These functions are a Swiss Army Knife of mathematical physics. Here is how to open the knife.
Laplace's Equation is a linear second-order differential equation that puts stringent restrictions on a function φ(x,y,z). Many physical functions obey, or satisfy, the equation, such as the electrostatic potential, the velocity potential of irrotational flow in an incompressible fluid, the gravitational potential, the displacement of an elastic membrane, and many others. The solutions of the wave equation, and of Schrödinger's equation, are very similar. A solution of Laplace's Equation is called a solid harmonic. When φ can be written as the product of a function of r, R(r), and a function of the angles θ and ψ, Θ(θ,ψ), the function Θ is called a spherical harmonic. These English names were given by Thomson and Tait, and have stuck. The German term is Kugelfunktion, sphere-function. I hope the electrostatic potential is familiar to the reader, so that it is a friendly example.
The derivative of a solid harmonic is also a solid harmonic, since the operation D of differentiation commutes with the Laplacian operator. This is a simple consequence of the fact that order of differentiation does not matter. The operator D defined in the Figure on the right is a differentiation in a certain direction, specified by the direction cosines l,m,n. These useful parameters are shown in the Figure on the left. More commonly these days, the concise vector notation for a directional derivative is used. The relation to the direction cosines should not be overlooked, since this shows explicitly what is meant, and must be used in numerical calculations in any case. A useful set of solid harmonics of order n = n1 + n2 + n3 is formed by the indicated differentiations in the axial directions (l = 1, m = 1, and n = 1, respectively). The n's must be integral, of course, but can be negative as well as positive, if D-1 is defined to be an integration.
The harmonics derived by differentiation have a clear and useful interpretation. The starting function 1/r is the potential of a point charge of unit strength at the origin. Suppose we want the potential at some point P due to a negative charge -q at distance r1, and a positive charge q at r2. This is simply q/r2 - q/r1. If the distance between the two charges is dr, then this difference is about (qdr)D(1/r), evaluated at the mean distance to the two charges (the midpoint of the line joining them). D includes the direction cosines of the line from -q to +q. As dr becomes smaller compared to r, this expression becomes more and more accurate. If q is constant, then the result approaches the uninteresting value zero as dr approaches 0. However, if we assume that qdr approaches a limit p as dr approaches 0, then the potential has the finite value pD(1/r). This result can also be expressed inv ector notation as the result of the scalar product of p and del acting upon 1/r. The constant p is called the dipole moment (the two charges are two 'poles'). The potential need not be the result of an actual limit (which is, actually, unrealistic), but simply a way to describe a particular potential. We are actually finding new solutions from old by the process of superposition, taking advantage of the linearity of Laplace's Equation.
Repeated differentiations give more complex superpositions. For example, a differentiation in one direction followed by differentiation in another superimposes the potentials of four charges, adding to zero, called a quadrupole. Since the two directions may be chosen arbitrarily, a variety of configurations is possible. If the charge is q, and the distances are a and b, we have a quadrupole moment Q = qab. In practice, what are called the quadrupole moments are defined in terms of functions of different normalizations and specific angular dependences, and our Q would be some linear combination of these conventional values. Generalizing, we have multipole moments of any order, in which the sum of the charges is zero.
For the sake of an example, suppose we want to have an expression for the potential of a point charge q that we perversely do not place at the origin, but a short distance r = (u,v,w) away, using u,v,w instead of x,y,z to avoid confusion with the coordinates of the point P. Exactly the same potential at P results from an equal charge q at the origin, together with a dipole of moment (qu, qv, qw) located at the origin. This will be a good approximation if the distance of q from the origin is small compared to the distance to P. Thus, we have φ = q/r + qr.del(1/r). This result suggests two useful extensions: first, if we have a number of charges, the potential of the assembly can be found by summing the charges q and the dipoles qr to find a total charge Q and a total dipole moment P that serve to describe the system. This is easily extended to a continuous charge distribution by integrating over the charge density ρ and the dipole density ρr. Second, we suspect that our expression may just be the first two terms in an infinite series that describes the potential exactly. Indeed, 1/r can be expanded in a Taylor's series about the origin in terms of its space derivatives, which are just the solid harmonics we have come to recognize, so the coefficients involve the multipole moments. Among the many uses of these concepts are the descriptions of the electric fields of atoms and nuclei, in which the lower moments are of most importance.
