7. Ch. 3, part 1

Lecture 7 - Legendre Functions, Spherical Harmonics

The Laplace equation can be separated in several coordinate systems, the most important of which are: Cartesian x, y, z, circular cylindrical ρ, θ, z and spherical polar r, θ, ϕ. Confocal elliptical coordinate systems are also useful, but are not in Jackson’s book. In all these systems the one-dimensional equations to be solved are similar to the familiar

\frac{d^{2}f}{dx^{2}} + k^{2}f = 0  (7.1)

but usually more complicated. For instance, in polar coordinates we write a solution of Laplace’s equation as Q(ϕ)P (θ)U(r)/r and we find that the equation in the azimuthal angle ϕ is the simple one

\frac{d^{2}Q}{dϕ^{2}} + m^{2}Q = 0  (7.2)

and the radial equation is almost as simple:

\frac{d^{2}U}{dr^{2}} − λ^{2} \frac{U}{r^{2}} = 0

but the equation in the polar angle θ is more complicated. If there is no ϕ dependence, i.e. if m = 0, it reduces to

\frac{1}{sin θ} \frac{d}{dθ} (sin θ \frac{dP}{dθ} ) + λ^{2}P = 0  (7.3)

The “separation constant” λ^{2} appears as an eigenvalue in the one-dimensional equations, and is to be determined with the help of the boundary conditions. The solutions of this and similar eigenvalue problems share many of the properties of the well-known solutions of the simple equation (7.1), namely:

• A discrete set of eigenvalues

• Series expansions in the appropriate variables

• Recursion relations

• Existence of a generating function

These properties can be used to compute

• The functions and their derivatives

• Normalization integrals

• Three-function integrals, and others

7.1. Azimuthal eigenfunctions

All the above properties are quite trivial in the case of the eigenfunctions of eq. (7.2) with the boundary condition of periodicity in (0, 2π). The (unnormalized) solutions are

U_{m}(ϕ) = exp(imϕ)

• The eigenvalues are m^{2} with m integer

• The series expansion in ϕ is

U_{m}(ϕ) = ∑_{n=0}^{∞} \frac{(imϕ)^{n}}{n!}

but it is more convenient to use the variable X = e^{iϕ}, in terms of which we simply have

U_{m}(X) = X^{m}

• There is a very simple one-step recursion relation

U_{m}(X) = XU_{m−1}(X)

and the derivative is also given by a simple step-down relation

dU_{m}/dX = mU_{m−1}

• The series expansion

\frac{1}{1 − aX} = ∑_{m=0}^{∞} a^{m}X^{m} = ∑_{m=0}^{∞} a^{m}U_{m}(X)

shows what is meant by a “generating function”

As for the use of these relations: if we take U_{0} = 1,we can trivially iterate U_{m}(X) = XU_{m−1}(X) to get U_{m}(X) = X^{m}, and the derivative is explicitly given. Normalization is trivial because ∫ |U_{m}|^{2} dϕ = ∫ dϕ = 2π. The three-function integrals

∫ U_{l}U_{m}U_{n}dϕ

are also computed directly in this case. They vanish unless l + m + n = 0, and have then the value 2π.

Remark

This case is overly simple, because the eigenfunctions obey in effect a first-order differential equation. More typical is the case of eq.(7.1) with zero boundary conditions at x = ±a/2 (particle in a box). If we put ϕ = πx/a, this becomes eq. (7.2) for ϕ in (−π/2, π/2). Now the eigenfunctions are U_{m} = cos(mϕ) with positive half-integer m. For any eigenvalue m^{2} there is a “second solution” sin(mϕ) that does not satisfy the boundary conditions. The recursion relations are now two-step, or involve derivatives. For instance

cos(m + 1)ϕ = cos mϕ cos ϕ − sin mϕ sin ϕ
= cos mϕ cos ϕ − \frac{sin^{2} ϕ}{m} \frac{d}{d cos ϕ} cos mϕ

The expansion of cos mϕ in the variable X = cos ϕ is just a polynomial of degree m, the standard Chebyshev polynomial T_{m}(X) (Arfken, page 741). This is just mentioned here as an example to make more palatable the Legendre functions (coming up).

