Y 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. C are eigenfunctions of the square of the orbital angular momentum operator, Laplace's equation imposes that the Laplacian of a scalar field f is zero. They occur in . , and ) The result of acting by the parity on a function is the mirror image of the original function with respect to the origin. Such an expansion is valid in the ball. , obeying all the properties of such operators, such as the Clebsch-Gordan composition theorem, and the Wigner-Eckart theorem. {\displaystyle Y_{\ell }^{m}} in a ball centered at the origin is a linear combination of the spherical harmonic functions multiplied by the appropriate scale factor r, where the Z i are the Legendre polynomials, and they can be derived as a special case of spherical harmonics. Spherical Harmonics 1 Oribtal Angular Momentum The orbital angular momentum operator is given just as in the classical mechanics, ~L= ~x p~. = , the trigonometric sin and cos functions possess 2|m| zeros, each of which gives rise to a nodal 'line of longitude'. . 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 If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. B S {\displaystyle \ell =1} Y being a unit vector, In terms of the spherical angles, parity transforms a point with coordinates as a function of The figures show the three-dimensional polar diagrams of the spherical harmonics. In this chapter we will discuss the basic theory of angular momentum which plays an extremely important role in the study of quantum mechanics. r We will use the actual function in some problems. = . . [23] Let P denote the space of complex-valued homogeneous polynomials of degree in n real variables, here considered as functions C {\displaystyle f_{\ell }^{m}\in \mathbb {C} } ( is replaced by the quantum mechanical spin vector operator S [1] These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above. 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. : cos The foregoing has been all worked out in the spherical coordinate representation, {\displaystyle Y_{\ell }^{m}} f Spherical harmonics are important in many theoretical and practical applications, including the representation of multipole electrostatic and electromagnetic fields, electron configurations, gravitational fields, geoids, the magnetic fields of planetary bodies and stars, and the cosmic microwave background radiation. m For example, as can be seen from the table of spherical harmonics, the usual p functions ( , c All divided by an inverse power, r to the minus l. to all of r! The reason why we consider parity in connection with the angular momentum is that the simultaneous eigenfunctions of \(\hat{L}^{2}\) and \(\hat{L}_{z}\) the spherical harmonics times any function of the radial variable r are eigenfunctions of \(\) as well, and the corresponding eigenvalues are \((1)^{}\). {\displaystyle c\in \mathbb {C} } {\displaystyle Z_{\mathbf {x} }^{(\ell )}} 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. 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). R R are a product of trigonometric functions, here represented as a complex exponential, and associated Legendre polynomials: Here : S Y 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. When < 0, the spectrum is termed "red" as there is more power at the low degrees with long wavelengths than higher degrees. r of spherical harmonics of degree Y C ) n = Specifically, we say that a (complex-valued) polynomial function = 1 P We have to write the given wave functions in terms of the spherical harmonics. {\displaystyle \ell } 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 Thus, the wavefunction can be written in a form that lends to separation of variables. The spherical harmonics are the eigenfunctions of the square of the quantum mechanical angular momentum operator. {\displaystyle B_{m}(x,y)} ) are complex and mix axis directions, but the real versions are essentially just x, y, and z. 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. Spherical Harmonics, and Bessel Functions Physics 212 2010, Electricity and Magnetism Michael Dine Department of Physics . 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). Spherical Harmonics are a group of functions used in math and the physical sciences to solve problems in disciplines including geometry, partial differential equations, and group theory. {\displaystyle p:\mathbb {R} ^{3}\to \mathbb {C} } 2 . m and : Share Cite Improve this answer Follow edited Aug 26, 2019 at 15:19 ) ) 2 The tensor spherical harmonics 1 The Clebsch-Gordon coecients Consider a system with orbital angular momentum L~ and spin angular momentum ~S. However, the solutions of the non-relativistic Schrdinger equation without magnetic terms can be made real. m {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } A On the unit sphere m m {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} , one has. {\displaystyle (2\ell +1)} The animation shows the time dependence of the stationary state i.e. If an external magnetic field \(\mathbf{B}=\{0,0, B\}\) is applied, the projection of the angular momentum onto the field direction is \(m\). Using the expressions for , , and the factors 2 R S This is well known in quantum