Bessel Functions - University of New Mexicoquantum.phys.unm.edu/466-17/ch9.pdf · 9 Bessel...

9

Bessel Functions

9.1 Bessel functions of the first kind

Friedrich Bessel (1784–1846) invented functions for problems with circularsymmetry. The most useful ones are defined for any integer n by the series

Jn(z) =zn

2nn!

1� z2

2(2n+ 2)+

z4

2 · 4(2n+ 2)(2n+ 4)� . . .

�

=⇣z2

⌘n 1X

m=0

(�1)m

m! (m+ n)!

⇣z2

⌘2m.

(9.1)

The first term of this series tells us that for small |z| ⌧ 1

Jn(z) ⇡zn

2nn!. (9.2)

The alternating signs in (9.1) make the waves plotted in Fig. 9.1, and we havefor big |z| � 1 the approximation (Courant and Hilbert, 1955, chap. VII)

Jn(z) ⇡r

2

⇡zcos

⇣z � n⇡

2� ⇡

4

⌘+O(|z|�3/2). (9.3)

The Jn(z) are entire transcendental functions. They obey Bessel’s equation

d2Jndz2

+1

z

dJndz

+

✓1� n2

z2

◆Jn = 0 (9.4)

(6.393) as one may show (exercise 9.1) by substituting the series (9.1) intothe di↵erential equation (9.4). Their generating function is

exphz2(u� 1/u)

i=

1X

n=�1un Jn(z) (9.5)

9.1 Bessel functions of the first kind 383

0 2 4 6 8 10 12−0.5

0

0.5

1

ρ

The Besse l Functions J0(ρ ) , J1(ρ ) , and J2(ρ )

0 2 4 6 8 10 12−0.4

−0.2

0

0.2

0.4

ρ

The Besse l Functions J3(ρ ) , J4(ρ ) , and J5(ρ )

Figure 9.1 Top: Plots of J0(⇢) (solid curve), J1(⇢) (dot-dash), and J2(⇢)(dashed) for real ⇢. Bottom: Plots of J3(⇢) (solid curve), J4(⇢) (dot-dash),and J5(⇢)(dashed). The points at which Bessel functions cross the ⇢-axisare called zeros or roots; we use them to satisfy boundary conditions.

from which one may derive (exercise 9.5) the series expansion (9.1) and(exercise 9.6) the integral representation (5.51)

Jn(z) =1

⇡

Z ⇡

0cos(z sin ✓ � n✓) d✓ = J�n(�z) = (�1)nJ�n(z) (9.6)

for all complex z. For n = 0, this integral is (exercise 9.7) more simply

J0(z) =1

2⇡

Z 2⇡

0eiz cos ✓ d✓ =

1

2⇡

Z 2⇡

0eiz sin ✓ d✓. (9.7)

These integrals (exercise 9.8) give Jn(0) = 0 for n 6= 0, and J0(0) = 1.

By di↵erentiating the generating function (9.5) with respect to u and

384 Bessel Functions

identifying the coe�cients of powers of u, one finds the recursion relation

Jn�1(z) + Jn+1(z) =2n

zJn(z). (9.8)

Similar reasoning after taking the z derivative gives (exercise 9.10)

Jn�1(z)� Jn+1(z) = 2 J 0n(z). (9.9)

By using the gamma function (section 5.15), one may extend Bessel’sequation (9.4) and its solutions Jn(z) to non-integral values of n

J⌫(z) =⇣z2

⌘⌫ 1X

m=0

(�1)m

m!�(m+ ⌫ + 1)

⇣z2

⌘2m. (9.10)

The self-adjoint form (6.392) of Bessel’s equation (9.4) with z = kx is(exercise 9.11)

� d

dx

✓xd

dxJn(kx)

◆+

n2

xJn(kx) = k2xJn(kx). (9.11)

In the notation of equation (6.365), p(x) = x, k2 is an eigenvalue, and⇢(x) = x is a weight function. To have a self-adjoint system (section 6.30)on an interval [0, b], we need the boundary condition (6.325)

0 =⇥p(Jnv

0 � J 0nv)

⇤b0=⇥x(Jn(kx)v

0(kx)� J 0n(kx)v(kx))

⇤b0

(9.12)

for all functions v(kx) in the domain D of the system. The definition (9.1)of Jn(z) implies that J0(0) = 1, and that Jn(0) = 0 for integers n > 0. Thussince p(x) = x, the terms in this boundary condition vanish at x = 0 aslong as the domain consists of functions v(kx) that are twice di↵erentiableon the interval [0, b]. To make these terms vanish at x = b, we require thatJn(kb) = 0 and that v(kb) = 0. Thus kb must be a zero zn,m of Jn(z), andso Jn(kb) = Jn(zn,m) = 0. With k = zn,m/b, Bessel’s equation (9.11) is

� d

dx

✓xd

dxJn (zn,mx/b)

◆+

n2

xJn (zn,mx/b) =

z2n,mb2

xJn (zn,mx/b) . (9.13)

For fixed n, the eigenvalue k2 = z2n,m/b2 is di↵erent for each positive integerm. Moreover as m ! 1, the zeros zn,m of Jn(x) rise as m⇡ as one mightexpect since the leading term of the asymptotic form (9.3) of Jn(x) is pro-portional to cos(x�n⇡/2�⇡/4) which has zeros at m⇡+(n+1)⇡/2+⇡/4. Itfollows that the eigenvalues k2 ⇡ (m⇡)2/b2 increase without limit as m ! 1in accordance with the general result of section 6.36. It follows then fromthe argument of section 6.37 and from the orthogonality relation (6.404)that for every fixed n, the eigenfunctions Jn(zn,mx/b), one for each zero, are


complete in the mean, orthogonal, and normalizable on the interval [0, b]with weight function ⇢(x) = xZ b

0xJn

⇣zn,mx

b

⌘Jn

⇣zn,m0x

b

⌘dx = �m,m0

b2

2J 02n (zn,m) = �m,m0

b2

2J2n+1(zn,m)

(9.14)and a normalization constant (exercise 9.12) that depends upon the firstderivative of the Bessel function or the square of the next Bessel function atthe zero.

Because they are complete, sums of Bessel functions Jn(zn,kx/b) can rep-resent Dirac’s delta function on the interval [0, b] as in the sum (6.451)

�(x� y) =�2x↵ y1�↵/b2

� 1X

k=1

Jn(zn,kx/b)Jn(zn,ky/b)

J2n+1(zn,k)

. (9.15)

For n = 0 and b = 1, this sum is

�(x� y) = 2x↵ y1�↵1X

k=1

J0(z0,kx)J0(z0,ky)

J21 (z0,k)

. (9.16)

Figure 9.2 plots the sum of the first 10,000 terms of this series for ↵ = 0and y = 0.47, for ↵ = 1/2 and y = 1/2, and for ↵ = 1 and y = 0.53.The integrals of these 10,000-term sums from 0 to 1 respectively are 0.9966,0.9999, and 0.9999. The expansion (9.15) of the delta function expresses thecompleteness of the eigenfunctions Jn(zn,kx/b) for fixed n and lets us writeany twice di↵erentiable function f(x) that vanishes at x = b as the Fourierseries

f(x) = x↵1X

k=1

ak Jn(zn,kx/b) (9.17)

where

ak =2

b2 J2n+1(zn,k)

Z b

0y1�↵ Jn(zn,ky/b) f(y) dy. (9.18)

The orthogonality relation on an infinite interval is

�(x� y) = x

Z 1

0k Jn(kx) Jn(ky) dk (9.19)

for positive values of x and y. One may generalize these relations (9.11–9.19)from integral n to complex ⌫ with Re ⌫ > �1.