7.2. Radial eigenfunctions

The solutions of the equation

\frac{d^{2}U}{dr^{2}} − λ^{2} \frac{U}{r^{2}} = 0

are manifestly of two types: r^{−l}, which for l > 0 is well behaved at ∞ but not at the origin, and r^{l+1}, which is well-behaved at the origin but not at ∞ . In either case the eigenvalue is λ^{2} = l(l + 1).

7.3. Polar eigenfunctions

7.3.1. Legendre functions

The differential equation in θ is conveniently studied in the variable x = cos θ, in the interval (-1,1). For m = 0 it is the Legendre equation:

\frac{d}{dx} [(1 − x^{2}) \frac{dP}{dx} ] + l (l + 1) P = 0

• The usual boundary conditions are that P should remain finite at the end points x = 1 and x = −1 (corresponding to θ = 0 and θ = π). They can be satisfied only if l is a positive integer or zero.

• The coefficients a_{l} of the series solution obey the recursion relation

a_{j+2} = \frac{j(j + 1) − l(l + 1)}{(j + 1) (j + 2)} a_{j}

• There are two solutions, one even in x, one odd.

– If l is an even integer or zero, the even solution reduces to the Legendre polynomial of order l and is well-behaved at x = ±1. The “other solution” is not well behaved.

– If l is an odd integer, the odd solution reduces to the Legendre polynomial of order l and is well-behaved at x = ±1. The “other solution” is not well behaved.

• The first few Legendre polynomials are

P_{0}(x) = 1
P_{1}(x) = x

P_{2}(x) = \frac{1}{2} (3x^{2} − 1)

P_{3}(x) = \frac{1}{2} (5x^{3} − 3x)

P_{4}(x) = \frac{1}{8} (35x^{4} − 30x^{2} + 3)

• To prove some properties of the P_{l} (especially to evaluate integrals) it is useful to know the Rodrigues formula

P_{l}(x) = \frac{1}{2^{l}l!} \frac{d^{l}}{dx^{l}} (x^{2} − 1)^{l}

• Even more useful, especially in electrostatics, is the expansion (generating function), valid for |a| < 1,

\frac{1}{\sqrt{1 + a^{2} − 2ax}} = ∑_{l=0}^{∞} a^{l}P_{l}(x)  (7.4)

If we go back to the variable θ and also put a = r′/r, this becomes

\frac{r}{\sqrt{r^{2} + r^{′2} − 2rr′ cos θ}} = ∑_{l=0}^{∞} (\frac{r′}{r} )^{l} P_{l}(cos θ)

or

\frac{1}{|x − x′|} = \frac{1}{r} ∑_{l=0}^{∞} (\frac{r′}{r} )^{l} P_{l}(cos θ)

We recognize an expansion of the Coulomb potential, valid for r′ < r.

• There are many recursion relations. The most useful are probably Lecture 7 (Ch 3 part 1) Legendre functions, spherical harmonics 6

– The step-up and step-down recursions, involving the derivative

(l + 1) P_{l+1} = (l + 1) xP_{l} + (x^{2} − 1)Pl^{′}

lP_{l−1} = lxP_{l} − (x^{2} − 1)Pl^{′}

– The two-step recursion

(l + 1) P_{l+1} = (2l + 1) xP_{l} − lP_{l−1}

which is the most efficient numerical way to generate the P_{l} • The orthogonality and normalization integrals are

∫_{−1}^{1} P_{l}(x)P_{l′}(x) dx = \frac{2}{2l + 1} δ_{ll′}

• The integrals

∫_{−1}^{1} P_{l}(x)P_{m}(x)P_{n}(x) dx

are also useful, especially in atomic and nuclear physics. It can be shown that they vanish unless

– l + m + n is even

– l, m, n satisfy “triangle” inequalities, i.e. |m − n| ≤ l ≤ m + n We probably need them only for n = 1 and n = 2. See (J3.31-32).

