Recalling that the spherical harmonics are eigenfunctions of the angular momentum operator: (r; ;) = R(r)Ym l ( ;) SeparationofVariables L^2Ym l ( ;) = h2l . m {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } m The spherical harmonics play an important role in quantum mechanics. : + , are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782. {\displaystyle Y_{\ell }^{m}} Therefore the single eigenvalue of \(^{2}\) is 1, and any function is its eigenfunction. In both classical and quantum mechanical systems, angular momentum (together with linear momentum and energy) is one of the three fundamental properties of motion. m The angular momentum operator plays a central role in the theory of atomic physics and other quantum problems involving rotational symmetry. R (Here the scalar field is understood to be complex, i.e. {\displaystyle Y_{\ell }^{m}} S . Analytic expressions for the first few orthonormalized Laplace spherical harmonics directions respectively. symmetric on the indices, uniquely determined by the requirement. C ( \left(\partial_{\theta \theta}^{2}+\cot \theta \partial_{\theta}+\frac{1}{\sin ^{2} \theta} \partial_{\phi \phi}^{2}\right) Y(\theta, \phi) &=-\ell(\ell+1) Y(\theta, \phi) , This constant is traditionally denoted by \(m^{2}\) and \(m^{2}\) (note that this is not the mass) and we have two equations: one for \(\), and another for \(\). {\displaystyle k={\ell }} {\displaystyle \mathbf {r} } + r \(\begin{aligned} 2 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). Statements relating the growth of the Sff() to differentiability are then similar to analogous results on the growth of the coefficients of Fourier series. ) Another way of using these functions is to create linear combinations of functions with opposite m-s. as a function of ( Spherical harmonics can be generalized to higher-dimensional Euclidean space J We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. m , the real and imaginary components of the associated Legendre polynomials each possess |m| zeros, each giving rise to a nodal 'line of latitude'. {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } {\displaystyle (r',\theta ',\varphi ')} (12) for some choice of coecients am. {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } 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. [14] An immediate benefit of this definition is that if the vector {\displaystyle v} 3 r {\displaystyle \langle \theta ,\varphi |lm\rangle =Y_{l}^{m}(\theta ,\varphi )} ( {\displaystyle P_{i}:[-1,1]\to \mathbb {R} } m , or alternatively where {\displaystyle {\mathcal {R}}} 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]. Given two vectors r and r, with spherical coordinates 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. m The same sine and cosine factors can be also seen in the following subsection that deals with the Cartesian representation. 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. and {\displaystyle f_{\ell }^{m}\in \mathbb {C} } {\displaystyle z} R R One can determine the number of nodal lines of each type by counting the number of zeros of The solution function Y(, ) is regular at the poles of the sphere, where = 0, . &\hat{L}_{z}=-i \hbar \partial_{\phi} {\displaystyle S^{2}} {\displaystyle Y_{\ell }^{m}} R 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. R from the above-mentioned polynomial of degree = z ), 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. 2 However, the solutions of the non-relativistic Schrdinger equation without magnetic terms can be made real. 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. 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 \(\). Spherical harmonics, as functions on the sphere, are eigenfunctions of the Laplace-Beltrami operator (see the section Higher dimensions below). ) Looking for the eigenvalues and eigenfunctions of \(\), we note first that \(^{2}=1\). 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. ( C [ > Using the expressions for and modelling of 3D shapes. f {\displaystyle P_{\ell }^{m}} With respect to this group, the sphere is equivalent to the usual Riemann sphere. 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 Y_{\ell m}} ) Nodal lines of &\Pi_{\psi_{-}}(\mathbf{r})=\quad \psi_{-}(-\mathbf{r})=-\psi_{-}(\mathbf{r}) Throughout the section, we use the standard convention that for ] 2 provide a basis set of functions for the irreducible representation of the group SO(3) of dimension S S J form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions {\displaystyle \mathbb {R} ^{3}} {\displaystyle {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })} {\displaystyle \Re [Y_{\ell }^{m}]=0} S ) R [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. {\displaystyle Y_{\ell m}} f : 0 Spherical Harmonics, and Bessel Functions Physics 212 2010, Electricity and Magnetism Michael Dine Department of Physics . by \(\mathcal{R}(r)\). terms (sines) are included: The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation. {\displaystyle \ell } Y , and The spherical harmonics are normalized . 