Nikiya Anton Bettey 2021: 09-23-2021-0550

Thursday, September 23, 2021

09-23-2021-0550 - spherical harmonics

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields.

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. 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. 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. Further, spherical harmonics are basis functions for irreducible representations of SO(3), the group of rotations in three dimensions, and thus play a central role in the group theoretic discussion of SO(3).

Spherical harmonics originates from solving Laplace's equation in the spherical domains. Functions that are solutions to Laplace's equation are called harmonics. Despite their name, spherical harmonics take their simplest form in Cartesian coordinates, where they can be defined as homogeneous polynomials of degree $\ell$ in $(x,y,z)$ that obey Laplace's equation. The connection with spherical coordinates arises immediately if one uses the homogeneity to extract a factor of radial dependence $r^\ell$ from the above-mentioned polynomial of degree $\ell$ ; the remaining factor can be regarded as a function of the spherical angular coordinates $\theta$ and $\varphi$ only, or equivalently of the orientational unit vector $\mathbf {r}$ specified by these angles. 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.

A specific set of spherical harmonics, denoted $Y_{\ell }^{m}(\theta ,\varphi )$ or $Y_{\ell }^{m}({\mathbf {r} })$ , are known as Laplace's spherical harmonics, as they were first introduced by Pierre Simon de Laplace in 1782.^[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.

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. In 3D computer graphics, spherical harmonics play a role in a wide variety of topics including indirect lighting (ambient occlusion, global illumination, precomputed radiance transfer, etc.) and modelling of 3D shapes.

Visual representations of the first few real spherical harmonics. Blue portions represent regions where the function is positive, and yellow portions represent where it is negative. The distance of the surface from the origin indicates the absolute value of

Y_{\ell }^{m}(\theta ,\phi )

in angular direction

(\theta ,\phi )

Symmetry properties[edit]

The spherical harmonics have deep and consequential properties under the operations of spatial inversion (parity) and rotation.

Parity[edit]

The spherical harmonics have definite parity. That is, they are either even or odd with respect to inversion about the origin. Inversion is represented by the operator $P\Psi (\mathbf {r} )=\Psi (-\mathbf {r} )$ . Then, as can be seen in many ways (perhaps most simply from the Herglotz generating function), with $\mathbf {r}$ being a unit vector,

Y_{\ell }^{m}(-\mathbf {r} )=(-1)^{\ell }Y_{\ell }^{m}(\mathbf {r} ).

In terms of the spherical angles, parity transforms a point with coordinates $\{\theta ,\phi \}$ to $\{\pi -\theta ,\pi +\phi \}$ . The statement of the parity of spherical harmonics is then

Y_{\ell }^{m}(\theta ,\phi )\rightarrow Y_{\ell }^{m}(\pi -\theta ,\pi +\phi )=(-1)^{\ell }Y_{\ell }^{m}(\theta ,\phi )

(This can be seen as follows: The associated Legendre polynomials gives (−1)^ℓ+m and from the exponential function we have (−1)^m, giving together for the spherical harmonics a parity of (−1)^ℓ.)

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 ℓ changes the sign by a factor of (−1)^ℓ.

Rotations[edit]

The rotation of a real spherical function with m = 0 and ℓ = 3. The coefficients are not equal to the Wigner D-matrices, since real functions are shown, but can be obtained by re-decomposing the complex functions

Consider a rotation $\mathcal R$ about the origin that sends the unit vector $\mathbf {r}$ to ${\mathbf r}'$ . Under this operation, a spherical harmonic of degree $\ell$ and order $m$ transforms into a linear combination of spherical harmonics of the same degree. That is,

Y_{\ell }^{m}({\mathbf {r} }')=\sum _{m'=-\ell }^{\ell }A_{mm'}Y_{\ell }^{m'}({\mathbf {r} }),

where $A_{mm'}$ is a matrix of order $(2\ell +1)$ that depends on the rotation $\mathcal R$ . However, this is not the standard way of expressing this property. In the standard way one writes,

Y_{\ell }^{m}({\mathbf {r} }')=\sum _{m'=-\ell }^{\ell }[D_{mm'}^{(\ell )}({\mathcal {R}})]^{*}Y_{\ell }^{m'}({\mathbf {r} }),

where $D_{mm'}^{(\ell )}({\mathcal {R}})^{*}$ is the complex conjugate of an element of the Wigner D-matrix. In particular when ${\mathbf r}'$ is a $\phi _{0}$ rotation of the azimuth we get the identity,

Y_{\ell }^{m}({\mathbf {r} }')=Y_{\ell }^{m}({\mathbf {r} })e^{im\phi _{0}}.