• By writing x^{2} − 1 = (x + 1) (x − 1) in the Rodrigues formula it is easy to see that

P_{l}(1) = 1

P_{l}(−1) = (−1)^{l}

One also can find, using the binomial expansion of (1 − x^{2})^{l} , that P_{l}(0) vanishes for l odd (obviously, by parity) and that for even l

P_{l}(0) = (−1)^{l} \frac{(2l + 1)!!}{2^{l}l!}

7.3.2. Linear multipoles

We now run some checks of the above relations, starting from the generating function expansion (7.4), and describe the physical meaning of some of the quantities.

We have already seen that for r′ < r

\frac{1}{|x − x′|} = \frac{1}{r} ∑_{l=0}^{∞} (\frac{r′}{r} )^{l} P_{l}(cos θ)

Similarly, by putting a = r/r′ in (7.4), we obtain for r < r′

\frac{1}{|x − x′|} = \frac{1}{r′} ∑_{l=0}^{∞} (\frac{r}{r′} )^{l} P_{l}(cos θ)

For r = r′ both these expansions converge, except at θ = 0, which of course corresponds to x = x′. We note that, where the series converges, both sides of these relations satisfy Laplace’s equation, because

\frac{1}{r^{l+1}} P_{l}(cos θ) and r^{l}P_{l}(cos θ)

are the basic separable solutions when there is no dependence on ϕ.

Generally, a ϕ-independent solution of Laplace’s equation in the whole region exterior to the sphere r = r′ has the Legendre expansion

Φ(x) =\frac{1}{r} ∑_{l=0}^{∞} C_{l} (\frac{r′}{r} )^{l} P_{l}(cos θ)

and the coefficients C_{l} can be found by equating the two sides for θ = 0, where P_{l}(1) = 1, or for some other convenient θ. This is a very useful trick. When Φ(x) =1/ |x − x′| one quickly finds C_{l} = 1 by this method and one can also turn the argument around to obtain the values of P_{l}(−1) and P_{l}(0) already quoted.

The explicit expressions of P_{l}(cos θ)/r^{l+1} for small l are

\frac{1}{r}

\frac{cos θ}{r^{2}} = \frac{z}{r^{3}}
\frac{1}{2r^{3}} (3 cos^{2} θ − 1) = \frac{1}{2r^{5}} (3z^{2} − r^{2})

We recognize the potential of a dipole oriented along the z axis and of a quadrupole, and we also note that

\frac{z}{r^{3}} = −\frac{∂}{∂z} \frac{1}{r}
\frac{1}{r^{5}} (3z^{2} − r^{2}) = −\frac{∂}{∂z} \frac{z}{r^{3}}

We can show that in general

−\frac{∂}{∂z} [\frac{1}{r^{l+1}} P_{l}(cos θ)] = \frac{l + 1}{r^{l+2}}P_{l+1}(cos θ)  (7.5)

so that the successive application of −∂/∂z (at constant x and y) generates the fields of linear multipoles consisting of charges aligned along z. As already stated, these are the basic fields for an exterior problem.

Similarly for an interior problem the basic fields are r^{l}P_{l}(cos θ). Their explicit expressions for small l are

1

r cos θ = z
\frac{1}{2}r^{2} (3 cos^{2} θ − 1) = \frac{1}{2} (3z^{2} − r^{2})

and we have the “step-down” relations

\frac{∂}{∂z} [r^{l}P_{l}(cos θ)] = lr^{l−1}P_{l−1}(cos θ)  (7.6)

7.3.3. Associated Legendre functions

For any m, the differential equation in θ is best studied using the variable x = cos θ, in the interval (−1, 1). It is the associated Legendre equation:

\frac{d}{dx} [(1 − x^{2}) \frac{dP}{dx} ] + l (l + 1) P − \frac{m^{2}}{1 − x^{2}} P = 0

