∞

e (t’ t ) =

x 1

tn Jn (x). (8.82)

2

n=’∞

8.3. BESSEL FUNCTIONS 233

Expanding the left-hand side and using the binomial theorem, we ¬nd

∞ m

x 1 (r + s)!

(’1)s tr t’s ,

LHS =

2 m! r!s!

r+s=m

m=0

∞∞ r+s tr’s

x

s

= (’1) ,

2 r!s!

r=0 s=0

∞ ∞

(’1)s 2s+n

x

n

= t . (8.83)

s=0 s!(s + n)! 2

n=’∞

We recognize that the sum in the braces is the series expansion de¬ning

Jn (x). This therefore proves the generating function formula.

Bessel Identies

There are many identies and integrals involving Bessel functions. The most

common can be found in in the monumental Treatise on the Theory of Bessel

Functions by G. N. Watson. Here are just a few for your delectation:

i) Starting from the generating function

∞

1

Jn (x)tn ,

1

t’

exp x = (8.84)

2

t n=’∞

we can, with a few lines of work, show that

2Jn (x) = Jn’1 (x) ’ Jn+1 (x), (8.85)

2n

Jn (x) = Jn’1 (x) + Jn+1 (x), (8.86)

x

J0 (x) = ’J1 (x), (8.87)

∞

Jn (x + y) = Jr (x)Jn’r (y). (8.88)

r=’∞

ii) From the series expansion for Jn (x) we ¬nd

d

{xn Jn (x)} = xn Jn’1 (x). (8.89)

dx

iii) By similar methods, we ¬nd

m

1d

x’n Jn (x) = (’1)m x’n’m Jn+m (x). (8.90)

x dx

234 CHAPTER 8. SPECIAL FUNCTIONS I

iv) Again from the series expansion, we ¬nd

1

∞

J0 (ax)e’px dx = √ . (8.91)

a2 + p2

0

Semi-classical picture

The Schr¨dinger equation

o

h2

¯ 2

’ ψ = Eψ (8.92)

2m

can be separated in cylindrical polars, and has eigenfunctions

ψk,l (r, θ) = Jl (kr)eilθ . (8.93)

The eigenvalues are E = h2 k 2 /2m. The quantity L = hl is the angular

¯ ¯

momentum of the Schr¨dinger particle about the origin. If we impose rigid-

o

wall boundary conditions that ψk,l (r, θ) vanish on the circle r = R, then the

allowed k form a discrete set kl,n , where Jl (kl,n R) = 0. To ¬nd the energy

eigenvalues we therefore need to know the location of the zeros of Jl (x).

There is no closed form eqution for these numbers, but they are tabulated.

The zeros for kR l are also approximated by the zeros of the asymptotic

expression

2 1 1

Jl (kR) ∼ cos(kR ’ lπ ’ π), (8.94)

πkR 2 4

which are located at

1 1 π

kl,n R = lπ + π + (2n + 1) . (8.95)

2 4 2

If we let R ’ ∞, then the spectrum becomes continuous and we are de-

scribing uncon¬ned scattering states. Since the particles are free, their classi-

cal motion is in a straight line at constant velocity. A classical particle mak-

ing a closest approach at a distance rmin , has angular momentum L = prmin .

Since p = hk is the particle™s linear momentum, we have l = krmin . Be-

¯

cause the classical particle is never closer than rmin , the quantum mechanical

wavefunction representing such a particle will become evanescent (i.e. tend

rapidly to zero) as soon as r is smaller than rmin . We therefore expect that

Jl (kr) ≈ 0 if kr < l. This e¬ect is dramatically illustrated by the following

MathematicaTM plot.

8.3. BESSEL FUNCTIONS 235

0.15

0.1

0.05

50 100 150 200

-0.05

-0.1

J100 (x).

An improved asymptotic expression, which gives a better estimate of the

zeros, is the approximation

2

Jn (kr) ≈ sin(kx ’ lθ ’ π/4), r rmin . (8.96)

πkx

Here x = r sin θ and θ = cos’1 (rmin /r) are functions of r. They have a

geometric interpretation in the right-angled triangle

θ

r