Example 9.1 (Bessel’s Drum). The top of a drum is a circular membrane


0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54

−2000

0

2000

4000

6000

8000

10000

x

J0Series

10,000-Term Besse l Series for Dirac Delta Function

Figure 9.2 The sum of the first 10,000 terms of the Bessel representation(9.16) for the Dirac delta function �(x � y) is plotted for y = 0.47 and↵ = 0, for y = 1/2 and ↵ = 1/2, and for y = 0.53 and ↵ = 1.

with a fixed circumference. The membrane’s potential energy is approxi-mately proportional to the extra area it has when it’s not flat. Let h(x, y)be the displacement of the membrane in the z direction normal to the x-yplane of the flat membrane, and let hx and hy denote its partial derivatives(6.20). The extra length of a line segment dx on the stretched membrane isp1 + h2x dx, and so the extra area of an element dx dy is

dA ⇡⇣p

1 + h2x

q1 + h2y � 1

⌘dx dy ⇡ 1

2

�h2x + h2y

�dx dy. (9.20)

The (nonrelativistic) kinetic energy of the area element is proportional tothe square of its speed. So if � is the surface tension and µ the surface massdensity of the membrane, then to lowest order in d the action functional(sections 6.15 and 6.42) is

S[h] =

Z hµ2h2t �

�

2

�h2x + h2y

�idx dy dt. (9.21)


We minimize this action for h’s that vanish on the boundary x2 + y2 = r2d

0 = �S[h] =

Z[µht �ht � � (hx �hx + hy �hy)] dx dy dt. (9.22)

Since (6.206) �ht = (�h)t, �hx = (�h)x, and �hy = (�h)y, we can integrateby parts and get

0 = �S[h] =

Z[� µhtt + � (hxx + hyy )] �h dx dy dt (9.23)

apart from a surface term proportional to �h which vanishes because �h = 0on the circumference of the membrane. The membrane therefore obeys thewave equation

µhtt = � (hxx + hyy ) ⌘ �4h. (9.24)

This equation is separable, and so letting h(x, y, t) = s(t) v(x, y), we have

stts

=�

µ

4v

v= � !2. (9.25)

The eigenvalues of the Helmholtz equation � 4v = � v give the angularfrequencies as ! =

p��/µ. The time dependence then is

s(t) = a sin(p��/µ (t� t0)) (9.26)

in which a and t0 are constants.In polar coordinates, Helmholtz’s equation is separable (6.66–6.69)

�4v = � vrr � r�1vr � r�2v✓✓ = �v. (9.27)

We set v(r, ✓) = u(r)h(✓) and find � u00h � r�1u0h � r�2uh00 = �uh. Aftermultiplying both sides by r2/uh, we get

r2u00

u+ r

u0

u+ �r2 = � h00

h= n2. (9.28)

The general solution for h then is h(✓) = b sin(n(✓ � ✓0)) in which b and✓0 are constants and n must be an integer so that h is single valued on thecircumference of the membrane.

The displacement u(r) must obey the di↵erential equation

��r u0

�0+ n2 u/r = � r u (9.29)

and vanish at r = rd, the rim of the drum. Therefore it is an eigenfunction


u(r) = Jn(zn,mr/rd) of the self-adjoint di↵erential equation (9.13)

� d

dr

✓rd

drJn (zn,mr/rd)

◆+n2

rJn (zn,mr/rd) =

z2n,mr2d

xJn (zn,mr/rd) (9.30)

with an eigenvalue � = z2n,m/r2d in which zn,m is the mth zero of the nthBessel function, Jn(zn,m) = 0. For each integer n � 0, there are infinitelymany zeros zn,m at which the Bessel function vanishes, and for each zero thefrequency is ! = (zn,m/rd)

p�/µ. The general solution to the wave equation

(9.24) of the membrane thus is

h(r, ✓, t) =1X

n=0

1X

m=1

cn,m sin

zn,mrd

r�

µ(t� t0)

�sin [n(✓ � ✓0)] Jn

✓zn,m

r

rd

◆

(9.31)in which t0 and ✓0 can depend upon n and m. For each n and big m, thezero zn,m is near m⇡ + (n+ 1)⇡/2 + ⇡/4.

We learned in section 6.5 that in three dimensions Helmholtz’s equation�4V = ↵2 V separates in cylindrical coordinates (and in several other co-ordinate systems). That is, the function V (⇢,�, z) = B(⇢)�(�)Z(z) satisfiesthe equation

�4V = �1

⇢

(⇢V,⇢),⇢ +

1

⇢V,�� + ⇢V,zz

�= ↵2 V (9.32)

if B(⇢) obeys Bessel’s equation

⇢d

d⇢

✓⇢dB

d⇢

◆+�(↵2 + k2)⇢2 � n2

�B = 0 (9.33)

and � and Z respectively satisfy

� d2�

d�2= n2�(�) and

d2Z

dz2= k2Z(z) (9.34)

or if B(⇢) obeys the Bessel equation

⇢d

d⇢

✓⇢dB

d⇢

◆+�(↵2 � k2)⇢2 � n2

�B = 0 (9.35)

and � and Z satisfy

� d2�

d�2= n2�(�) and

d2Z

dz2= �k2Z(z). (9.36)

In the first case (9.33 & 9.34), the solution V is

Vk,n(⇢,�, z) = Jn(p↵2 + k2 ⇢)e±in�e±kz (9.37)


while in the second case (9.35 & 9.36), it is

Vk,n(⇢,�, z) = Jn(p↵2 � k2 ⇢)e±in�e±ikz. (9.38)

In both cases, n must be an integer if the solution is to be single valued onthe full range of � from 0 to 2⇡.

When ↵ = 0, the Helmholtz equation reduces to the Laplace equation4V = 0 of electrostatics which the simpler functions

Vk,n(⇢,�, z) = Jn(k⇢)e±in�e±kz and Vk,n(⇢,�, z) = Jn(ik⇢)e

±in�e±ikz

(9.39)satisfy.