We consider only the case of integer m, and we can take m ≥ 0. Then:

• The usual boundary conditions are that P should remain finite at the end points x = 1 and x = −1 (corresponding to θ = 0 and θ = π). They can be satisfied only if l is a positive integer or zero, and m ≤ l

• The solutions can be written as power series, but it is easier to derive them from the Legendre functions P_{l}(x) by the formula

P_{l}^{m}(x) = (−1)^{m} (1 − x^{2})^{m/2} \frac{d^{m}}{dx^{m}} P_{l}(x)

• For any l, m pair there are two solutions, one even in x, one odd. If l is an integer, one solution is a polynomial of order l − m in x = cos θ times the factor sin^{m} θ. This solution, which has parity l − m, is well-behaved at x = ±1. The “other solution” is not well behaved.

• From the Rodriguez formula for P_{l} one obtains at once the Rodrigues formula for P_{l}^{m}

P_{l}^{m}(x) = \frac{(−1)^{m}}{2^{l}l!} (1 − x^{2})^{m/2} \frac{d^{l+m}}{dx^{l+m}} (x^{2} − 1)^{l}

• Clearly P_{l}^{0}(x) = P_{l}(x). Also, from the Rodrigues formula:

P_{l}^{l}(cos θ) = (−1)^{l} \frac{(2l)!}{2^{l} l!} sin^{l} θ

P_{l}^{l−1}(cos θ) = (−1)^{l−1} \frac{(2l)!}{2^{l} l!} sin^{l} θ cos θ = −\frac{d}{dθ}P_{l}^{l}(cos θ)

• The first few associated Legendre functions with m > 0 are

P_{1}^{1}(x) = − (1 − x^{2})^{1/2} = − sin θ
P_{2}^{1}(x) = 3x (1 − x^{2})^{1/2} = −3 cos θ sin θ
P_{2}^{2}(x) = 3 (1 − x^{2}) = 3 sin^{2} θ

P_{3}^{1}(x) = −\frac{3}{2} (5x^{2} − 1) (1 − x^{2})^{1/2} = −\frac{3}{2} (5 cos^{2} θ − 1) sin θ

P_{3}^{2}(x) = 15x (1 − x^{2}) = 15 cos θ sin^{2} θ
P_{3}^{3}(x) = −15 (1 − x^{2})^{3/2} = −15 sin^{3} θ

• Also useful is the expansion (generating function), valid for |a| < 1,

\frac{(2m)!}{2^{m}m!} \begin{matrix}(1 − x^{2}) & m/2 \\ (1 + a^{2} − 2ax) & m+1/2 \end{matrix} = ∑_{l=0}^{∞} a^{l}P_{l+m}^{m}(x)

However, it does not have a direct physical meaning.

• There are very many recursion relations. Recursions in m are considered more fully in the next section. We list here only

– the two-step recursion in l, which is the efficient numerical way to generate P_{l+m}^{m} moving up from P_{m}^{m} and P_{m+1}^{m}

(l − m + 1) P_{l+1}^{m} = (2l + 1) xP_{l}^{m} − (l + m)P_{l−1}^{m}

– the two-step recursion in m, which can generate P_{l}^{m} moving down from P_{l}^{l} and P_{l}^{l−1}

P_{l}^{m+1} + \frac{2mx}{\sqrt{1 − x^{2}}} P_{l}^{m} + [l(l + 1) − m(m + 1)] P_{l}^{m−1} = 0

• The orthogonality and normalization integrals are

∫_{−1}^{1} P_{l}^{m}(x)P_{l′}^{m}(x) dx = \frac{2}{2l + 1} \frac{(l + m)!}{(l − m)!} δ_{ll′}

Three-function integrals are considered in the next section.

• Associated Legendre functions for a negative upper index are given by

P_{l}^{−m}(x) = (−1)^{m} \frac{(l + m)!}{(l − m)!}P_{l}^{m}(x)

With this definition, the Rodrigues formula, recursion relations, and normalization integral are valid for all integer values of m. Note that we have not introduced new eigenfunctions, simply relabeled the ones we had. Why we want to do this is discussed next.

