Spherical Harmonics - Higher Dimensions

Higher Dimensions

The classical spherical harmonics are defined as functions on the unit sphere S2 inside three-dimensional Euclidean space. Spherical harmonics can be generalized to higher dimensional Euclidean space Rn as follows. Let P denote the space of homogeneous polynomials of degree ℓ in n variables. That is, a polynomial P is in P provided that

Let A denote the subspace of P consisting of all harmonic polynomials; these are the solid spherical harmonics. Let H denote the space of functions on the unit sphere

obtained by restriction from A.

The following properties hold:

  • The spaces H are dense in the set of continuous functions on Sn−1 with respect to the uniform topology, by the Stone-Weierstrass theorem. As a result, they are also dense in the space L2(Sn−1) of square-integrable functions on the sphere.
  • For all ƒH, one has
where ΔSn−1 is the Laplace–Beltrami operator on Sn−1. This operator is the analog of the angular part of the Laplacian in three dimensions; to wit, the Laplacian in n dimensions decomposes as
  • It follows from the Stokes theorem and the preceding property that the spaces H are orthogonal with respect to the inner product from L2(Sn−1). That is to say,
for ƒH and gHk for k ≠ ℓ.
  • Conversely, the spaces H are precisely the eigenspaces of ΔSn−1. In particular, an application of the spectral theorem to the Riesz potential gives another proof that the spaces H are pairwise orthogonal and complete in L2(Sn−1).
  • Every homogeneous polynomial PP can be uniquely written in the form
P(x) = P_\ell(x) + |x|^2P_{\ell-2} + \cdots + \begin{cases}
|x|^\ell P_0 & \ell \rm{\ even}\\
|x|^{\ell-1} P_1(x) & \ell\rm{\ odd}
\end{cases}
where PjAj. In particular,

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

where φ is the axial coordinate in a spherical coordinate system on Sn−1. The end result of such a procedure is


Y_{l_1, \dots l_{n-1}} (\theta_1, \dots \theta_{n-1}) = \frac{1}{\sqrt{2\pi}} e^{i l_1 \theta_1} \prod_{j = 2}^{n-1} {}_j \bar{P}^{l_{n-1} - 1}_{l_j} (\theta_j)

where the indices satisfy |ℓ1| ≤ ℓ2 ≤ ... ≤ ℓn−1 and the eigenvalue is −ℓn−1(ℓn−1 + n−2). The functions in the product are defined in terms of the Legendre function


{}_j \bar{P}^l_{L} (\theta) = \sqrt{\frac{2L+j-1}{2} \frac{(L+l+j-2)!}{(L-l)!}} \sin^{\frac{2-j}{2}} (\theta) P^{-(l + \frac{j-2}{2})}_{L+\frac{j-2}{2}} (\cos \theta)

Read more about this topic:  Spherical Harmonics

Famous quotes containing the words higher and/or dimensions:

    The foot of the heavenly ladder, which we have got to mount in order to reach the higher regions, has to be fixed firmly in every-day life, so that everybody may be able to climb up it along with us. When people then find that they have got climbed up higher and higher into a marvelous, magical world, they will feel that that realm, too, belongs to their ordinary, every-day life, and is, merely, the wonderful and most glorious part thereof.
    —E.T.A.W. (Ernst Theodor Amadeus Wilhelm)

    The truth is that a Pigmy and a Patagonian, a Mouse and a Mammoth, derive their dimensions from the same nutritive juices.... [A]ll the manna of heaven would never raise the Mouse to the bulk of the Mammoth.
    Thomas Jefferson (1743–1826)