When we are presented with the problem of determining values for φ, knowing that it satisfies Laplace's Equation and certain boundary conditions, we can proceed either by using known functions, such as the harmonics, or by numerical methods. Computers have given numerical methods real power in practical problems, and an equality with analytical methods. Analytical methods can be applied only to a few problems that have symmetry and simple geometry, but they furnish us with deep insight and understanding. Numerical methods can be applied to any problem, but they give little insight and understanding, and may be limited by incorrect assumptions, roundoff error or excessive computation.
The method of differentiation, which we have used to introduce harmonic functions, is not convenient for problems with spherical symmetry, which happen to be very common. Laplace proceeded by expressing his equation in spherical polar coordinate r,θψ defined such that z = r cos θ, x = r sin θ cos ψ, and y = r sin θ sin ψ. The easiest way to do this is to use the fact that the Laplacian (del squared) of φ is the net flux of -grad φ out of a small volume, divided by the small volume. Try this in rectangular coordinates; it is easy to do, and the familiar sum of the three second derivatives is the result. When this is done for a small volume in spherical coordinates, the result is shown in the Figure. Now we assume that φ = R(r)Θ(θ,ψ), and separate the radial and angular variables. This can also be done for equations related to Laplace's, where R(r) satisfies a different equation, but Θ(θ,ψ) the same one. Θ is called a surface harmonic, since it involves only the angles.
For Laplace's Equation, R(r) = rn, where n is real, and need not be integral nor positive. This gives an equation for Θ that depends on n(n + 1). If -(n+1) is substituted for n in this expression, we recover the same value. Therefore, for any Θ depending on n, both rn and r-n-1 are possible R(r), where n is positive. The first is finite at the origin and large at large distances, the second infinite at the origin and vanishing at large distances. If Θ is to be finite on the axis at θ = 0 and π, n is restricted to integral values. It requires a good deal of mathematics to prove this, so we shall simply accept it.
Θ is a product of a function of θ called the associated Legendre polynomial, and a function of ψ that is the familar exponential. Single-valuedness restricts m to an integer value, positive or negative. For m = 0, the θ function is the Legendre polynomial in cos θ This is not a mysterious function, just a polynomial. The first four are given in the Figure on the right, and will do for most applications. The harmonics for m = 0 consist of these polynomials alone, and are called zonal harmonics, because the circles Pn = 0 divide the sphere into zones. At the other end of the series of values of m, m = n and m = -n make the θ function simply (1 - cos2θ)m/2, which is zero only at θ = 0 and π. The zeros are only the zeros of the ψ function, A cos nψ + B sin nψ, which are meridians. These harmonics are called sectoral harmonics. All the harmonics in between have both circles of latitude and meridians as zeros, and are called tesseral harmonics, since they seem to divide the spherical surface into tesserae. Examples are shown below. The harmonic has opposite signs in regions of different colour. The boundaries of the regions are the nodal lines.
Although any spherical harmonics will have the same general behaviour and form, they may be defined with slightly different constants, which can make the comparison of references difficult. This is really only of great concern in quantum mechanics, when the phases of the different harmonics make a difference, so care must be taken with definitions in that case. Although the functions may look complicated, they are really quite simple, and you should satisfy yourself that you can get explicit forms for small values of n. There are second solutions, as in the case of Bessel functions, that are not finite at the poles.