The product i�⌫ J⌫(ik⇢) is real and is known as the modified Besselfunction

I⌫(k⇢) ⌘ i�⌫ J⌫(ik⇢). (9.40)

It occurs in various solutions of the di↵usion equation D4� = �. Thefunction V (⇢,�, z) = B(⇢)�(�)Z(z) satisfies

4V =1

⇢

(⇢V,⇢),⇢ +

1

⇢V,�� + ⇢V,zz

�= ↵2 V (9.41)

if B(⇢) obeys Bessel’s equation

⇢d

d⇢

✓⇢dB

d⇢

◆��(↵2 � k2)⇢2 + n2

�B = 0 (9.42)

and � and Z respectively satisfy

� d2�

d�2= n2�(�) and

d2Z

dz2= k2Z(z) (9.43)

or if B(⇢) obeys the Bessel equation

⇢d

d⇢

✓⇢dB

d⇢

◆��(↵2 + k2)⇢2 + n2

�B = 0 (9.44)

and � and Z satisfy

� d2�

d�2= n2�(�) and

d2Z

dz2= �k2Z(z). (9.45)

In the first case (9.42 & 9.43), the solution V is

Vk,n(⇢,�, z) = In(p↵2 � k2 ⇢)e±in�e±kz (9.46)

while in the second case (9.44 & 9.45), it is

Vk,n(⇢,�, z) = In(p↵2 + k2 ⇢)e±in�e±ikz. (9.47)


In both cases, n must be an integer if the solution is to be single valued onthe full range of � from 0 to 2⇡.

Example 9.2 (Charge near a Membrane). We will use ⇢ to denote thedensity of free charges—those that are free to move in or out of a dielectricmedium, as opposed to those that are part of the medium, bound in it bymolecular forces. The time-independent Maxwell equations are Gauss’s lawr · D = ⇢ for the divergence of the electric displacement D, and the staticform r ⇥ E = 0 of Faraday’s law which implies that the electric field E isthe gradient of an electrostatic potential E = �rV .Across an interface between two dielectrics with normal vector n, the

tangential electric field is continuous, n ⇥ E2 = n ⇥ E1, while the normalcomponent of the electric displacement jumps by the surface density � of freecharge, n ·(D2 �D1) = �. In a linear dielectric, the electric displacement Dis the electric field multiplied by the permittivity ✏ of the material, D = ✏E.The membrane of a eukaryotic cell is a phospholipid bilayer whose area

is some 3 ⇥ 108 nm2, and whose thickness t is about 5 nm. On a scale ofnanometers, the membrane is flat. We will take it to be a slab extending toinfinity in the x and y directions. If the interface between the lipid bilayer andthe extracellular salty water is at z = 0, then the cytosol extends thousandsof nm down from z = �t = �5 nm. We will ignore the phosphate headgroups and set the permittivity ✏` of the lipid bilayer to twice that of thevacuum ✏` ⇡ 2✏0; the permittivity of the extracellular water and that of thecytosol are ✏w ⇡ ✏c ⇡ 80✏0.We will compute the electrostatic potential V due to a charge q at a point

(0, 0, h) on the z-axis above the membrane. This potential is cylindricallysymmetric about the z-axis, so V = V (⇢, z). The functions Jn(k⇢) ein� e±kz

form a complete set of solutions (9.39) of Laplace’s equation, but due to thesymmetry, we only need the n = 0 functions J0(k⇢) e±kz. Since there areno free charges in the lipid bilayer or in the cytosol, we may express thepotential in the lipid bilayer V` and in the cytosol Vc as

V`(⇢, z) =

Z 1

0dk J0(k⇢)

hm(k) ekz + f(k) e�kz

i

Vc(⇢, z) =

Z 1

0dk J0(k⇢) d(k) e

kz.(9.48)

The Green’s function (3.112) for Poisson’s equation �4G(x) = �(3)(x) incylindrical coordinates is (5.147)

G(x) =1

4⇡|x| =1

4⇡p⇢2 + z2

=

Z 1

0

dk

4⇡J0(k⇢) e

�k|z|. (9.49)


Thus we may expand the potential in the salty water as

Vw(⇢, z) =

Z 1

0dk J0(k⇢)

q

4⇡✏we�k|z�h| + u(k) e�kz

�. (9.50)

Using n⇥E2 = n⇥E1 and n ·(D2 �D1) = �, suppressing k, and setting� ⌘ qe�kh/4⇡✏w and y = e2kt, we get four equations

m+ f � u = � and ✏`m� ✏`f + ✏wu = ✏w�

✏`m� ✏`yf � ✏cd = 0 and m+ yf � d = 0.(9.51)

In terms of the abbreviations ✏w` = (✏w + ✏`) /2 and ✏c` = (✏c + ✏`) /2 as wellas p = (✏w � ✏`)/(✏w + ✏`) and p0 = (✏c � ✏`)/(✏c + ✏`), the solutions are

u(k) = �p� p0/y

1� pp0/yand m(k) = �

✏w✏w`

1

1� pp0/y

f(k) = � �✏w✏w`

p0/y

1� pp0/yand d(k) = �

✏w✏`✏w`✏c`

1

1� pp0/y.

(9.52)

Inserting these solutions into the Bessel expansions (9.48) for the poten-tials, expanding their denominators

1

1� pp0/y=

1X

0

(pp0)n e�2nkt (9.53)

and using the integral (9.49), we find that the potential Vw in the extracel-lular water of a charge q at (0, 0, h) in the water is

Vw(⇢, z) =q

4⇡✏w

"1

r+

pp⇢2 + (z + h)2

�1X

n=1

p0�1� p2

�(pp0)n�1

p⇢2 + (z + 2nt+ h)2

#

(9.54)in which r =

p⇢2 + (z � h)2 is the distance to the charge q. The principal

image charge pq is at (0, 0,�h). Similarly, the potential V` in the lipid bilayeris

V`(⇢, z) =q

4⇡✏w`

1X

n=0

"(pp0)np

⇢2 + (z � 2nt� h)2� pnp0n+1

p⇢2 + (z + 2(n+ 1)t+ h)2

#

(9.55)and that in the cytosol is

Vc(⇢, z) =q ✏`

4⇡✏w`✏c`

1X

n=0

(pp0)np⇢2 + (z � 2nt� h)2

. (9.56)

These potentials are the same as those of example 4.19, but this derivationis much simpler and less error prone than the method of images.


Since p = (✏w� ✏`)/(✏w+ ✏`) > 0, the principal image charge pq at (0, 0, �h) has the same sign as the charge q and so contributes a positive termproportional to pq2 to the energy. So a lipid slab repels a nearby charge inwater no matter what the sign of the charge.A cell membrane is a phospholipid bilayer. The lipids avoid water and form

a 4-nm-thick layer that lies between two 0.5-nm layers of phosphate groupswhich are electric dipoles. These electric dipoles cause the cell membrane toweakly attract ions that are within 0.5 nm of the membrane.