7.3.4. Spherical harmonics

It is often better to deal with the θ and ϕ dependence at the same time by introducing the normalized eigenfunctions

Y_{lm}(θ, ϕ) = \sqrt{\frac{2l + 1}{4π} \frac{(l − m)!}{(l + m)!}} P_{l}^{m}(cos θ) e^{imϕ}

Note that Y_{lm} and Y_{l,−m} are different eigenfunctions, related by

Y_{l,−m} = (−1)^{m} Y_{l,m}^{∗}

These spherical harmonics form an orthogonal and complete set. The first few should be memorized, at least in their θ, ϕ dependence.

Multipole fields and “ladder” operations

It is actually easiest to memorize the basic interior solutions, r^{l}Y_{l,m}(θ, ϕ), as a function of the Cartesian coordinates x, y, z. For l = 0

\sqrt{4π} Y_{00} = 1

For l = 1

−\sqrt{4π} rY_{1,1} = \sqrt{\frac{3}{2}} r sin θ e^{iϕ} = \sqrt{\frac{3}{2}} (x + iy)

\sqrt{4π} rY_{10} = \sqrt{3}r cos θ = \sqrt{3} z

For l = 2

\sqrt{4π} r^{2}Y_{22} = \sqrt{\frac{15}{8}} r^{2} sin^{2} θ e^{2iϕ} = \sqrt{\frac{15}{8}} (x + iy)^{2}

Lecture 7 (Ch 3 part 1) Legendre functions, spherical harmonics 11

−\sqrt{4π} r^{2}Y_{21} = \sqrt{\frac{15}{2}} r^{2} sin θ cos θ e^{iϕ} = \sqrt{\frac{15}{2}} (x + iy) z

\sqrt{4π} r^{2}Y_{20} = \sqrt{\frac{5}{4}} r^{2}(3 cos^{2} θ − 1) = \sqrt{\frac{5}{4}} (3z^{2} − r^{2})

For l = 3

r^{3}Y_{33} ≈ (x + iy)^{3}
r^{3} Y_{32} ≈ (x + iy)^{2}z
r^{3} Y_{31} ≈ (x + iy) (5z^{2} − r^{2})
r^{3} Y_{30} ≈ z(5z^{2} − 3r^{2})

A look at these expressions suggests that it is convenient to introduce

x_{+} = x + iy
x_{−} = x − iy

with the inverse transformation

x = \frac{1}{2} (x_{+} + x_{−})
y = \frac{1}{2i} (x_{+} − x_{−})

which gives

\frac{∂}{∂x_{+}} = \frac{1}{2} (\frac{∂}{∂x} − i \frac{∂}{∂y} )
\frac{∂}{∂x_{−}} = \frac{1}{2} (\frac{∂}{∂x} + i \frac{∂}{∂y} )

Then apart from factors, when applied to r^{l}Y_{l,m}(θ, ϕ):

• \frac{∂}{∂z} steps down from l, m to l − 1, m

• \frac{∂}{∂x_{+}} steps down from l, m to l − 1, m − 1

• \frac{∂}{∂x_{−}} steps from l, m to l − 1, m + 1

• z \frac{∂}{∂x_{+}} − x_{−} \frac{∂}{∂z} steps down from l, m to l, m − 1

• x_{+} \frac{∂}{∂z} − z \frac{∂}{∂x_{−}} steps up from l, m to l, m + 1

As we did earlier in the case m = 0, we can also consider the multipole fields Y_{l,m}(θ, ϕ)/r^{l+1}, which are the basic exterior solutions. The “ladder” operations for these fields are

• \frac{∂}{∂z} steps up from l, m to l + 1, m

• \frac{∂}{∂x_{+}} steps from l, m to l + 1, m − 1

• \frac{∂}{∂x_{−}} steps up from l, m to l + 1, m + 1

• The other two operations, which are very familiar in the quantum theory of angular momentum, have the same effect as before.