For each value of n, there is a family of 2n + 1 funtions. In the case of rectangular coordinates, for each n there was a family member for each of the ways of expressing n as the sum of three integers. For example, for n = 2 we have 2,0,0; 0,2,0; 0,0,2; 1,1,0; 1,0,1; 0,1,1--six possibilities in all. Now for n = 2 we have only 5 functions. The spherical representation is in a certain way simpler, for no linear combinations of its family functions reduces to a function of lower order. In fact, the spherical families are irreducible representations of the rotation group, and this give them a certain brilliance. Step-up and step-down operators can be defined that increase or decrease the value of m, going from one member to another, and giving zero when operating on the last one. All this is extensively developed in the theory of angular momentum in quantum mechanics, where spherical harmonics play a major role. As an example, for n = 1, there are three functions in either family, so they must be linear combinations of each other. This is easily verified by writing them out explicitly. We find z/r, x/r and y/r in one case, and cos θ, sin θeiψ and sin θe-iψ in the other. In quantum mechanics, the latter correspond to quantum numbers L = 1 and M = 0, 1 and -1. The nodal lines for the real-valued harmonics, with sin mψ and cos nψ, are shown in the Figure below for n = 0 to 3. A top view and a side view are shown for each harmonic, and the x-axis points outwards from the centre of the side view. Note that a harmonic of order n has n nodal lines, m meridional and n - m horizontal.
As an example of how this diagram can be used, suppose you wanted to express the gravitational potential of the earth as a function of distance and angle. If you assume axial symmetry, a series of zonal harmonics would be appropriate. The largest term would have n = 0, of course, and vary as 1/r. The polar flattening of the earth would be reflected in a term that added mass near the equator, and subtracted it near the poles. Examining the diagram, we see that a term with n = 2 and m = 0 is appropriate, with its radial dependence of 1/r3, and angular dependence of P2(sin φ), where now φ is the latitude, 90° - θ. This is the lowest-order term. A series of higher ones (n = 4, 6, ...) can be made to fit any spheroidal distribution of mass. We now know something about this effect simply by recognizing the appropriate spherical harmonics. The constant is given in astronomical reference works. As an exercise for the reader, suppose you wanted to represent a pear-shaped distortion of the earth, with more mass in the northern hemisphere and less in the southern. Which harmonic would be appropriate in this case?
The most useful mathematical property of the spherical harmonics is their orthogonality 'over the sphere,' that is, integrated over all angles. Each of the Legendre polynomials is orthogonal to all the others when integrated over d cos θ from θ = 0 to π, or, as P(x), from x = -1 to x = +1. The associated functions of the same m but different n are also orthogonal. When combined with the orthogonality of the sine and cosine, every spherical harmonic is orthogonal to every other. The square of a Legendre polynomial integrated over the same interval is 2/2n + 1. The square of an associated Legendre polynomial integrates to 2(n + s)!/(2n + 1)(n - s)!. These facts allow one to expand any function of θ,ψ in a Fourier series of spherical harmonics, provided one can do the integrals.
We can now reveal an astonishing result. Suppose we have a problem with axial symmetry, so that only m = 0 terms are necessary in φ. Then the general form of φ is the sum of terms Pn(cos θ)( Anrn + Bn/rn+1), where An and Bn are constants to be determined. Along the axis, θ = 0, cos θ = 1, and Pn(1) = 1. For this special case, φ = Σ (Anzn + Bn/zn+1). We may be able to find a power series for φ in this special case quite easily. For example, suppose we have charge q at z = a/2, and -q at z = -a/2. Then, φ = qa/z2, so, to lowest order, An = 0, Bn = 0 except B1 = qa. We could get higher-order terms easily. But now we have found the potential not just on the axis, but everywhere, since the same constants are in the original series. For our approximation, we have φ = qa cos θ/r2, which is the dipole potential for a dipole directed along the z-axis. The values of φ on the axis determine its values everywhere! This is the pervasive influence of Laplace's Equation, analogous to the connections inherent in an analytic complex function.
H. Lamb, Hydrodynamics 6th ed. (Cambridge: Cambridge Univ. Press, 1953), pp. 110-120.
J. D. Jackson, Classical Electrodynamics, 2nd ed. (New York: John Wiley, 1985). Care is taken so the phases are those used in quantum mechanics. This is the best modern reference on the use of spherical harmonics in physics, by far.
Composed by J. B. Calvert
Created 6 July 2000
Last revised 12 March 2001