Example 9.3 (Cylindrical Wave Guides). An electromagnetic wave travel-ing in the z-direction down a cylindrical wave guide looks like

E ein� ei(kz�!t) and B ein� ei(kz�!t) (9.57)

in which E and B depend upon ⇢

E = E⇢⇢+ E��+ Ezz and B = B⇢⇢+B��+Bzz (9.58)

in cylindrical coordinates (section 6.4). If the wave guide is an evacuated,perfectly conducting cylinder of radius r, then on the surface of the waveguide the parallel components of E and the normal component of B mustvanish which leads to the boundary conditions

Ez(r) = 0, E�(r) = 0, and B⇢(r) = 0. (9.59)

Since the E and B fields have subscripts, we will use commas to denotederivatives as in @(⇢E�)/@⇢ ⌘ (⇢E�),⇢ and so forth. In this notation, thevacuum forms r ⇥ E = � B and r ⇥ B = E/c2 of the Faraday andMaxwell-Ampere laws give us (exercise 9.15) the field equations

Ez,�/⇢� ikE� = i!B⇢ inBz/⇢� ikB� = �i!E⇢/c2

ikE⇢ � Ez,⇢ = i!B� ikB⇢ �Bz,⇢ = �i!E�/c2 (9.60)

[(⇢E�),⇢ � inE⇢] /⇢ = i!Bz [(⇢B�),⇢ � inB⇢] /⇢ = �i!Ez/c2.

Solving them for the ⇢ and � components of E and B in terms of their zcomponents (exercise 9.16), we find

E⇢ =�ikEz,⇢ + n!Bz/⇢

k2 � !2/c2E� =

nkEz/⇢+ i!Bz,⇢

k2 � !2/c2

B⇢ =�ikBz,⇢ � n!Ez/c2⇢

k2 � !2/c2B� =

nkBz/⇢� i!Ez,⇢/c2

k2 � !2/c2.

(9.61)

The fields Ez and Bz obey the wave equations (11.90, exercise 6.6)

�4Ez = �Ez/c2 = !2Ez/c

2 and �4Bz = �Bz/c2 = !2Bz/c

2. (9.62)


Because their z-dependence (9.57) is periodic, they are (exercise 9.17) linearcombinations of Jn(

p!2/c2 � k2 ⇢)ein�ei(kz�!t).

Modes with Bz = 0 are transverse magnetic or TM modes. For themthe boundary conditions (9.59) will be satisfied if

p!2/c2 � k2 r is a zero

zn,m of Jn. So the frequency !n,m(k) of the n,m TM mode is

!n,m(k) = cqk2 + z2n,m/r2. (9.63)

Since first zero of a Bessel function is z0,1 ⇡ 2.4048, the minimum frequency!0,1(0) = c z0,1/r ⇡ 2.4048 c/r occurs for n = 0 and k = 0. If the radius ofthe wave-guide is r = 1 cm, then !0,1(0)/2⇡ is about 11 GHz, which is amicrowave frequency with a wavelength of 2.6 cm. In terms of the frequencies(9.63), the field of a pulse moving in the +z-direction is

Ez(⇢,�, z, t) =1X

n=0

1X

m=1

Z 1

0cn,m(k) Jn

⇣zn,m ⇢r

⌘ein� exp i [kz � !n,m(k)t] dk.

(9.64)Modes with Ez = 0 are transverse electric or TE modes. For them the

boundary conditions (9.59) will be satisfied (exercise 9.19) ifp!2/c2 � k2 r

is a zero z0n,m of J 0n. Their frequencies are !n,m(k) = c

qk2 + z02n,m/r2. Since

first zero of a first derivative of a Bessel function is z01,1 ⇡ 1.8412, the mini-mum frequency !1,1(0) = c z01,1/r ⇡ 1.8412 c/r occurs for n = 1 and k = 0. Ifthe radius of the wave-guide is r = 1 cm, then !1,1(0)/2⇡ is about 8.8 GHz,which is a microwave frequency with a wavelength of 3.4 cm.

Example 9.4 (Cylindrical Cavity). The modes of an evacuated, perfectlyconducting cylindrical cavity of radius r and height h are like those of acylindrical wave guide (example 9.3) but with extra boundary conditions

Bz(⇢,�, 0, t) = 0 and Bz(⇢,�, h, t) = 0

E⇢(⇢,�, 0, t) = 0 and E⇢(⇢,�, h, t) = 0

E�(⇢,�, 0, t) = 0 and E�(⇢,�, h, t) = 0 (9.65)

at the two ends of the cylinder. If ` is an integer and ifp!2/c2 � ⇡2`2/h2 r

is a zero z0n,m of J 0n, then the TE fields Ez = 0 and

Bz = Jn(z0n,m ⇢/r) e

in� sin(⇡`z/h) e�i!t (9.66)

satisfy both these (9.65) boundary conditions at z = 0 and h and those (9.59)at ⇢ = r as well as the separable wave equations (9.62). The frequencies of

the resonant TE modes then are !n,m,` = cqz02n,m/r2 + ⇡2`2/h2.


The TM modes are Bz = 0 and

Ez = Jn(zn,m ⇢/r) ein� sin(⇡`z/h) e�i!t (9.67)

with resonant frequencies !n,m,` = cqz2n,m/r2 + ⇡2`2/h2.

9.2 Spherical Bessel functions of the first kind

If in Bessel’s equation (9.4), one sets n = ` + 1/2 and j` =p⇡/2xJ`+1/2,

then one may show (exercise 9.22) that

x2 j00` (x) + 2x j0`(x) + [x2 � `(`+ 1)] j`(x) = 0 (9.68)

which is the equation for the spherical Bessel function j`.We saw in example 6.7 that by setting V (r, ✓,�) = Rk,`(r)⇥`,m(✓)�m(�)

we could separate the variables of Helmholtz’s equation �4V = k2V inspherical coordinates

r24V

V=

(r2R0k,`)

0

Rk,`+

(sin ✓⇥0`,m)0

sin ✓⇥`,m+

�00

sin2 ✓�= �k2r2. (9.69)

Thus if �m(�) = eim� so that �00m = �m2�m, and if ⇥`,m satisfies the

associated Legendre equation (8.92)

sin ✓�sin ✓⇥0

`,m

�0+ [`(`+ 1) sin2 ✓ �m2]⇥`,m = 0 (9.70)

then the product V (r, ✓,�) = Rk,`(r)⇥`,m(✓)�m(�) will obey (9.69) becausein view of (9.68) the radial function Rk,`(r) = j`(kr) satisfies

(r2R0k,`)

0 + [k2r2 � `(`+ 1)]Rk,` = 0. (9.71)

In terms of the spherical harmonic Y`,m(✓,�) = ⇥`,m(✓)�m(�), the solutionis V (r, ✓,�) = j`(kr)Y`,m(✓,�).Rayleigh’s formula gives the spherical Bessel function

j`(x) ⌘r

⇡

2xJ`+1/2(x) (9.72)

as the `th derivative of sinx/x

j`(x) = (�1)` x`✓1

x

d

dx

◆` ✓sinx

x

◆(9.73)