2 ) {\displaystyle \ell } {\displaystyle \ell } {\displaystyle {\bar {\Pi }}_{\ell }^{m}(z)} . v One sees at once the reason and the advantage of using spherical coordinates: the operators in question do not depend on the radial variable r. This is of course also true for \(\hat{L}^{2}=\hat{L}_{x}^{2}+\hat{L}_{y}^{2}+\hat{L}_{z}^{2}\) which turns out to be \(^{2}\) times the angular part of the Laplace operator \(_{}\). Find the first three Legendre polynomials \(P_{0}(z)\), \(P_{1}(z)\) and \(P_{2}(z)\). R J {\displaystyle Y_{\ell }^{m}} Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree m n Essentially all the properties of the spherical harmonics can be derived from this generating function. The spherical harmonics have definite parity. and Spherical Harmonics 11.1 Introduction Legendre polynomials appear in many different mathematical and physical situations: . Laplace's spherical harmonics [ edit] Real (Laplace) spherical harmonics for (top to bottom) and (left to right). 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. There are two quantum numbers for the rigid rotor in 3D: \(J\) is the total angular momentum quantum number and \(m_J\) is the z-component of the angular momentum. f 3 Such spherical harmonics are a special case of zonal spherical functions. One concludes that the spherical harmonics in the solution for the electron wavefunction in the hydrogen atom identify the angular momentum of the electron. A , the trigonometric sin and cos functions possess 2|m| zeros, each of which gives rise to a nodal 'line of longitude'. that obey Laplace's equation. z S {\displaystyle \mathbb {R} ^{3}\to \mathbb {R} } 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 ). 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. is called a spherical harmonic function of degree and order m, The eigenfunctions of \(\hat{L}^{2}\) will be denoted by \(Y(,)\), and the angular eigenvalue equation is: \(\begin{aligned} In both definitions, the spherical harmonics are orthonormal, The disciplines of geodesy[10] and spectral analysis use, The magnetics[10] community, in contrast, uses Schmidt semi-normalized harmonics. { cos L 's, which in turn guarantees that they are spherical tensor operators, http://en.Wikipedia.org/wiki/Spherical_harmonics. (See Applications of Legendre polynomials in physics for a more detailed analysis. are associated Legendre polynomials without the CondonShortley phase (to avoid counting the phase twice). : . ) , we have a 5-dimensional space: For any {\displaystyle m<0} ) \end{aligned}\) (3.30). m to correspond to a (smooth) function f : by setting, The real spherical harmonics 1 2 m S Y In spherical coordinates this is:[2]. Spherical harmonics are ubiquitous in atomic and molecular physics. R R {\displaystyle f:S^{2}\to \mathbb {R} } {\displaystyle {\mathcal {Y}}_{\ell }^{m}({\mathbf {J} })} 3 (see associated Legendre polynomials), In acoustics,[7] the Laplace spherical harmonics are generally defined as (this is the convention used in this article). {\displaystyle A_{m}(x,y)} Y 0 Then That is. S ( {\displaystyle P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )} {\displaystyle \mathbb {R} ^{3}\to \mathbb {C} } {\displaystyle f:\mathbb {R} ^{3}\to \mathbb {C} } 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. {\displaystyle \mathbb {R} ^{3}} , S But when turning back to \(cos=z\) this factor reduces to \((\sin \theta)^{|m|}\). , Y 2 This parity property will be conrmed by the series 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? C P f \end{aligned}\) (3.8). S 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 {\displaystyle p:\mathbb {R} ^{3}\to \mathbb {C} } R = ) do not have that property. are constants and the factors r Ym are known as (regular) solid harmonics He discovered that if r r1 then, where is the angle between the vectors x and x1. [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. \(Y_{\ell}^{0}(\theta)=\sqrt{\frac{2 \ell+1}{4 \pi}} P_{\ell}(\cos \theta)\) (3.28). 1 0 These angular solutions , with The angular components of . {\displaystyle \lambda \in \mathbb {R} } 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. The animation shows the time dependence of the stationary state i.e. k , S {\displaystyle \ell =4} 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. being a unit vector, In terms of the spherical angles, parity transforms a point with coordinates {\displaystyle \mathbf {a} =[{\frac {1}{2}}({\frac {1}{\lambda }}-\lambda ),-{\frac {i}{2}}({\frac {1}{\lambda }}+\lambda ),1].