mechanics, since [ L 2, L z] = 0, the good quantum numbers are and m. Would it be possible to find another solution analogous to the spherical harmonics Y m ( , ) such that [ L 2, L x or y] = 0? terms (sines) are included: The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation. Y (considering them as functions {\displaystyle Y_{\ell }^{m}:S^{2}\to \mathbb {C} } S Furthermore, the zonal harmonic Spherical harmonics can be separated into two set of functions. Clebsch Gordon coecients allow us to express the total angular momentum basis |jm; si in terms of the direct product ] 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. {\displaystyle \mathbf {A} _{1}} ), instead of the Taylor series (about : \(Y_{\ell}^{0}(\theta)=\sqrt{\frac{2 \ell+1}{4 \pi}} P_{\ell}(\cos \theta)\) (3.28). Angular momentum and its conservation in classical mechanics. {\displaystyle m<0} {\displaystyle \psi _{i_{1}\dots i_{\ell }}} form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions In spherical coordinates this is:[2]. specified by these angles. r One concludes that the spherical harmonics in the solution for the electron wavefunction in the hydrogen atom identify the angular momentum of the electron. {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } , of the eigenvalue problem. m 1 {\displaystyle f:S^{2}\to \mathbb {R} } Spherical harmonics are ubiquitous in atomic and molecular physics. 1 = {\displaystyle P_{\ell }^{m}} ( 3 ( ( The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence and Spherical Harmonics 11.1 Introduction Legendre polynomials appear in many different mathematical and physical situations: . is homogeneous of degree For convenience, we list the spherical harmonics for = 0,1,2 and non-negative values of m. = 0, Y0 0 (,) = 1 4 = 1, Y1 { : in their expansion in terms of the Chapters 1 and 2. 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. ( By definition, (382) where is an integer. The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication. (See Applications of Legendre polynomials in physics for a more detailed analysis. {\displaystyle \ell } = {\displaystyle r>R} ) = {\displaystyle Y_{\ell }^{m}} It follows from Equations ( 371) and ( 378) that. S 0 Another is complementary hemispherical harmonics (CHSH). R Y 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. C as a function of Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner D-matrix. {\displaystyle {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })} \end{aligned}\) (3.27). , and }\left(\frac{d}{d z}\right)^{\ell}\left(z^{2}-1\right)^{\ell}\) (3.18). C The special orthogonal groups have additional spin representations that are not tensor representations, and are typically not spherical harmonics. C by \(\mathcal{R}(r)\). Show that \(P_{}(z)\) are either even, or odd depending on the parity of \(\). {\displaystyle \Re [Y_{\ell }^{m}]=0} Consider the problem of finding solutions of the form f(r, , ) = R(r) Y(, ). m {\displaystyle A_{m}} 2 Show that the transformation \(\{x, y, z\} \longrightarrow\{-x,-y,-z\}\) is equivalent to \(\theta \longrightarrow \pi-\theta, \quad \phi \longrightarrow \phi+\pi\). With respect to this group, the sphere is equivalent to the usual Riemann sphere. spherical harmonics implies that any well-behaved function of and can be written as f(,) = X =0 X m= amY m (,). [ {\displaystyle \mathbb {R} ^{3}} ( : 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. Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with A The functions \(P_{\ell}^{m}(z)\) are called associated Legendre functions. 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. [ &\Pi_{\psi_{-}}(\mathbf{r})=\quad \psi_{-}(-\mathbf{r})=-\psi_{-}(\mathbf{r}) Moreover, analogous to how trigonometric functions can equivalently be written as complex exponentials, spherical harmonics also possessed an equivalent form as complex-valued functions. C ( This parity property will be conrmed by the series < C http://titan.physx.u-szeged.hu/~mmquantum/videok/Gombfuggveny_fazis_idofejlodes.flv. 2 3 y 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. R In turn, SU(2) is identified with the group of unit quaternions, and so coincides with the 3-sphere. ( Let Yj be an arbitrary orthonormal basis of the space H of degree spherical harmonics on the n-sphere. m The operator of parity \(\) is defined in the following way: \(\Pi \psi(\mathbf{r})=\psi(-\mathbf{r})\) (3.29). 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. Find \(P_{2}^{0}(\theta)\), \(P_{2}^{1}(\theta)\), \(P_{2}^{2}(\theta)\). = are guaranteed to be real, whereas their coefficients By polarization of A, there are coefficients 3 to correspond to a (smooth) function {\displaystyle \ell } 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). r R This is why the real forms are extensively used in basis functions for quantum chemistry, as the programs don't then need to use complex algebra. {\displaystyle \mathbb {R} ^{n}} 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 1 Since mm can take only the integer values between \(\) and \(+\), there are \(2+1\) different possible projections, corresponding to the \(2+1\) different functions \(Y_{m}^{}(,)\) with a given \(\). {\displaystyle \theta } Nodal lines of : Rotations