(Lord Rayleigh (John William Strutt) 1842–1919). In particular, j0(x) =

9.2 Spherical Bessel functions of the first kind 395

sinx/x and j1(x) = sinx/x2 � cosx/x. Rayleigh’s formula leads to the re-cursion relation (exercise 9.23)

j`+1(x) =`

xj`(x)� j0`(x) (9.74)

with which one can show (exercise 9.24) that the spherical Bessel functionsas defined by Rayleigh’s formula do satisfy their di↵erential equation (9.71)with x = kr.The spherical Bessel functions j`(kr) satisfy the self-adjoint Sturm-Liouville

(6.415) equation (9.71)

� (r2j0`)0 + `(`+ 1)j` = k2r2j` (9.75)

with eigenvalue k2 and weight function ⇢ = r2. If j`(z`,n) = 0, then thefunctions j`(kr) = j`(z`,nr/a) vanish at r = a and form an orthogonal basis

Z a

0j`(z`,nr/a) j`(z`,mr/a) r2 dr =

a3

2j2`+1(z`,n) �n,m (9.76)

for a self-adjoint system on the interval [0, a]. Moreover, since as n ! 1the eigenvalues k2`,n = z2`,n/a

2 ⇡ [(n + `/2)⇡]2/a2 ! 1, the eigenfunctionsj`(z`,nr/a) also are complete in the mean (section 6.37) as shown by theexpansion of the delta function

�(r � r0) =2r↵r02�↵

a3

1X

n=1

j`(z`,nr/a) j`(z`,mr0/a)

j2`+1(z`,n)(9.77)

for 0 ↵ 2. This formula lets us expand any twice-di↵erentiable functionf(r) that vanishes at r = a as

f(r) = r↵1X

n=1

an j`(z`,nr/a) (9.78)

where

an =2

a3 j2`+1(z`,n)

Z a

0r02�↵ j`(z`,mr0/a) f(r0) dr0. (9.79)

On an infinite interval, the delta-function formulas are for positive valuesof k and k0

�(k � k0) =2kk0

⇡

Z 1

0j`(kr) j`(k

0r) r2 dr (9.80)

and for r, r0 > 0

�(r � r0) =2rr0

⇡

Z 1

0j`(kr) j`(kr

0) k2 dk. (9.81)


If we write the spherical Bessel function j0(x) as the integral

j0(z) =sin z

z=

1

2

Z 1

�1eizx dx (9.82)

and use Rayleigh’s formula (9.73), we may find an integral for j`(z)

j`(z) = (�1)` z`✓1

z

d

dz

◆` ✓sin z

z

◆= (�1)` z`

✓1

z

d

dz

◆` 1

2

Z 1

�1eizx dx

=z`

2

Z 1

�1

(1� x2)`

2``!eizx dx =

(�i)`

2

Z 1

�1

(1� x2)`

2``!

d`

dx`eizx dx

=(�i)`

2

Z 1

�1eizx

d`

dx`(x2 � 1)`

2``!dx =

(�i)`

2

Z 1

�1P`(x) e

izx dx (9.83)

(exercise 9.25) that contains Rodrigues’s formula (8.8) for the Legendre poly-nomial P`(x). With z = kr and x = cos ✓, this formula

i` j`(kr) =1

2

Z 1

�1P`(cos ✓)e

ikr cos ✓ d cos ✓ (9.84)

and the Fourier-Legendre expansion (8.31) give

eikr cos ✓ =1X

`=0

2`+ 1

2P`(cos ✓)

Z 1

�1P`(cos ✓

0) eikr cos ✓0d cos ✓0

=1X

`=0

(2`+ 1)P`(cos ✓) i` j`(kr).

(9.85)

If ✓,� and ✓0,�0 are the polar angles of the vectors r and k, then by usingthe addition theorem (8.122) we get the plane-wave expansion

eik·r =1X

`=0

4⇡ i` j`(kr)Y`,m(✓,�)Y ⇤`,m(✓0,�0). (9.86)

The series expansion (9.1) for Jn and the definition (9.72) of j` give us forsmall |⇢| ⌧ 1 the approximation

j`(⇢) ⇡` ! (2⇢)`

(2`+ 1)!=

⇢`

(2`+ 1)!!. (9.87)

To see how j`(⇢) behaves for large |⇢| � 1, we use Rayleigh’s formula(9.73) to compute j1(⇢) and notice that the derivative d/d⇢

j1(⇢) = � d

d⇢

✓sin ⇢

⇢

◆= � cos ⇢

⇢+

sin ⇢

⇢2(9.88)


0 2 4 6 8 10 12−0.5

0

0.5

ρ

j1(ρ

)

The Spherical Besse l Function j1(ρ ) for ρ ≪ 1 and ρ ≫ 1

0 2 4 6 8 10 12−0.5

0

0.5

ρ

j2(ρ

)

The Spherical Besse l Function j2(ρ ) for ρ ≪ 1 and ρ ≫ 1

Figure 9.3 Top: Plot of j1(⇢) (solid curve) and its approximations ⇢/3 forsmall ⇢ (9.87, dashes) and sin(⇢�⇡/2)/⇢ for big ⇢ (9.89, dot-dash). Bottom:Plot of j2(⇢) (solid curve) and its approximations ⇢2/15 for small ⇢ (9.87,dashed) and sin(⇢ � ⇡)/⇢ for big ⇢ (9.89, dot-dash). The values of ⇢ atwhich j`(⇢) = 0 are the zeros or roots of j`; we use them to fit boundaryconditions.

adds a factor of 1/⇢ when it acts on 1/⇢ but not when it acts on sin ⇢. Thusthe dominant term is the one in which all the derivatives act on the sine,and so for large |⇢| � 1, we have approximately

j`(⇢) = (�1)`⇢`✓1

⇢

d

d⇢

◆`✓sin ⇢

⇢

◆⇡ (�1)`

⇢

d` sin ⇢

d⇢`=

sin (⇢� `⇡/2)

⇢(9.89)

with an error that falls o↵ as 1/⇢2. The quality of the approximation, whichis exact for ` = 0, is illustrated for ` = 1 and 2 in Fig. 9.3.

Example 9.5 (Partial Waves). Spherical Bessel functions occur in the wavefunctions of free particles with well-defined angular momentum.

The hamiltonian H0 = p2/2m for a free particle of mass m and the square

L2 of the orbital angular-momentum operator are both invariant under ro-


tations; thus they commute with the orbital angular momentum operator L.Since the operators H0, L

2, and Lz commute with each other, simultaneouseigenstates |k, `,mi of these compatible operators (section 1.31) exist

H0 |k, `,mi = p2

2m|k, `,mi =

(~ k)22m

|k, `,mi (9.90)

L2 |k, `,mi = ~2 `(`+ 1) |k, `,mi and Lz |k, `,mi = ~m |k, `,mi.