The rotational behavior of the spherical harmonics is perhaps their quintessential feature from the viewpoint of group theory. The $Y_{\ell }^{m}$ 's of degree $\ell$ provide a basis set of functions for the irreducible representation of the group SO(3) of dimension $(2\ell +1)$ . Many facts about spherical harmonics (such as the addition theorem) that are proved laboriously using the methods of analysis acquire simpler proofs and deeper significance using the methods of symmetry.

Spherical harmonics expansion[edit]

The Laplace spherical harmonics $Y_{\ell }^{m}:S^{2}\to \mathbb {C}$ form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions $L_{\mathbb {C} }^{2}(S^{2})$ . On the unit sphere $S^{2}$ , any square-integrable function $f:S^{2}\to \mathbb {C}$ can thus be expanded as a linear combination of these:

f(\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }f_{\ell }^{m}\,Y_{\ell }^{m}(\theta ,\varphi ).

This expansion holds in the sense of mean-square convergence — convergence in L² of the sphere — which is to say that

\lim _{N\to \infty }\int _{0}^{2\pi }\int _{0}^{\pi }\left|f(\theta ,\varphi )-\sum _{\ell =0}^{N}\sum _{m=-\ell }^{\ell }f_{\ell }^{m}Y_{\ell }^{m}(\theta ,\varphi )\right|^{2}\sin \theta \,d\theta \,d\varphi =0.

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 Ω, and utilizing the above orthogonality relationships. This is justified rigorously by basic Hilbert space theory. For the case of orthonormalized harmonics, this gives:

f_{\ell }^{m}=\int _{\Omega }f(\theta ,\varphi )\,Y_{\ell }^{m*}(\theta ,\varphi )\,d\Omega =\int _{0}^{2\pi }d\varphi \int _{0}^{\pi }\,d\theta \,\sin \theta f(\theta ,\varphi )Y_{\ell }^{m*}(\theta ,\varphi ).

If the coefficients decay in ℓ sufficiently rapidly — for instance, exponentially — then the series also converges uniformly to f.

A square-integrable function $f:S^{2}\to \mathbb {R}$ can also be expanded in terms of the real harmonics $Y_{\ell m}:S^{2}\to \mathbb {R}$ above as a sum

f(\theta ,\varphi )=\sum _{\ell =0}^{\infty }\sum _{m=-\ell }^{\ell }f_{\ell m}\,Y_{\ell m}(\theta ,\varphi ).

The convergence of the series holds again in the same sense, namely the real spherical harmonics $Y_{\ell m}:S^{2}\to \mathbb {R}$ form a complete set of orthonormal functions and thus form an orthonormal basis of the Hilbert space of square-integrable functions $L_{\mathbb {R} }^{2}(S^{2})$ . The benefit of the expansion in terms of the real harmonic functions $Y_{\ell m}$ is that for real functions $f:S^{2}\to \mathbb {R}$ the expansion coefficients $f_{\ell m}$ are guaranteed to be real, whereas their coefficients $f_{\ell }^{m}$ in their expansion in terms of the $Y_{\ell}^m$ (considering them as functions $f:S^{2}\to \mathbb {C} \supset \mathbb {R}$ ) do not have that property.

