Homogeneous harmonic polynomials

Physicists are familiar with many special functions that arise over and over again in solutions to various problems. The analysis of problems with spherical symmetry in P often appeal to the spherical harmonic functions, often called simply spherical harmonics. Spherical harmonics are the restrictions of homogeneous harmonic polynomials of three variables to the sphere S. In this section we will give a typical physics-style introduction to spherical harmonics. Here we state, but do not prove, their relationship to homogeneous harmonic polynomials a formal statement and proof are given Proposition A. 2 of Appendix A. 

Relating the spherical harmonic functions introduced here to the homogeneous harmonic polynomials is not logically necessary in this book. Morally, however, the calculation is well worth doing, in the name of better communication between mathematics and physics. Because this calculation is a bit tricky, we have postponed it to Appendix A. 

Restrictions of homogeneous harmonic polynomials play an important role in our analysis. 

Definition 2.6 Suppose f is a nonnegative integer. Define the vector space of homogeneous harmonic polynomials of degree f in three variables... 

In Section 7.1 we will use this characterization of homogeneous harmonic polynomials as a kernel of a linear transformation (along with the Fundamental Theorem of Linear Algebra, Proposition 2.5) to calculate the dimensions of the spaces of the spherical harmonics. 

All of the concepts of this section — kernel, image. Fundamental Theorem, homogeneous harmonic polynomials and isomorphisms — come up repeatedly in the rest of the text. 

Proof. Consider any function y e. By Definition 2,6, there is a homogeneous harmonic polynomial p of degree f such that y = PI52. Now rotating a polynomial preserves its degree (by Exercise 4.14), and the Laplacian is invariant imder rotation (by Exercise 3.11). So for any g e SO(3 ) the function g p is a homogeneous harmonic polynomial of degree . Hence g-y = g s2-p is an element of 3. ... 

In this proposition, as in Proposition 6.14, it is possible to drop the hypothesis that the representation be unitary. See Exercise 6.13. We will apply this classification of irreducibles of 50(3) in Section 7.1 to show that for each nonnegative integer n the set of homogeneous harmonic polynomials of degree n forms an irreducible representation of 50(3). 

Combining this last result with our knowledge of the classification of the irreducible representations of the group 50(3), we can show that the representation of the rotation group on homogeneous harmonic polynomials of any fixed degree is irreducible. 

Recall the vector space y of spherical harmonics from Definition 2.6 y is the set of restrictions to of homogeneous harmonic polynomials. Recall also the definition of spanning (Defiiution 3.7). The set y of spherical harmoiucs spans... 

In four dimensions, as in three dimensions, the restrictions of homogeneous harmonic polynomials of degree n to the unit sphere are called spherical harmonic functions of degree n. The analysis in four dimensions proceeds much as it did in three dimensions, although the dimension counts change. 

Anticipating Fock s notation, let us call our variables Xi, 3 and %<. An example of a homogeneous harmonic polynomial of degree four is... 

One can rotate this picture to give the other two easy harmonic polynomials of degree 2, yz and xz. But this is not the whole story because the dimension of the space of homogeneous polynomials of degree two is 6 (given by combinations of xy, xz, yz, xx, yy, zz) and we only lose one dimension by setting... 

Such a homogeneous polynomial is said to be harmonic. We would like to resolve / into a series of harmonic polynomials of the form... 

Recall from Section 1.5 that any function in the kernel of the Laplacian (on any space of functions) is called a harmonic function. In other words, a function f is harmonic if V / = 0. The harmonic functions in the example just above are the harmonic homogeneous polynomials of degree two. We call this vector space In Exercise 2,23 we invite the reader to check that the following set is a basis of H/ ... 

If we pass to polynomials of degree 2, we have to be careful. They do not always look homogeneous when we finish setting x2 + y2 + z2 = 1. Let us look at z2, for example, which is not harmonic and so should not be an eigenfunction of A. Passing to spherical coordinates, and setting r = 1, this gives the function... 

It is often of interest to perform an expansion of the type of Eq. (65). Hobson has developed a projection operator which extracts (projects) the harmonic hi out of the homogenous polynomial /j. This operator S is an irreducible tensorial operator of the degree zero for the three-dimensional rotation group and has the form [Ref. [18) p. 127)]... 

It is sometimes necessary to obtain the content of a harmonic of the degree / in a homogeneous polynomial/j+2n of the degree I 2 n. In this case one has to annihilate the terms of higher degree than I by operating on the polynomial fi+zn with the totally symmetric operator F2 . This can be done because... 

Solid harmonics (r) are homogeneous polynomials of degree / in the coordinates... 

Here y m are the spherical harmonic functions Q m = yj47r/(2k + 1) y, m is the Racah tensor operator = rk Ykm is the irreducible tensor operator Pk m (not to be confused with the Legendre polynomials) are unnormalised homogeneous polynomials of Cartesian coordinates proportional to the function rk Ykm + Yk m) Ok are referred to as equivalent operators which are constructed of only the angular momentum operators. 

The electric field in the axially homogeneous colimm plasma consists of a superposition of the constmit axial electric field and of the radially varying radial space charge field, i.e., E r) = E rjer + E e. Thus, the direction of the total electric field E r) is different from the radial direction in which the inhomogeneity of the plasma column occurs. Therefore, the expansion of the velocity distribution in Legendre polynomials can no longer be used and has to be replaced by an expansion in spherical harmonics (Uhrlandt and Winkler, 1996). 

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