By (9.68–9.71), their wave functions are products of spherical Bessel func-tions and spherical harmonics (8.111)

hr|k, `,mi = hr, ✓,�|k, `,mi =r

2

⇡k j`(kr)Y`,m(✓,�). (9.91)

They satisfy the normalization condition

hk, `,m|k0, `0,m0i = 2kk0

⇡

Z 1

0j`(kr)j`(k

0r) r2 dr

ZY ⇤`,m(✓,�)Y`0,m0(✓,�) d⌦

= �(k � k0) �`,`0 �m,m0 (9.92)

and the completeness relation

1 =

Z 1

0dk

1X

`=0

X

m=�`

|k, `,mihk, `,m|. (9.93)

Their inner products with an eigenstate |k0i of a free particle of momentump0 = ~k0 are

hk, `,m|k0i = i`

k�(k � k0)Y ⇤

`,m(✓0,�0) (9.94)

in which the polar coordinates of k0 are ✓0,�0.Using the resolution (9.93) of the identity operator and the inner-product

formulas (9.91 & 9.94), we recover the expansion (9.86)

eik0·r

(2⇡)3/2= hr|k0i =

Z 1

0dk

1X

`=0

X

m=�`

hr|k, `,mihk, `,m|k0i

=1X

`=0

r2

⇡i` j`(kr) Y`,m(✓,�) Y ⇤

`,m(✓0,�0).

(9.95)

The small kr approximation (9.87), the definition (9.91), and the normal-ization (8.114) of the spherical harmonics tell us that the probability that a


particle with angular momentum ~` about the origin has r = |r| ⌧ 1/k is

P (r) =2k2

⇡

Z r

0j2` (kr

0)r02dr0 ⇡ 2

⇡[(2`+ 1)!!]2

Z r

0(kr)2`+2dr =

(4`+ 6)(kr)2`+3

⇡[(2`+ 3)!!]2k(9.96)

which is very small for big ` and tiny k. So a short-range potential can onlya↵ect partial waves of low angular momentum. When physicists found thatnuclei scattered low-energy hadrons into s-waves, they knew that the rangeof the nuclear force was short, about 10�15m.If the potential V (r) that scatters a particle is of short range, then at big

r the radial wave function u`(r) of the scattered wave should look like thatof a free particle (9.95) which by the big kr approximation (9.89) is

u(0)` (r) = j`(kr) ⇡sin(kr � `⇡/2)

kr=

1

2ikr

hei(kr�`⇡/2) � e�i(kr�`⇡/2)

i.

(9.97)

Thus at big r the radial wave function u`(r) di↵ers from u(0)` (r) only by aphase shift �`

u`(r) ⇡sin(kr � `⇡/2 + �`)

kr=

1

2ikr

hei(kr�`⇡/2+�`) � e�i(kr�`⇡/2+�`)

i.

(9.98)The phase shifts determine the cross-section � to be (Cohen-Tannoudjiet al., 1977, chap. VIII)

� =4⇡

k2

1X

`=0

(2`+ 1) sin2 �`. (9.99)

If the potential V (r) is negligible for r > r0, then for momenta k ⌧ 1/r0the cross-section is � ⇡ 4⇡ sin2 �0/k2.

Example 9.6 (Quantum dots). The active region of some quantum dots isa CdSe sphere whose radius a is less than 2 nm. Photons from a laser exciteelectron-hole pairs which fluoresce in nanoseconds.

I will model a quantum dot simply as an electron trapped in a sphere ofradius a. Its wave function (r, ✓,�) satisfies Schrodinger’s equation

� ~22m

4 + V = E (9.100)

with the boundary condition (a, ✓,�) = 0 and the potential V constantand negative for r < a and infinitely positive for r > a. With k2 = 2m(E �V )/~2 = z2`,n/a

2, the unnormalized eigenfunctions are

n,`,m(r, ✓,�) = j`(z`,nr/a)Y`,m(✓,�) ✓(a� r) (9.101)


in which the Heaviside function ✓(a � r) makes vanish for r > a, and `and m are integers with �` m ` because must be single valued forall angles ✓ and �.The zeros z`,n of j`(x) fix the energy levels as En,`,m = (~z`,n/a)2/2m+V .

For j0(x) = sinx/x, they are z0,n = n⇡. So En,0,0 = (~n⇡/a)2/2m + V . Ifthe coupling to a photon is via a term like p · A, then one expects �` = 1.The energy gap from the n, ` = 1 state to the n = 1, ` = 0 ground state is

�En = En,1,0 � E1,0,0 = (z21,n � ⇡2)~2

2ma2. (9.102)

Inserting factors of c2 and using ~c = 197 eV nm, and mc2 = 0.511 MeV, wefind from the zero z1,2 = 7.72525 that �E2 = 1.89 (nm/a)2 eV, which is redlight if a = 1 nm. The next zero z1,3 = 10.90412 gives �E3 = 4.14 (nm/a)2

eV, which is in the visible if 1.2 < a < 1.5 nm. The Mathematica commandDo[Print[N[BesselJZero[1.5, k]]], {k, 1, 5, 1}] gives the first five zeros of j1(x)to 6 significant figures.

9.3 Bessel Functions of the Second Kind

In section 7.5, we derived integral representations (7.55 & 7.56) for the

Hankel functions H(1)� (z) and H(2)

� (z) for Re z > 0. One may analyticallycontinue them (Courant and Hilbert, 1955, chap. VII) to the upper halfz-plane

H(1)� (z) =

1

⇡ie�i�/2

Z 1

�1eiz coshx��x dx Imz � 0 (9.103)

and to the lower half z-plane

H(2)� (z) = � 1

⇡ie+i�/2

Z 1

�1e�iz coshx��x dx Imz 0. (9.104)

When both z = ⇢ and � = ⌫ are real, the two Hankel functions are complexconjugates of each other

H(1)⌫ (⇢) = H(2)⇤

⌫ (⇢). (9.105)

Hankel functions, called Bessel functions of the third kind, are linearcombinations of Bessel functions of the first J�(z) and second Y�(z) kind

H(1)� (z) = J�(z) + iY�(z)

H(2)� (z) = J�(z)� iY�(z).

(9.106)

9.3 Bessel Functions of the Second Kind 401

Bessel functions of the second kind also are called Neumann functions;the symbols Y�(z) = N�(z) refer to the same function. They are infinite atz = 0 as illustrated in Fig. 9.4.When z = ix is imaginary, we get the modified Bessel functions

I↵(x) = i�↵J↵(ix) =1X

m=0

1

m!�(m+ ↵+ 1)

⇣x2

⌘2m+↵

K↵(x) =⇡

2i↵+1H(1)

↵ (ix) =

Z 1

0e�x cosh t cosh↵t dt.

(9.107)

Some simple cases are

I�1/2(z) =

r2

⇡zcosh z, I1/2(z) =

r2

⇡zsinh z, and K1/2(z) =

r⇡

2ze�z.

