9 Bessel Functions

9.1 Bessel functions of the first kind Friedrich Bessel (1784–1846) invented functions for problems with circular symmetry. The most useful ones are defined for any integer n by the series zn z2 z4 J (z)= 1 + ... n 2nn! 2(2n + 2) 2 4(2n + 2)(2n + 4)  · (9.1) z n 1 ( 1)m z 2m = . 2 m!(m + n)! 2 m=0 ⇣ ⌘ X ⇣ ⌘ The first term of this series tells us that for small z 1 | |⌧ zn J (z) . (9.2) n ⇡ 2nn! The alternating signs in (9.1) make the waves plotted in Fig. 9.1, and we have for big z 1 the approximation (Courant and Hilbert, 1955, chap. VII) | | 2 n⇡ ⇡ 3/2 J (z) cos z + O( z ). (9.3) n ⇡ ⇡z 2 4 | | r ⇣ ⌘ The Jn(z) are entire transcendental functions. They obey Bessel’s equation d2J 1 dJ n2 n + n + 1 J = 0 (9.4) dz2 z dz z2 n ✓ ◆ (6.393) as one may show (exercise 9.1) by substituting the series (9.1) into the di↵erential equation (9.4). Their generating function is

z 1 exp (u 1/u) = un J (z) (9.5) 2 n n= h i X1 9.1 Bessel functions of the ﬁrst kind 383

The Bessel Functions J0(ρ), J1(ρ), and J2(ρ) 1

0.5

−0.5 0 2 4 6 8 10 12 ρ

The Bessel Functions J3(ρ), J4(ρ), and J5(ρ)

0.4

0.2

−0.2

−0.4 0 2 4 6 8 10 12 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 ⇢-axis are 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)

⇡ 1 n Jn(z)= cos(z sin ✓ n✓) d✓ = J n( z)=( 1) J n(z) (9.6) ⇡ Z0 for all complex z. For n = 0, this integral is (exercise 9.7) more simply

1 2⇡ 1 2⇡ J (z)= eiz cos ✓ d✓ = eiz sin ✓ d✓. (9.7) 0 2⇡ 2⇡ Z0 Z0 These integrals (exercise 9.8) give J (0) = 0 for n = 0, and J (0) = 1. n 6 0 By di↵erentiating the generating function (9.5) with respect to u and 384 Bessel Functions identifying the coecients of powers of u, one ﬁnds the recursion relation 2n Jn 1(z)+Jn+1(z)= Jn(z). (9.8) z Similar reasoning after taking the z derivative gives (exercise 9.10)

Jn 1(z) Jn+1(z)=2Jn0 (z). (9.9) By using the gamma function (section 5.15), one may extend Bessel’s equation (9.4) and its solutions Jn(z) to non-integral values of n

z ⌫ 1 ( 1)m z 2m J (z)= . (9.10) ⌫ 2 m!(m + ⌫ + 1) 2 m=0 ⇣ ⌘ X ⇣ ⌘ The self-adjoint form (6.392) of Bessel’s equation (9.4) with z = kx is (exercise 9.11) d d n2 x J (kx) + J (kx)=k2xJ (kx). (9.11) dx dx n x n n ✓ ◆ 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) b b 0= p(J v0 J 0 v) = x(J (kx)v0(kx) J 0 (kx)v(kx)) (9.12) n n 0 n n 0 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. Thus since p(x)=x, the terms in this boundary condition vanish at x = 0 as long as the domain consists of functions v(kx) that are twice di↵erentiable on the interval [0,b]. To make these terms vanish at x = b, we require that Jn(kb) = 0 and that v(kb) = 0. Thus kb must be a zero zn,m of Jn(z), and so Jn(kb)=Jn(zn,m) = 0. With k = zn,m/b, Bessel’s equation (9.11) is d d n2 z2 x J (z x/b) + J (z x/b)= n,m xJ (z x/b) . (9.13) dx dx n n,m x n n,m b2 n n,m ✓ ◆ 2 2 2 For fixed n, the eigenvalue k = zn,m/b is di↵erent for each positive integer m. Moreover as m , the zeros z of J (x) rise as m⇡ as one might !1 n,m n expect since the leading term of the asymptotic form (9.3) of Jn(x) is proportional to cos(x n⇡/2 ⇡/4) which has zeros at m⇡+(n+1)⇡/2+⇡/4. It follows that the eigenvalues k2 (m⇡)2/b2 increase without limit as m ⇡ !1 in accordance with the general result of section 6.36. It follows then from the 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 9.1 Bessel functions of the first kind 385 complete in the mean, orthogonal, and normalizable on the interval [0,b] with weight function ⇢(x)=x

