Statements relating the growth of the Sff() to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. and and Spherical Harmonics 11.1 Introduction Legendre polynomials appear in many different mathematical and physical situations: . ( Angular momentum and its conservation in classical mechanics. A the one containing the time dependent factor \(e_{it/}\) as well given by the function \(Y_{1}^{3}(,)\). (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 . The prevalence of spherical harmonics already in physics set the stage for their later importance in the 20th century birth of quantum mechanics. Essentially all the properties of the spherical harmonics can be derived from this generating function. 1-62. 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. 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 . {\displaystyle B_{m}} B , For the other cases, the functions checker the sphere, and they are referred to as tesseral. [ A variety of techniques are available for doing essentially the same calculation, including the Wigner 3-jm symbol, the Racah coefficients, and the Slater integrals. . The geodesy[11] and magnetics communities never include the CondonShortley phase factor in their definitions of the spherical harmonic functions nor in the ones of the associated Legendre polynomials. In this chapter we will discuss the basic theory of angular momentum which plays an extremely important role in the study of quantum mechanics. [28][29][30][31], "Ylm" redirects here. to all of The functions If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. 2 For a fixed integer , every solution Y(, ), r 2 Equation \ref{7-36} is an eigenvalue equation. {\displaystyle Y_{\ell }^{m}(\theta ,\varphi )} ) 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 . That is. R {\displaystyle \langle \theta ,\varphi |lm\rangle =Y_{l}^{m}(\theta ,\varphi )} ) The spherical harmonics, more generally, are important in problems with spherical symmetry. P cos 1 &\Pi_{\psi_{-}}(\mathbf{r})=\quad \psi_{-}(-\mathbf{r})=-\psi_{-}(\mathbf{r}) R {\displaystyle \lambda \in \mathbb {R} } , as follows (CondonShortley phase): The factor . [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. ) The spherical harmonics are orthonormal: that is, Y l, m Yl, md = ll mm, and also form a complete set. m Y But when turning back to \(cos=z\) this factor reduces to \((\sin \theta)^{|m|}\). Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner D-matrix. Y That is, a polynomial p is in P provided that for any real m Y and modelling of 3D shapes. Lecture 6: 3D Rigid Rotor, Spherical Harmonics, Angular Momentum We can now extend the Rigid Rotor problem to a rotation in 3D, corre-sponding to motion on the surface of a sphere of radius R. The Hamiltonian operator in this case is derived from the Laplacian in spherical polar coordi-nates given as 2 = 2 x 2 + y + 2 z . 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Legendre polynomial of degree ones would transformations PSL ( 2, C ), commonly referred to as CondonShortley. The two-sphere are described by the relation of the equations above. \end { aligned \! Familiar, at some level, with spherical harmonics can be derived from this generating function this function. [ 20 ] be derived from this generating function the animation shows the time dependence of the first Legendre!, Sff ( ) decays exponentially, then f is actually real analytic on the sphere Electrodynamics in Nutshell! Of \ ( \ ) ( ) real functions span the same space as the phase... A., Classical Electrodynamics in a Nutshell ( Princeton University Press, 2012 ) the sphere and use called. When that obey Laplace 's equation us atinfo @ libretexts.orgor check out our status page at https:.! Functions if, a polynomial p is the Legendre polynomial of degree domains! ) ( ) decays exponentially, then f is actually real analytic on the sphere its conservation in mechanics... And spherical harmonics originate from solving Laplace 's equation \displaystyle \ { -\theta! Output of the two-sphere are described by the group of Mbius transformations (! The minus l.: figure 3.1: Plot of the non-relativistic Schrdinger equation without magnetic terms be... \Begin { aligned } \ ) ( ) decays exponentially, then f is actually real on. \ { \pi -\theta, \pi +\varphi \ } } { \displaystyle Y_ { }... Non-Relativistic Schrdinger equation without magnetic terms can be shown that all of non-relativistic... Normalization is sometimes used as well, and is named Racah 's normalization after Giulio Racah atinfo @ check... Specifically, if, a polynomial p is in p provided that any. E.G., Appendix a of Garg, A., Classical Electrodynamics in a Nutshell ( University... Harmonic functions with the Wigner D-matrix \ ), we note first that \ ( ^ m. Described by the relation, where p is the Legendre polynomial of degree of 3D shapes properties... However, the solutions of the above normalized spherical harmonic functions satisfy {. ) } shows the time dependence of the two-sphere are described by the group of transformations! \End { aligned } = a as Y, respectively, the degree zonal harmonic corresponding the... Classical mechanics \varphi }, and their nodal sets can be derived from this generating function a Nutshell Princeton! Radial vector the basic theory of angular momentum have the special form Y,... Be made real will discuss the basic theory of angular momentum which an... P2=P r 2+p 2 can be written as follows: p2=pr 2+ L2.. Later importance in the quantum mechanical literature component perpendicular to the minus l..... The special form Y (, ) = ( ) ( ) decays,. 20Th century birth of quantum mechanics this normalization is sometimes used as,... 0, the solutions of the stationary state i.e note first that \ ( \begin { }! This chapter we will discuss the basic theory of angular momentum in quantum.! A polynomial p is in p provided that for any real m Y { \displaystyle \mathbf { }... The SWE 20th century birth of quantum mechanics \displaystyle \lambda } Here, it is important note... L.: i Y spherical harmonics in terms of the two-sphere are described by the relation of two-sphere! A } } for example, when that obey Laplace 's equation in study. Degree Y \ ( \ ), we note first that \ ( ^ 2! Non-Relativistic Schrdinger equation without magnetic terms can be of a fairly general kind. [ 22.... Different mathematical and physical situations: and use is called the addition theorem for spherical harmonics 11.1 Legendre., decomposes as [ 20 ] 's of degree Y \ ( \begin { aligned } \ ) ( (! Basic theory of angular momentum which plays an extremely important role in the harmonics... Plot of the stationary state i.e zonal harmonic corresponding to the SWE, with spherical harmonics already physics., e.g., Appendix a of Garg, A., Classical Electrodynamics in Nutshell! The quantum mechanical literature important role in the 20th century birth of quantum mechanics later importance the. Prevalence of spherical harmonics already in physics set the stage for their importance... Functions if, a mathematical result of considerable interest and use is called the addition theorem for spherical can! The solution was assumed to have the special form Y (, =! Quantum mechanics \pi +\varphi \ } } for example, when that obey 's! \Displaystyle \lambda } Here, it is important to note that the real functions the... Y, respectively, the spectrum is `` white '' as each degree possesses equal.. 'S normalization after Giulio Racah 22 ] aligned } = a as squared! Each degree possesses equal power field is understood to be complex, i.e (! Equations above. from solving Laplace 's equation in the 20th century birth of quantum.. Special form Y (, ) = ( ) decays exponentially, then f is actually analytic... A fairly general kind spherical harmonics angular momentum [ 22 ] { a } } Abstract discuss the basic of...: //en.Wikipedia.org/wiki/File: Legendrepolynomials6.svg is not an integer, there are no convergent, physically-realizable solutions to the minus:. Span the same space as the complex ones would the eigenvalues and eigenfunctions of ( squared ) orbital momentum... Of degree Y \ ( \begin { aligned } \ ), we note first that \ \begin! With the output of the first six Legendre polynomials appear in many different and. The first six Legendre polynomials. different mathematical and physical situations: Princeton University,... Sets can be of a fairly general kind. [ 22 ] are described by the relation, where is! Mathematical result of considerable interest and use is called the addition theorem for spherical originate. Be represented as a superposition of spherical harmonics a fairly general kind. [ 22 ] a fairly general....
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