Harmonic tensors[edit]

Formula[edit]

As a rule, harmonic functions are useful in theoretical physics to consider fields in the far field when the distance from charges is much farther than the size of their location. In this case, the radius R is constant and coordinates (θ,φ) are convenient to use. Theoretical physics considers many problems when a solution of Laplace's equation is needed as a function of Сartesian coordinates. At the same time, it is important to get an invariant form of solutions relative to the rotation of space or, generally speaking, relative to group transformations.^[15]^[16]^[17]^[18] The simplest tensor solutions – dipole, quadrupole and octupole potentials – are fundamental concepts of general physics:

{\begin{aligned}T_{i}^{(1)}=x_{i},\\T_{ik}^{(2)}=3x{_{i}}x{_{k}}-\delta _{ik}r^{2},\\T_{ikn}^{(3)}=15x{_{i}}x{_{k}}x{_{n}}-3\delta _{ik}{r^{2}}x{_{n}}-3\delta _{kn}{r^{2}}x{_{i}}-3\delta _{ni}{r^{2}}x{_{k}}.\end{aligned}}

It is easy to verify that they are the harmonic functions. The total set of tensors is defined by the Taylor series of a point charge field potential for $r_{0}<r$ :

{\frac {1}{\left|{\boldsymbol {r-r}}{_{0}}\right|}}=\sum _{l}(-1)^{l}{\frac {{({\boldsymbol {r_{0}}}\nabla )}^{l}}{l!}}{\frac {1}{r}}=\sum _{l}{\frac {x_{0i}\ldots x_{0k}}{l!\,r^{2l+1}}}T_{i\ldots k}^{(l)}({\boldsymbol {r}})=\sum _{l}{\frac {\left[\otimes {\boldsymbol {{r_{0}}^{l}T^{(l)}}}\right]}{l!\,r^{2l+1}}},

where tensor $x_{0i}\ldots x_{0k}$ is denoted by symbol $\otimes {\boldsymbol {r_{0}}}^{l}$ and contraction of the tensors is in the brackets [...]. Therefore, the tensor ${\boldsymbol {T}}^{(l)}$ is defined by the $l$ -th tensor derivative:

{\frac {\boldsymbol {T^{(l)}}}{r^{(2l+1)}}}=(-\otimes {\boldsymbol {\nabla }})^{l}{\frac {1}{r}}

James Clerk Maxwell used similar considerations without tensors naturally.^[19] E. W. Hobson analysed Maxwell's method as well.^[20] One can see from the equation the following properties that repeat mainly those of solid and spherical functions.

The tensor is the harmonic polynomial i. e. $\Delta {\boldsymbol {T}}^{(l)}=0$ .
The trace over each pair of indices is zero, as far as ${\textstyle \Delta {\frac {1}{r}}=0}$ .
The tensor is a homogeneous polynomial of degree $l$ i.e. summed degree of variables x, y, z of each item is equal to $l$ .
The tensor has invariant form under rotations of variables x,y,z i.e. of vector $\mathbf {r}$ .
The total set of potentials $\mathbf {T} ^{(l)}$ is complete.
Contraction of $\mathbf {T} ^{(l)}(\mathbf {r} )$ with a tensor $\otimes {\boldsymbol {\rho }}^{l}$ is proportional to contraction of two harmonic potentials: $\left[\mathbf {T} ^{(l)}(\mathbf {r} )\otimes {\boldsymbol {\rho }}^{l}\right]={\frac {1}{(2l-1)!!}}\left[\mathbf {T} ^{(l)}(\mathbf {r} )\mathbf {T} ^{(l)}({\boldsymbol {\rho }})\right]$

The formula for a harmonic invariant tensor was found in paper.^[21] A detailed description is given in the monography.^[22] 4-D harmonic tensors are important in Fock symmetry found in quantum the Coulomb problem.^[23] The formula contains products of tensors $x_{i}\ldots x_{k}=\mathbf {\otimes } r^{m}$ and Kronecker symbols $\delta _{ik}$ :

\mathbf {T} ^{(l)}=(2l-1)!!\ \mathbf {(} \otimes r^{l})-(2l-3)!!\ r^{2}\left\langle \otimes \mathbf {r} ^{(l-2)}\otimes {\mathsf {\boldsymbol {\delta }}}^{1}\right\rangle +(2l-5)!!\ r^{4}\left\langle \otimes \mathbf {r} ^{(l-4)}\otimes {\mathsf {\boldsymbol {\delta }}}^{2}\right\rangle -\cdots

