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## Factorization Method for Solving a Partial Differential Equation: Spherical Harmonics

Now back to the remaining eigenvalues, . The radial equation can be changed into a familiar one by letting

This results in

which is the familiar Bessel equation of order . Its solutions are and , where is to be determined.

Lecture 48

The value of is not arbitrary. It is one of the (degenerate) eigenvalues of Eq.(5.74), the two-dimensional Helmholtz equation on the unit two-sphere. As already observed on page , for this equation reduces to Helmholtz's equation on the Euclidean plane. This observation is very useful for several reasons. One of them is that it implies, as shown on page , that there is a simple algebraic way of generating a complete basis for each degenerate eigenspace of

We shall now extend this algebraic method from the eigenfunctions of on the two-dimensional Euclidean plane to those of on the two-dimensional surface of a unit sphere.

The factorization method of solving a partial (or ordinary) differential equation is remarkable. This method differs from a power series or a numerical approach in that one solves a calculus problem without the use of calculus: one obtains the linear algebraic aspects of the problem (eigenvalues, all normalized eigenvectors, their properties, etc.) in one fell swoop without ever having to determine explicitly the detailed functional form ( i.e. local behaviour) of the solutions. To be sure, one can readily determine and exhibit these solutions in explicit form in terms of Legendre and associated Legendre polynomials, and we shall do so. However, this is a straight forward, but secondary, calculus exercise which is an implied but not an integral part of the algebraic approach.

Important Reminder: Throughout the ensuing discussion an eigenfunction on the unit sphere refers to a function which is square-integrable, i.e.

 (576)

One will see that the very existence of the eigenvalue spectrum of on the unit sphere hinges on this fact. For this reason, the extension of this algebraic method is considerably more powerful. It yields not only the basis for each eigenspace of , but also the actual value for each allowed degenerate eigenvalue.

Global Analysis: Algebra

Global analysis deals with the solutions of a differential equation wholesale''. It characterizes them in relationship to one another without specifying their individual behaviour on their domain of definition. Thus one focusses via algebra, linear or otherwise, on the space of solutions'', its subspaces, bases etc.

Local analysis (next subsubsection), by contrast, deals with the solutions of a differential equation retail''. Using differential calculus, numerical analysis, one zooms in on individual functions and characterizes them by their local values, slopes, location of zeroes, etc.

1. Factorization

The algebraic method depends on factoring

into a pair of first order operators which are adjoints of each other. The method is analogous to factoring a quadratic polynomial, except that here one has differential operators and instead of the variables and . Taking our cue from Properties 16 and 17, one attempts

However, one immediately finds that this factorization yields for a cross term. This is incorrect. What one needs instead is . This leads us to consider
 (577)

Here we have introduced the self-adjoint operator

It generates rotations around the polar axis of a sphere. This operator, together with the two mutually adjoint operators

are of fundamental importance to the factorization method of solving the given differential equation. In terms of them the factorized Eq.(5.77) and its complex conjugate have the form

 (578)

This differs from Eq.(5.16), (Property 17 on page ), the factored Laplacian on the Euclidean plane.

2. Fundamental Relations

In spite of this difference, the commutation relations corresponding to Eqs.(5.18), (5.19), and (5.20) are all the same, except one. Thus, instead of Eq.(5.19), for a sphere one has

 (579)

This is obtained by subtracting the two Eqs.(5.78). However, the commutation relations corresponding to the other two equations remain the same. Indeed, a little algebraic computation yields

or

 (580)

Furthermore, using Eq.(5.78) one finds
 (581)

The last equality was obtained with the help of Eqs.(5.79) and (5.80). Together with the complex conjugate of this equation, one has therefore

 (582)

In addition, one has quite trivially

 (583)

The three algebraic relations, Eqs.(5.79)-(5.80) and their consequences, Eq.(5.82)-(5.83), are the fundamental equations from which one deduces the allowed degenerate eigenvalues of Eq.(5.74) as well as the corresponding normalized eigenfunctions.

3. The Eigenfunctions

One starts by considering a function which is a simultaneous solution to the two eigenvalue equations

This is a consistent system, and it is best to postpone until later the easy task of actually exhibiting non-zero solutions to it. First we deduce three properties of any given solution .

The first property is obtained by applying the operator to this solution. One finds that

Similarly one finds

Thus and are again eigenfunctions of , but having eigenvalues and . One is, therefore, justified in calling and raising and lowering operators. The raised'' and lowered'' functions have the additional property that they are still eigenfunctions of belonging to the same eigenvalue . Indeed, with the help of Eq.(5.82) one finds

Thus, if belongs to the eigenspace of , then so do and .

4. Normalization and the Eigenvalues

The second and third properties concern the normalization of and the allowed values of . One obtains them by examining the sequence of squared norms of the sequence of eigenfunctions

All of them are square-integrable. Hence their norms are non-negative. In particular, for one has
 (584)

This is the second property. It is a powerful result for two reasons:

First of all, if has been normalized to unity, then so will be

 (585)

This means that once the normalization integral has been worked out for any one of the 's, the already normalized are given by Eq.(5.85); no additional normalization integrals need to be evaluated. By repeatedly applying the operator one can extend this result to , , etc. They all are already normalized if is. No extra work is necessary.