}. ( {\displaystyle Y_{\ell m}:S^{2}\to \mathbb {R} } to m ) are complex and mix axis directions, but the real versions are essentially just x, y, and z. p. The cross-product picks out the ! f and r m transforms into a linear combination of spherical harmonics of the same degree. The spaces of spherical harmonics on the 3-sphere are certain spin representations of SO(3), with respect to the action by quaternionic multiplication. When the spherical harmonic order m is zero (upper-left in the figure), the spherical harmonic functions do not depend upon longitude, and are referred to as zonal. 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 Y 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). and 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? By definition, (382) where is an integer. form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions When = 0, the spectrum is "white" as each degree possesses equal power. m In the form L x; L y, and L z, these are abstract operators in an innite dimensional Hilbert space. 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 . The total angular momentum of the system is denoted by ~J = L~ + ~S. p is ! 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 . Y 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. where the superscript * denotes complex conjugation. Rotations and Angular momentum Intro The material here may be found in Sakurai Chap 3: 1-3, (5-6), 7, (9-10) . Accessibility StatementFor more information contact us atinfo@libretexts.orgor check out our status page at https://status.libretexts.org. ,[15] one obtains a generating function for a standardized set of spherical tensor operators, {\displaystyle Z_{\mathbf {x} }^{(\ell )}} m as a function of For a given value of , there are 2 + 1 independent solutions of this form, one for each integer m with m . ( We will use the actual function in some problems. L | If, furthermore, Sff() decays exponentially, then f is actually real analytic on the sphere. m of the elements of Y ) {\displaystyle \theta } = The half-integer values do not give vanishing radial solutions. Parity continues to hold for real spherical harmonics, and for spherical harmonics in higher dimensions: applying a point reflection to a spherical harmonic of degree changes the sign by a factor of (1). x [27] One is hemispherical functions (HSH), orthogonal and complete on hemisphere. 2 ( The result of acting by the parity on a function is the mirror image of the original function with respect to the origin. In that case, one needs to expand the solution of known regions in Laurent series (about to Laplace's equation C R 0 C ( {\displaystyle P_{\ell }^{m}(\cos \theta )} {\displaystyle f_{\ell }^{m}} {\displaystyle \theta } 2 } : , such that 2 The general, normalized Spherical Harmonic is depicted below: Y_ {l}^ {m} (\theta,\phi) = \sqrt { \dfrac { (2l + 1) (l - |m|)!} S } of spherical harmonics of degree Consider the problem of finding solutions of the form f(r, , ) = R(r) Y(, ). {\displaystyle S^{2}\to \mathbb {C} } The solid harmonics were homogeneous polynomial solutions setting, If the quantum mechanical convention is adopted for the . Y Y they can be considered as complex valued functions whose domain is the unit sphere. Functions that are solutions to Laplace's equation are called harmonics. The Laplace spherical harmonics -\Delta_{\theta \phi} Y(\theta, \phi) &=\ell(\ell+1) Y(\theta, \phi) \quad \text { or } \\ : ( For example, as can be seen from the table of spherical harmonics, the usual p functions ( k B We demonstrate this with the example of the p functions. specified by these angles. m As to what's "really" going on, it's exactly the same thing that you have in the quantum mechanical addition of angular momenta. Alternatively, this equation follows from the relation of the spherical harmonic functions with the Wigner D-matrix. = ( is just the space of restrictions to the sphere {\displaystyle {\mathcal {Y}}_{\ell }^{m}} 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} {\displaystyle f:S^{2}\to \mathbb {C} \supset \mathbb {R} } 2 p m When = |m| (bottom-right in the figure), there are no zero crossings in latitude, and the functions are referred to as sectoral. 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Operator ( see Applications of Legendre polynomials appear in many different mathematical and physical situations: sphere are!