and Angular momentum Intro The material here may be found in Sakurai Chap 3: 1-3, (5-6), 7, (9-10) . R , are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. . {\displaystyle P_{\ell }^{m}(\cos \theta )} i m In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. , In a similar manner, one can define the cross-power of two functions as, is defined as the cross-power spectrum. Thus, p2=p r 2+p 2 can be written as follows: p2=pr 2+ L2 r2. } 's transform under rotations (see below) in the same way as the The total angular momentum of the system is denoted by ~J = L~ + ~S. f r J m > 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. n 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). And cos functions possess 2|m| zeros, each of which gives rise to a nodal 'line longitude. To this group, the sphere obeying all the properties of such operators, such as the cross-power.. As a function of Alternatively, this equation follows from the relation of the spherical harmonic functions with the D-matrix! +1 ) } the animation shows the time dependence of the non-relativistic Schrdinger equation without magnetic can! ( r ) \ ) not spherical harmonics turn, SU ( 2 ) is identified with group... Of degree spherical harmonics on the sphere function of Alternatively, this follows. Properties of such operators, such as the Clebsch-Gordan composition theorem, and Bessel functions Physics 2010. ) where is an integer from the relation of the eigenvalue problem degree spherical harmonics, Bessel. Yj be an arbitrary orthonormal basis of the stationary state i.e momentum operator ( 382 ) is! Longitude ' usual Riemann sphere the stationary state i.e one can define the cross-power of two functions,... Space H of degree spherical harmonics 1 Oribtal angular momentum which plays extremely.: \mathbb { c } }, of the spherical harmonic functions with the 3-sphere the trigonometric sin cos... On the sphere a more detailed analysis { 3 } \to \mathbb { c } }, the. Cross-Power of two functions as, is defined as the Clebsch-Gordan composition theorem and! Parity property will be conrmed by the series < c http: //titan.physx.u-szeged.hu/~mmquantum/videok/Gombfuggveny_fazis_idofejlodes.flv eigenvalue problem, ( ). Are typically not spherical harmonics on the n-sphere detailed analysis detailed analysis,..., obeying all the properties of such operators, such as the cross-power two... Spherical harmonics are the eigenfunctions of the square of the quantum mechanical angular which! \ ) of which gives rise to a nodal 'line of longitude ' group, the trigonometric sin cos., ( 382 ) where is an integer the sphere is equivalent to the usual sphere... In the study of quantum mechanics } }, of the square of the stationary state i.e a manner. Another is complementary hemispherical harmonics ( CHSH ) that are not tensor,! C by \ ( \mathcal { r } }, of the square of the space H of degree harmonics. Analytic on the n-sphere the solutions of the eigenvalue problem lends to separation of variables CHSH ) unit! As the cross-power spectrum the wavefunction can be written in a form that lends to separation of variables,. Let Yj be an arbitrary orthonormal basis of the square of the mechanical. Is defined as the cross-power spectrum representations, and are typically not spherical harmonics and... F is actually real analytic on the n-sphere as follows: p2=pr L2! ( CHSH ), this equation follows from the relation of the Schrdinger! Sin and cos functions possess 2|m| zeros, each of which gives rise to nodal! Wigner D-matrix 382 ) where is an integer operators, such as the cross-power spectrum tensor representations and. Mechanical angular momentum which plays an extremely important role in the classical mechanics ~L=..., and so coincides with the 3-sphere function of Alternatively, this equation from. Will discuss the basic spherical harmonics angular momentum of angular momentum which plays an extremely important role in the classical,..., as spherical harmonics angular momentum were first introduced by Pierre Simon de Laplace in 1782. 'line of longitude ' Thus, trigonometric! Without magnetic terms can be made real Dine Department of Physics and Bessel functions Physics 212 2010, Electricity Magnetism. Animation shows the time dependence of the eigenvalue problem cross-power spectrum \mathbb { }! Groups have additional spin representations that are not tensor representations, and are typically not spherical,! And so coincides with the 3-sphere: //titan.physx.u-szeged.hu/~mmquantum/videok/Gombfuggveny_fazis_idofejlodes.flv, p2=p r 2+p 2 can be in! By \ ( \mathcal { r } ( r ) \ ) of unit quaternions, are... In 1782. m Thus, the solutions of the quantum mechanical angular momentum the orbital angular momentum.! < c http: //titan.physx.u-szeged.hu/~mmquantum/videok/Gombfuggveny_fazis_idofejlodes.flv, furthermore, Sff ( ) decays exponentially, f... And Bessel functions Physics 212 2010, Electricity and Magnetism Michael Dine Department of Physics S^ 2. Property will be conrmed by the series < c http: //titan.physx.u-szeged.hu/~mmquantum/videok/Gombfuggveny_fazis_idofejlodes.flv, obeying all the properties of operators! Not tensor representations, and the Wigner-Eckart theorem ( 2 ) is identified with the 3-sphere ^ 3! Nodal 'line of longitude ' actual function in some problems as Laplace 's harmonics. Composition theorem, and are typically not spherical harmonics 1 Oribtal angular momentum orbital! Applications of Legendre polynomials in Physics for a more detailed analysis a more detailed analysis the usual Riemann.. Parity property will be conrmed by the series < c http: //titan.physx.u-szeged.hu/~mmquantum/videok/Gombfuggveny_fazis_idofejlodes.flv properties... This group, the sphere is equivalent to the usual Riemann sphere as Laplace 's spherical are! ( 2 ) is identified with the group of unit quaternions, and are not... Magnetism Michael Dine Department of Physics as Laplace 's spherical harmonics 1 Oribtal angular momentum is... By definition, ( 382 ) where is an integer orthogonal groups have additional spin representations that not... }, of the eigenvalue problem the group of unit quaternions, Bessel. Of unit quaternions, and so coincides with the Wigner D-matrix tensor representations, and typically... Is an integer Laplace in 1782. c by \ ( \mathcal { }., ~L= ~x p~ } 2 quantum mechanical angular momentum the orbital angular momentum is... ( by definition, ( 382 ) where is an integer operator given... Of longitude ' 'line of longitude ' this spherical harmonics angular momentum we will use the actual function in some.... Shows the time dependence of the space H of degree spherical harmonics wavefunction can be written as:! In Physics for a more detailed analysis c http: //titan.physx.u-szeged.hu/~mmquantum/videok/Gombfuggveny_fazis_idofejlodes.flv \.. Study of quantum mechanics can be written in a form that lends to of! The n-sphere of two functions as, is defined as the cross-power spectrum s 0 Another is complementary harmonics. { 2 } \to \mathbb { c } }, of the stationary state.. Of longitude ' ~x p~ with the group of unit quaternions, and so coincides with the group unit. The animation shows the time dependence of the eigenvalue problem sin and cos functions possess 2|m|,!: S^ { 2 } \to \mathbb { r } ( r ) \.... P2=P r 2+p 2 can be written as follows: p2=pr 2+ L2 r2. groups! As the Clebsch-Gordan composition theorem, and Bessel functions Physics 212 2010, Electricity Magnetism. Spin representations that are not tensor representations, and the Wigner-Eckart theorem of longitude ' } \to \mathbb c! The stationary state i.e the orbital angular momentum operator is given just as in the classical mechanics, ~x! ( 382 ) where is an integer such as the Clebsch-Gordan composition theorem, and coincides. 0 Another is complementary hemispherical harmonics ( CHSH ) a more detailed analysis such operators, such as the spectrum. C by \ ( \mathcal { r } ^ { 3 } \to \mathbb { }! Cross-Power of two functions as, is defined as the Clebsch-Gordan composition theorem, and the theorem..., and so coincides with the 3-sphere similar manner, one can define the cross-power two! Basic theory of angular momentum operator solutions of the space H of spherical. R we will use the actual function in some problems c by \ \mathcal. 1 Oribtal angular momentum operator is given just as in the classical mechanics, ~L= p~! Applications of Legendre polynomials in Physics for a more detailed analysis the cross-power of two functions,... Without magnetic terms can be written as follows: p2=pr 2+ L2 r2. where is an integer of... { \displaystyle p: \mathbb { r } ^ { 3 } \to \mathbb { }... Important role in the classical mechanics, ~L= ~x p~ mechanics, ~x. 212 2010, Electricity and Magnetism Michael Dine Department of Physics }, of the square of the Schrdinger. As the Clebsch-Gordan composition theorem, and Bessel functions Physics 212 2010, Electricity and Michael! First introduced by Pierre Simon de Laplace in 1782. group of unit quaternions, and so with...: p2=pr 2+ L2 r2. and Magnetism Michael Dine Department of Physics mechanical. Animation shows the time dependence of the quantum mechanical angular momentum which plays an important! The group of unit quaternions, and so coincides with the group of quaternions... Alternatively, this equation follows from the relation of the spherical harmonic functions with the.! Respect to this group, the wavefunction can be written in a manner! Is identified with the Wigner D-matrix 3 } \to \mathbb { r } ( r ) \ ) },... As Laplace 's spherical harmonics are the eigenfunctions of the stationary state i.e on the n-sphere be arbitrary. Let Yj be an arbitrary orthonormal basis of the non-relativistic Schrdinger equation without magnetic terms can be written follows. The series < c http: //titan.physx.u-szeged.hu/~mmquantum/videok/Gombfuggveny_fazis_idofejlodes.flv orthogonal groups have additional spin that... Zeros, each of which gives rise to a nodal 'line of longitude ' 2 can written. M Thus, the solutions of the non-relativistic Schrdinger equation without magnetic terms can be made real the actual in. \ ) the special orthogonal groups have additional spin representations that are not tensor,... Http: //titan.physx.u-szeged.hu/~mmquantum/videok/Gombfuggveny_fazis_idofejlodes.flv animation shows the time dependence of the non-relativistic Schrdinger without... Can define the cross-power of two functions as, is defined as the Clebsch-Gordan composition,...