Secondly, repeated use of the relation (5.84) yields

This relation implies that for sufficiently large integer the leading factor in square brackets must vanish. If it did not, the squared norm of would become negative. To prevent this from happening, must have very special values. This is the third property: The only allowed values of are necessarily

(Note that would give nothing new.) Any other value for would yield a contradiction, namely a negative norm for some integer . As a consequence, one has the result that for each allowed eigenvalue there is a sequence of eigenfunctions

 (586)

(Nota bene: Note that these eigenfunctions are now labelled by the non-negative integer instead of the corresponding eigenvalue .) Of particular interest are the two eigenfunctions and . The squared norm of ,

is not positive. It vanishes. This implies that

 (587)

In other words, and all subsequent members of the above sequence, Eq.(5.86) vanish, i.e. they do not exist. Similarly one finds that

 (588)

Thus members of the sequence below do not exist either. It follows that the sequence of eigenfunctions corresponding to is finite. The sequence has only members, namely

for each integer . The union of these sequences forms a semi-infinite lattice in the as shown in Figure 5.20.

For obvious reasons it is appropriate to refer to this sequence as a ladder with elements, and to call the top, and the bottom of the ladder. The raising and lowering operators are the ladder operators which take us up and down the -element ladder. It is easy to determine the elements at the top and the bottom, and to use the ladder operators to generate any element in between.

5. Orthonormality and Completeness

The operators form a complete set of commuting operators. This means that their eigenvalues serve as sufficient labels to uniquely identify each of their (common) eigenbasis elements for the vector space of solutions to the Hermholtz equation

on the two-sphere. No additional labels are necessary. The fact that these operators are self-adjoint relative to the inner product, Eq.(5.76), implies that these eigenvectors (a.k.a spherical harmonics) are orthonormal:

The semi-infinite set is a basis for the vector space of functions square-integrable on the unit two-sphere. Let be any such function. Then

In other words, the spherical harmonics are the basis elements for a generalized double Fourier series representation of the function . If one leaves this function unspecified, then this completeness relation can be restated in the equivalent form

in terms of the Dirac delta functions on the compact domains and .

Local Analysis: Calculus

What is the formula for a harmonics ? An explicit functional form determines the graph, the location of its zeroes, and other aspects of its local behaviour.

1. Spherical Harmonics: Top and Bottom of the Ladder

Each member of the ladder sequence satisfies the differential equation

Consequently, all eigenfunctions have the form

 (589)

Here is a normalization factor. The two eigenfunctions and at the top and the bottom of the ladder satisfy Eqs.(5.87) and (5.88) respectively, namely

 (590)

It is easy to see that their solutions are

The normalization condition

implies that

 (591)

The phase factor is not determined by the normalization. Its form is chosen so as to simplify the to-be-derived formula for the Legendre polynomials, Eq.(5.95).

2. Spherical harmonics: Legendre and Associated Legendre polynomials

The functions are obtained by applying the lowering operator to . A systematic way of doing this is first to apply repeatedly the lowering relation

 (592)

to until one obtains the azimuthally invariant harmonic . Then continue applying this lowering relation, or alternatively the raising relation
 (593)

until one obtains the desired harmionic . The execution of this two-step algorithm reads as follows:

Step 1: Letting , apply Eq.(5.92) times and obtain

which, because of Eq.(5.89), is independent of . Now let , use Eq.(5.91), and obtain
 (594)

The polynomials in the variable

 (595)

are called the Legendre polynomials. They have the property that at the North pole they have the common value unity, while at the South pole their value is whenever is an even polynomial and whenever it is odd:

Step 2: To obtain the harmonics having positive azimuthal integer , apply the raising operator times to . With the help of Eq.(5.93) one obtains (for )
 (596)

The polynomials in the variable

 (597)

are called the associated Legendre polynomials. Inserting Eq.(5.96) into Eq.(5.74), one finds that they satisfy the differential equation

Also note that satisfies the same differential equation. In other words, and must be proportional to each other. (Why?) Indeed,

 (598)

This one sees by comparing the right hand side of Eq.(5.96) with the right hand side of
 (599)

This validates Eq.(5.98), but only for . One sees, however, that this formula is also true for . Could it be that formulas Eqs.(5.96) and (5.97) are also true whenever ? The answer is `yes'. This follows from considering the harmonics. They are obtained by using Eq.(5.92) times starting with :

 (5100) (5101)

(Nota bene: The first line was obtained by using Eq.(5.92) and letting , the third, fourth, and fifth line used Eqs.(5.94), (5.97), and (5.98), respectively. ) Comparison with Eq.(5.96) verifies that the spherical harmonic

 (5102)

is indeed correct for all positive and negative integers that satisfy .

A second result is obtained by comparing Eq.(5.100) with Eq.(5.96). This comparison yields the complex conjugation formula

which holds for both for positive and negative azimuthal integers .

Next: Static Solutions Up: Wave Equation for Spherically Previous: Spherically Symmetric Solutions   Contents   Index
Ulrich Gerlach 2010-12-09