(9.108)

When do we need to use these functions? If we are representing functionsthat are finite at the origin ⇢ = 0, then we don’t need them. But if the point⇢ = 0 lies outside the region of interest or if the function we are representingis infinite at that point, then we do need the Y⌫(⇢)’s.

Example 9.7 (Coaxial Wave Guides). An ideal coaxial wave guide is per-fectly conducting for ⇢ < r0 and ⇢ > r, and the waves occupy the regionr0 < ⇢ < r. Since points with ⇢ = 0 are not in the physical domain of theproblem, the electric field E(⇢,�) exp(i(kz � !t)) is a linear combination ofBessel functions of the first and second kinds with

Ez(⇢,�) = a Jn(p!2/c2 � k2 ⇢) + b Yn(

p!2/c2 � k2 ⇢) (9.109)

in the notation of example 9.3. A similar equation represents the magneticfield Bz. The fields E and B obey the equations and boundary conditionsof example 9.3 as well as

Ez(r0,�) = 0, E�(r0,�) = 0, and B⇢(r0,�) = 0 (9.110)

at ⇢ = r0. In TM modes with Bz = 0, one may show (exercise 9.28) that theboundary conditions Ez(r0,�) = 0 and Ez(r,�) = 0 can be satisfied if

Jn(x)Yn(vx)� Jn(vx)Yn(x) = 0 (9.111)

in which v = r/r0 and x =p!2/c2 � k2 r0. One can use the Matlab code

n = 0.; v = 10.;

f=@(x)besselj(n,x).*bessely(n,v*x)-besselj(n,v*x).*bessely(n,x)

x=linspace(0,5,1000);


0 2 4 6 8 10 12−1

−0.5

0

0.5

ρ

The Besse l Functions of the Second Kind Y 0(ρ ) , Y 1(ρ ) , Y 2(ρ )

2 4 6 8 10 12 14−1

−0.5

0

0.5

ρ

The Besse l Functions of the Second Kind Y 3(ρ ) , Y 4(ρ ) , Y 5(ρ )

Figure 9.4 Top: Y0(⇢) (solid curve), Y1(⇢) (dot-dash), and Y2(⇢) (dashed)for 0 < ⇢ < 12. Bottom: Y3(⇢) (solid curve), Y4(⇢) (dot-dash), and Y5(⇢)(dashed) for 2 < ⇢ < 14. The points at which Bessel functions cross the ⇢-axis are called zeros or roots; we use them to satisfy boundary conditions.

figure

plot(x,f(x)) % we use the figure to guess at the roots

grid on

options=optimset(’tolx’,1e-9);

fzero(f,0.3) % we tell fzero to look near 0.3

fzero(f,0.7)

fzero(f,1)

to find that for n = 0 and v = 10, the first three solutions are x0,1 = 0.3314,x0,2 = 0.6858, and x0,3 = 1.0377. Setting n = 1 and adjusting the guessesin the code, one finds x1,1 = 0.3941, x1,2 = 0.7331, and x1,3 = 1.0748. The

corresponding dispersion relations are !n,i(k) = cqk2 + x2n,i/r

20.

9.4 Spherical Bessel Functions of the Second Kind 403

9.4 Spherical Bessel Functions of the Second Kind

Spherical Bessel functions of the second kind are defined as

y`(⇢) =

r⇡

2⇢Y`+1/2(⇢) (9.112)

and Rayleigh formulas express them as

y`(⇢) = (�1)`+1⇢`✓

d

⇢ d⇢

◆` ✓cos ⇢

⇢

◆. (9.113)

The term in which all the derivatives act on the cosine dominates at big ⇢

y`(⇢) ⇡ (�1)`+1 1

⇢

d` cos ⇢

d⇢`= � cos (⇢� `⇡/2) /⇢. (9.114)

The second kind of spherical Bessel functions at small ⇢ are approximately

y`(⇢) ⇡ � (2`� 1)!!/⇢`+1. (9.115)

They all are infinite at x = 0 as illustrated in Fig. 9.5.

Example 9.8 (Scattering o↵ a Hard Sphere). In the notation of exam-ple 9.5, the potential of a hard sphere of radius r0 is V (r) = 1 ✓(r0 � r) inwhich ✓(x) = (x+ |x|)/2|x| is Heaviside’s function. Since the point r = 0 isnot in the physical region, the scattered wave function is a linear combina-tion of spherical Bessel functions of the first and second kinds

u`(r) = c` j`(kr) + d` y`(kr). (9.116)

The boundary condition u`(kr0) = 0 fixes the ratio v` = d`/c` of theconstants c` and d`. Thus for ` = 0, Rayleigh’s formulas (9.73 & 9.113) andthe boundary condition say that kr0 u0(r0) = c0 sin(kr0) � d0 cos(kr0) = 0or d0/c0 = tan kr0. The s-wave then is u0(kr) = c0 sin(kr�kr0)/(kr cos kr0),which tells us that the tangent of the phase shift is tan �0(k) = � kr0. By(9.99), the cross-section at low energy is � ⇡ 4⇡r20 or four times the classicalvalue.Similarly, one finds (exercise 9.29) that the tangent of the p-wave phase

shift is

tan �1(k) =kr0 cos kr0 � sin kr0cos kr0 � kr0 sin kr0

. (9.117)

For kr0 ⌧ 1, we have �1(k) ⇡ �(kr0)3/3; more generally, the `th phase shiftis �`(k) ⇡ �(kr0)2`+1/

�(2`+ 1)[(2`� 1)!!]2

for a potential of range r0 at

low energy k ⌧ 1/r0.


1 2 3 4 5 6 7 8 9 10 11 12−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

ρ

The Spherical Besse l Function y1(ρ ) for ρ ≪ 1 and ρ ≫ 1

1 2 3 4 5 6 7 8 9 10 11 12−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

ρ

The Spherical Besse l Function y2(ρ ) for ρ ≪ 1 and ρ ≫ 1

Figure 9.5 Top: Plot of y1(⇢) (solid curve) and its approximations �1/⇢2

for small ⇢ (9.115, dot-dash) and � cos(⇢�⇡/2)/⇢ for big ⇢ (9.114, dashed).Bottom: Plot of y2(⇢) (solid curve) and its approximations �3/⇢3 for small⇢ (9.115, dot-dash) and � cos(⇢�⇡)/⇢ for big ⇢ (9.114, dashed). The valuesof ⇢ at which y`(⇢) = 0 are the zeros or roots of y`; we use them to fitboundary conditions. All six plots run from ⇢ = 1 to ⇢ = 12.

Further Reading

A great deal is known about Bessel functions. Students may find Mathe-matical Methods for Physics and Engineering (Riley et al., 2006) as well asthe classics A Treatise on the Theory of Bessel Functions (Watson, 1995), ACourse of Modern Analysis (Whittaker and Watson, 1927, chap. XVII), andMethods of Mathematical Physics (Courant and Hilbert, 1955) of special in-terest. The NIST Digital Library of Mathematical Functions (dlmf.nist.gov)and the companionNIST Handbook of Mathematical Functions (Olver et al.,2010) are superb.