The number of Kronecker symbols is increased by two in the product of each following item when the range of tensors $x_{i}\dots x_{k}$ is reduced by two accordingly. The operation $\left\langle \cdot \right\rangle$ symmetrizes a tensor by means of summing all independent permutations of indices. Particularly, each $\delta _{ik}$ does not need to be transformed into $\delta _{ki}$ and tensors $x_{i}x_{k}$ do not become $x_{k}x_{i}$ .

These tensors are convenient to substitute into Laplace's equation:

\Delta \left\langle \otimes \mathbf {r} ^{(l-2k)}\otimes {\mathsf {\boldsymbol {\delta }}}^{k}\right\rangle =2\left\langle \otimes \mathbf {r} ^{(l-2k-2)}\otimes {\mathsf {\boldsymbol {\delta }}}^{(k+2)}\right\rangle ,\quad (\mathbf {r} \mathbf {\nabla } )\left\langle \otimes \mathbf {r} ^{(l-2k)}\otimes {\mathsf {\boldsymbol {\delta }}}^{(k)}\right\rangle =l\left\langle \otimes \mathbf {r} ^{(l-2k)}\otimes {\mathsf {\boldsymbol {\delta }}}^{(k)}\right\rangle .

The last relation is Euler's formula for homogeneous polynomials. The Laplace operator $\Delta$ does not affect the index symmetry of tensors. The two relations allow substitution of a tensor into Laplace's equation to check directly that the tensor is a harmonic function:

\Delta \mathbf {T} ^{(l)}=0.

Simplified moments[edit]

The last property is important for theoretical physics for the following reason. Potential of charges outside of their location is integral to be equal to the sum of multipole potentials:

\iiint {\frac {f({\boldsymbol {r}})}{\left||\mathbf {r-r} {_{0}}\right|}}\,dx\,dy\,dz=\sum _{l}\iiint f(\mathbf {r} )\left[\mathbf {T} ^{(l)}(\mathbf {r} )dx\,dy\,dz{\frac {\mathbf {T} ^{(l)}(\mathbf {r} _{0})}{(2l-1)!!\,l!\,r_{0}^{(2l+1)}}}\right],

where ${f}(\mathbf {r} )$ is the charge density. The convolution is applied to tensors in the formula naturally. Integrals in the sum are called in physics as multipole moments. Three of them are used actively while others applied less often as their structure (or that of spherical functions) is more complicated. Nevertheless, last property gives the way to simplify calculations in theoretical physics by using integrals with tensor ${\boldsymbol {\otimes }}r^{l}$ instead of harmonic tensor $\mathbf {T} ^{(l)}$ . Therefore, simplified moments give the same result and there is no need to restrict calculations for dipole, quadrupole and octupole potentials only. It is the advantage of the tensor point of view and not the only that.

Efimov's ladder operator[edit]

Spherical functions have a few recurrent formulas.^[24] In quantum mechanics recurrent formulas plays a role when they connect $\psi -$ functions of quantum states by means of a ladder operator. The property is occurred due to symmetry group of considered system. The vector ladder operator for the invariant harmonic states found in paper^[21] and detailed in.^[22]

For that purpose, transformation of

{\boldsymbol {\rho }}

-space is applied that conserves form of Laplace equation:

{\boldsymbol {\rho }}={\frac {\ \mathbf {r} }{r^{2}}}.

Operator ${\boldsymbol {\nabla }}$ applying to the harmonic tensor potential in ${\boldsymbol {\rho }}$ -space goes into Efimov's ladder operator acting on transformed tensor in $\mathbf {r}$ -space:

{\hat {\mathbf {D} }}=(2{\hat {l}}-1)\mathbf {r} -r^{2}{\boldsymbol {\nabla }},

where ${\hat {l}}$ is operator of module of angular momentum:

{\hat {l}}=({\boldsymbol {\nabla }}\mathbf {r} ).

