x

The parameter x has the physical interpretation of being the distance along

the straight-line semiclassical trajectory. The approximation is quite accurate

once r exceeds rmin by more than a few percent.

Exercise: Show that that this expression in the WKB approximation to the

solution of Bessel™s equation. It is therefore accurate once we are away from

the classical turning point at r = rmin

The asymptotic r’1/2 fall-o¬ of the Bessel function is also understandable in

the semiclassical picture.

236 CHAPTER 8. SPECIAL FUNCTIONS I

An ensemble of trajectories, each missing the origin by rmin , leaves a “hole”.

60

40

20

0

-20

-40

-60

-60 -40 -20 0 20 40 60

The hole is visible in the real part of ψk,20 (rθ) = ei20θ J20 (kr)

By the uncertainly principle, a particle with de¬nite angular momentum must

have completely uncertain angular position. The wavefunction Jl (kr)eilθ

8.3. BESSEL FUNCTIONS 237

therefore represents an ensemble of particles approaching from all directions,

but all missing the origin by the same distance. The density of classical par-

ticle trajectories is in¬nite at r = rmin , forming a caustic. By “conservation

of lines”, the particle density falls o¬ as 1/r as we move outwards. The par-

ticle density is proportional to |•|2 , so • itself decreases as r’1/2 . In contrast

to the classical particle density, the quantum mechanical wavefunction am-

plitude remains ¬nite at the caustic ” the “geometric optics” in¬nity being

tempered by di¬raction e¬ects.

8.3.2 Orthogonality and Completeness

We can write the equation obeyed by Jn (kr) in Sturm-Liouville form. We

have

m2

1d dy 2

+ k ’ 2 y = 0.

r (8.97)

r dr dr r

Comparison with the standard Sturm-Liouville equation shows that the weight

function, w(r), is r, and the eigenvalues are k2 .

From Lagrange™s identity we obtain

R

2 2

(k1 ’k2 ) Jm (k1 r)Jm (k2 r)r dr = R [k2 Jm (k1 R)Jm (k2 R) ’ k1 Jm (k2 R)Jm (k1 R)] .

0

(8.98)

We have no contribution from the origin on the right-hand side because all

Jm Bessel functions except J0 vanish there, whilst J0 (0) = 0. For each m we

get get a set of orthogonal functions, Jm (kn x), provided the kn R are chosen

to be roots of Jm (kn R) = 0 or Jm (kn R) = 0.

We can ¬nd the normalization constants by di¬erentiating with respect

to k1 and then setting k1 = k2 in the result. We ¬nd

m2

12

R 2 2 2

R Jm (kR) + 1 ’ 2 2 Jm (kR)

Jm (kr) r dr = ,

2 kR

0

12

R [Jn (kR)]2 ’ Jn’1 (kR)Jn+1 (kR) .

= (8.99)

2

(The second equality follows on applying the recurrence relations for the

Jn (kr), and provides an expression that is perhaps easier to remember.) For

Dirichlet boundary conditions we will require kn R to be zero of Jm , and so

we have

1

R 2 2

Jm (kr) r dr = R2 Jm (kR) . (8.100)

2

0

238 CHAPTER 8. SPECIAL FUNCTIONS I

For Neumann boundary conditions we require kn R to be a zero of Jm . In

this case

m2

12

R 2 2

Jm (kr) r dr = R 1 ’ 2 2 Jm (kR) . (8.101)

2 kR

0

Example: Harmonic function in cylinder.

z

L

r

a

We wish to solve 2 V = 0 within a cylinder of hight L and radius a. The volt-

age is prescribed on the upper surface of the cylinder: V (r, θ, L) = U (r, θ).

We are told that V = 0 on all other parts of boundary.

The general solution of Laplace™s equation in will be sum of terms such

as

sinh(kz) Jm (kr) sin(mθ)

— — , (8.102)

cosh(kz) Nm (kr) cos(mθ)

where the braces indice a choice of upper or lower functions. We must take

only the sinh(kz) terms because we know that V = 0 at z = 0, and only the

Jm (kr) terms because V is ¬nite at r = 0. The k™s are also restricted by the

boundary condition on the sides of the cylinder to be such that Jm (ka) = 0.

We therefore expand the prescribed voltage as

U (r, θ) = sinh(knm L)Jm (kmn r) [Anm sin(mθ) + Bnm cos(mθ)] , (8.103)

m,n

and use the orthonormality of the trigonometric and Bessel function to ¬nd

the coe¬cients to be

2cosech(knm L) 2π a

Anm =2 dθ U (r, θ)Jm (knm r) sin(mθ) rdr, (8.104)

πa [Jm (knm a)]2 0 0

8.3. BESSEL FUNCTIONS 239