Exercises 405

Exercises

9.1 Show that the series (9.1) for Jn(⇢) satisfies Bessel’s equation (9.4).9.2 Show that the generating function exp(z(u � 1/u)/2) for Bessel func-

tions is invariant under the substitution u ! �1/u.9.3 Use the invariance of exp(z(u� 1/u)/2) under u ! �1/u to show that

J�n(z) = (�1)nJn(z).9.4 By writing the generating function (9.5) as the product of the expo-

nentials exp(zu/2) and exp(�z/2u), derive the expansion

exphz2

�u� u�1

�i=

1X

n=0

1X

m=�n

⇣z2

⌘m+n um+n

(m+ n)!

⇣�z

2

⌘n u�n

n!.

(9.118)9.5 From this expansion (9.118) of the generating function (9.5), derive the

power-series expansion (9.1) for Jn(z).9.6 In the formula (9.5) for the generating function exp(z(u � 1/u)/2),

replace u by exp i✓ and then derive the integral representation (9.6) forJn(z). Start with the interval [�⇡,⇡].

9.7 From the general integral representation (9.6) for Jn(z), derive the twointegral formulas (9.7) for J0(z).

9.8 Show that the integral representations (9.6 & 9.7) imply that for anyinteger n 6= 0, Jn(0) = 0, while J0(0) = 1.

9.9 By di↵erentiating the generating function (9.5) with respect to u andidentifying the coe�cients of powers of u, derive the recursion relation

Jn�1(z) + Jn+1(z) =2n

zJn(z). (9.119)

9.10 By di↵erentiating the generating function (9.5) with respect to z andidentifying the coe�cients of powers of u, derive the recursion relation

Jn�1(z)� Jn+1(z) = 2 J 0n(z). (9.120)

9.11 Change variables to z = ax and turn Bessel’s equation (9.4) into theself-adjoint form (9.11).

9.12 If y = Jn(ax), then equation (9.11) is (xy0)0 + (xa2 � n2/x)y = 0.Multiply this equation by xy0, integrate from 0 to b, and so show thatif ab = zn,m and Jn(zn,m) = 0, then

2

Z b

0xJ2

n(ax) dx = b2J 02n (zn,m) (9.121)

which is the normalization condition (9.14).


9.13 Use the expansion (9.15) of the delta function to find formulas for thecoe�cients ak in the expansion for 0 < ↵ < 1

f(x) =1X

k=1

ak x↵ Jn(zn,kx/b) (9.122)

of twice-di↵erentiable functions that vanish at at x = b.

9.14 Show that with � ⌘ z2/r2d, the change of variables ⇢ = zr/rd andu(r) = Jn(⇢) turns � (r u0)0 + n2 u/r = � r u into Bessel’s equation(9.30).

9.15 Use the formula (6.52) for the curl in cylindrical coordinates and thevacuum forms r⇥E = �B and r⇥B = E/c2 of the laws of Faradayand Maxwell-Ampere to derive the field equations (9.60).

9.16 Derive equations (9.61) from (9.60).

9.17 Show that Jn(p!2/c2 � k2 ⇢)ein�ei(kz�!t) is a traveling-wave solution

(9.57) of the wave equations (9.62).

9.18 Find expressions for the nonzero TM fields in terms of the formula(9.64) for Ez.

9.19 Show that the TE field Ez = 0 andBz = Jn(p!2/c2 � k2 ⇢)ein�ei(kz�!t)

will satisfy the boundary conditions (9.59) ifp!2/c2 � k2 r is a zero

z0n,m of J 0n.

9.20 Show that if ` is an integer and ifp!2/c2 � ⇡2`2/h2 r is a zero z0n,m of

J 0n, then the fields Ez = 0 and Bz = Jn(z0n,m⇢/r) e

in� sin(`⇡z/h) e�i!t

satisfy both the boundary conditions (9.59) at ⇢ = r and those (9.65) atz = 0 and h as well as the wave equations (9.62). Hint: Use Maxwell’sequations r⇥E = � B and r⇥B = E/c2 as in (9.60).

9.21 Show that the resonant frequencies of the TM modes of the cavity of

example 9.4 are !n,m,` = cqz2n,m/r2 + ⇡2`2/h2.

9.22 By setting n = ` + 1/2 and j` =p⇡/2xJ`+1/2, show that Bessel’s

equation (9.4) implies that the spherical Bessel function j` satisfies(9.68).

9.23 Show that Rayleigh’s formula (9.73) implies the recursion relation (9.74).

9.24 Use the recursion relation (9.74) to show by induction that the sphericalBessel functions j`(x) as given by Rayleigh’s formula (9.73) satisfy theirdi↵erential equation (9.71) which with x = kr is

� x2j00` � 2xj0` + `(`+ 1)j` = x2j`. (9.123)

Hint: start by showing that j0(x) = sin(x)/x satisfies this equation.This problem involves some tedium.

Exercises 407

9.25 Iterate the trick

d

zdz

Z 1

�1eizx dx =

i

z

Z 1

�1xeizx dx =

i

2z

Z 1

�1eizx d(x2 � 1)

= � i

2z

Z 1

�1(x2 � 1)deizx =

1

2

Z 1

�1(x2 � 1)eizxdx

(9.124)

to show that (Schwinger et al., 1998, p. 227)✓

d

zdz

◆` Z 1

�1eizx dx =

Z 1

�1

(x2 � 1)`

2``!eizx dx. (9.125)

9.26 Use the expansions (9.85 & 9.86) to show that the inner product of theket |ri that represents a particle at r with polar angles ✓ and � and theone |ki that represents a particle with momentum p = ~k with polarangles ✓0 and �0 is with k · r = kr cos ✓

hr|ki = 1

(2⇡)3/2eikr cos ✓ =

1

(2⇡)3/2

1X

`=0

(2`+ 1)P`(cos ✓) i` j`(kr)

=1

(2⇡)3/2eik·r =

r2

⇡

1X

`=0

i` j`(kr)Y`,m(✓,�)Y ⇤`,m(✓0,�0).

(9.126)

9.27 Show that (�1)`d` sin ⇢/d⇢` = sin(⇢�⇡`/2) and so complete the deriva-tion of the approximation (9.89) for j`(⇢) for big ⇢.

9.28 In the context of examples 9.3 and 9.7, show that the boundary con-ditions Ez(r0,�) = 0 and Ez(r,�) = 0 imply (9.111).

9.29 Show that for scattering o↵ a hard sphere of radius r0 as in example 9.8,the p-wave phase shift is given by (9.117).

Bessel Functions - University of New Mexicoquantum.phys.unm.edu/466-17/ch9.pdf · 9 Bessel...

Documents

Transcript of Bessel Functions - University of New Mexicoquantum.phys.unm.edu/466-17/ch9.pdf · 9 Bessel...