Operator ${\hat {l}}$ multiplies harmonic tensor by its degree i.e. by $l$ if to recall according spherical function for quantum numbers $l$ , $m$ . To check action of the ladder operator $\mathbf {\hat {D}}$ , one can apply it to dipole and quadrupole tensors:

\mathbf {\hat {D}} _{i}x_{k}=3x_{i}x_{k}-\delta _{ik},

\mathbf {\hat {D}} _{i}x_{k}x_{n}=15x_{i}x_{k}x_{n}-3\delta _{ik}x_{n}-3\delta _{kn}x_{i}-3\delta _{ni}x_{k},

Applying successively $\mathbf {\hat {D}}$ to $\mathbf {1}$ we get general form of invariant harmonic tensors:

\mathbf {T} ^{(l)}=(\otimes \mathbf {\hat {D}} )^{l}\mathbf {1} .

The operator $\mathbf {\hat {D}}$ analogous to the oscillator ladder operator. To trace relation with a quantum operator it is useful to multiply it by $i\hbar$ to go to reversed space:

{\boldsymbol {\rho }}={\frac {\mathbf {r} }{r^{2}}}.

As a result, operator goes into the operator of momentum in ${\boldsymbol {\rho }}$ -space:

\mathbf {\hat {D}} \Rightarrow \mathbf {\hat {p}} .

It is useful to apply the following properties of $\mathbf {\hat {D}}$ .

Commutator of the coordinate operators is zero: ${\hat {D}}_{i}{\hat {D}}_{k}-{\hat {D}}_{k}{\hat {D}}_{i}=0.$

The property is utterly convenient for calculations.

The scalar operator product is zero in the space of harmonic functions: ${\hat {D}}_{i}{\hat {D}}_{i}={\hat {\mathbf {D} }}{\hat {\mathbf {D} }}=r^{4}\Delta .$

The property gives zero trace of the harmonic tensor $\mathbf {T} ^{l}$ over each two indices.

The ladder operator is analogous for that in problem of the quantum oscillator. It generates Glauber states those are created in the quantum theory of electromagnetic radiation fields.^[25] It was shown later as theoretical result that the coherent states are intrinsic for any quantum system with a group symmetry to include the rotational group.^[26]

Invariant form of spherical harmonics[edit]

Spherical harmonics accord with the system of coordinates. Let be $\mathbf {n} _{x},\mathbf {n} _{y},\mathbf {n} _{z}$ the unit vectors along axes X, Y, Z. Denote following unit vectors as $\mathbf {n} _{+}$ and $\ \mathbf {n} _{-}$ :

\mathbf {n} _{\pm }={\frac {(\mathbf {n} _{x}\pm i\mathbf {n} _{y})}{\sqrt {2}}}.

Using the vectors, the solid harmonics are equal to:

r^{l}Y_{(l\pm m)}=C_{l,m}(\mathbf {n} _{z}\mathbf {\hat {D)}} ^{(l-m)}(\mathbf {n} _{\pm }\mathbf {\hat {D)}} ^{m}\mathbf {1} =C_{l,m}\left[\mathbf {M} ^{(l)}\otimes \mathbf {n_{z}} ^{(l-m)}\otimes \mathbf {n_{\pm }} ^{m}\right]

where $C_{l,m}$ is the constant:

C_{l,m}={\frac {2^{m\setminus 2}{\sqrt {2l+1}}}{\sqrt {(l+m)!(l-m)!}}}

Angular momentum $\mathbf {\hat {L}}$ is defined by the rotational group. The mechanical momentum $\mathbf {\hat {p}}$ is related to the translation group. The ladder operator is the mapping of momentum upon inversion 1/r of 3-d space. It is raising operator. Lowering operator here is the gradient naturally together with partial contraction over pair indices $i$ to leave others:

\left[\partial x_{i}\mathbf {T} _{i}^{(l-1)}\right]=(2l+1)l\,\mathbf {T} ^{(l-1)}

https://en.wikipedia.org/wiki/Spherical_harmonics

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