The integrals of three spherical harmonics

∫ Y_{lm}Y_{l′m′}Y_{l”m”} dΩ

are often useful, especially in radiation problems (next semester). It can be shown that they vanish unless

– m + m′ + m′′ = 0 and l + l′ + l′′ is even

– l, l′, l′′ satisfy “triangle” inequalities, i.e. |l − l′| ≤ l′′ ≤ l + l′

7.3.5. The addition theorem

This is a generalization of the formula

cos γ = cos θ cos θ′ + sin θ sin θ′ cos (ϕ − ϕ′)

for the angle γ between the vectors in the directions of x and x′. It is

P_{l}(cos γ) = \frac{4π}{2l + 1} ∑_{m=−l}^{l} Y_{lm}^{∗}(θ′, ϕ′)Y_{lm}(θ, ϕ)

When inserted in the generating function formula it gives the extremely useful expansion

\frac{1}{|x − x′|} = ∑_{l=0}^{∞} ∑_{m=−l}^{l} \frac{4π}{2l + 1} \frac{r_{<}^{l}}{r_{>}^{l+1}} Y_{lm}^{∗}(θ′, ϕ′)Y_{lm}(θ, ϕ)

where r_{>} is the larger of r, r′.

7.4. Fields in a hole or near a point

The field is very strong near a sharp conducting point and does not penetrate well inside holes, with a power-law behavior similar to that near edges and inside corners. Near edges and in corners (2-d) we found Φ ∝ ρ^{α}, E ∝ ρ^{α−1} with α = π/θ_{0}; near conical points and in conical holes (3-d) one finds Φ ∝ r^{ν}, E ∝ r^{ν−1}, but the relation between ν and the cone angle β (see Fig. 3.5) is not as simple as ν = π/2β, where 2β in 3-d is the same as θ_{0} in 2-d. However, for a hole (0 < β < π/2) there is the excellent approximation (see Fig. 3.6)

v ≃ \frac{x_{01}}{β} − \frac{1}{2} − c_{0} β

where x_{01} is the first zero of the Bessel function J_{0}(x) and c_{0} is chosen so that ν = 1 for a flat surface (β = π/2), c_{0} = 4x_{01}/π^{2} − 3/π. Numerically, x_{01} ≃ 2.405 and c_{0} ≃ 0.01978. Actually, this approximation is also good for a point that is not too sharp and fails only when π − β < 10^{∘}, in which case

ν ≃ (2 ln \frac{2}{π − β} )^{−1}

To obtain these results one must construct solutions of the Legendre equation that are regular at the North pole (θ = 0) and are not necessarily regular at the South pole (θ = π). These solutions are known as the Legendre functions P_{ν}(x), where x = cos θ: they are conventionally normalized by setting P_{ν}(1) = 1 and reduce to the Legendre polynomials for integer ν. For a given β, one picks ν so that cos β is the first zero of P_{ν}(x). The dominant behavior near the apex is then

Φ = A_{ν}r^{ν}P_{ν}(cos θ)

and the field components are given by (J3.46). There is a simple expansion of P_{ν} in powers of the variable (1 − x)/2 = sin^{2}(θ/2) that is quite useful for the fields in shallow holes (ν near 1). The recursion relations for the Legendre polynomials are also valid for the Legendre functions, so in practice we need only tables of P_{ν}(x) for 0 < ν < 1.

7.5. Other Legendre functions

Jackson confines himself, wisely, to problems defined on the whole spherical surface or on a spherical cap, with azimuthal symmetry. The most general case of separable boundary value problem on the sphere involves the functions P_{ν}^{μ} for non-integer μ and ν, if the North pole is included in the domain, and in addition the “second” Legendre functions Q_{ν}^{μ} if the North pole is excluded. For instance, problems on an equatorial belt involve P_{ν}^{m} and Q_{ν}^{m} for m integer, problems in a watermelon slice involve P_{n}^{μ} for n integer.