b 2 2 zn,mx zn,m0 x b 2 b 2 xJn Jn dx = m,m0 Jn0 (zn,m)=m,m0 Jn+1(zn,m) 0 b b 2 2 Z ⇣ ⌘ ⇣ ⌘ (9.14) and a normalization constant (exercise 9.12) that depends upon the ﬁrst derivative of the Bessel function or the square of the next Bessel function at the 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)

↵ 1 ↵ 2 1 Jn(zn,kx/b)Jn(zn,ky/b) (x y)= 2 x y /b . (9.15) J 2 (z ) k=1 n+1 n,k X For n = 0 and b = 1, this sum is

↵ 1 ↵ 1 J0(z0,kx)J0(z0,ky) (x y)=2x y 2 . (9.16) J1 (z0,k) Xk=1 Figure 9.2 plots the sum of the ﬁrst 10,000 terms of this series for ↵ =0 and 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 the completeness of the eigenfunctions Jn(zn,kx/b) for ﬁxed n and lets us write any twice di↵erentiable function f(x) that vanishes at x = b as the Fourier series

↵ 1 f(x)=x ak Jn(zn,kx/b) (9.17) Xk=1 where b 2 1 ↵ a = y J (z y/b) f(y) dy. (9.18) k b2 J 2 (z ) n n,k n+1 n,k Z0 The orthogonality relation on an inﬁnite interval is

(x y)=x 1 kJ (kx) J (ky) dk (9.19) n n Z0 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 386 Bessel Functions

10,000-Term Bessel Series for Dirac Delta Function

10000

8000

6000

4000 Series 0 J

2000

−2000

0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54 x

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 approximately 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-y plane 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 is 2 1+hx dx, and so the extra area of an element dx dy is p 1 dA 1+h2 1+h2 1 dx dy h2 + h2 dx dy. (9.20) ⇡ x y ⇡ 2 x y ⇣p q ⌘ The (nonrelativistic) kinetic energy of the area element is proportional to the square of its speed. So if is the surface tension and µ the surface mass density of the membrane, then to lowest order in d the action functional (sections 6.15 and 6.42) is µ S[h]= h2 h2 + h2 dx dy dt. (9.21) 2 t 2 x y Z h i 9.1 Bessel functions of the first kind 387

2 2 2 We minimize this action for h’s that vanish on the boundary x + y = rd

0=S[h]= [µh h (h h + h h )] dx dy dt. (9.22) t t x x y y Z Since (6.206) ht =(h)t, hx =(h)x, and hy =(h)y, we can integrate by parts and get

0=S[h]= [ µh + (h + h )] hdxdy dt (9.23) tt xx yy Z apart from a surface term proportional to h which vanishes because h =0 on the circumference of the membrane. The membrane therefore obeys the wave equation µh = (h + h ) h. (9.24) tt xx yy ⌘ 4

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

s v tt = 4 = !2. (9.25) s µ v The eigenvalues of the Helmholtz equation v = v give the angular 4 frequencies as ! = /µ.Thetimedependencethenis

p s(t)=a sin( /µ (t t )) (9.26) 0 in which a and t0 are constants. p In polar coordinates, Helmholtz’s equation is separable (6.66–6.69)

1 2 v = v r v r v = v. (9.27) 4 rr r ✓✓ We set v(r, ✓)=u(r)h(✓) and ﬁnd u h r 1u h r 2uh = uh. After 00 0 00 multiplying both sides by r2/uh, we get u u h r2 00 + r 0 + r2 = 00 = n2. (9.28) u u h The general solution for h then is h(✓)=b sin(n(✓ ✓ )) in which b and 0 ✓0 are constants and n must be an integer so that h is single valued on the circumference of the membrane. The displacement u(r) must obey the di↵erential equation

2 ru0 0 + n u/r = ru (9.29) and vanish at r = rd, the rim of the drum. Therefore it is an eigenfunction 388 Bessel Functions u(r)=Jn(zn,mr/rd) of the self-adjoint di↵erential equation (9.13) d d n2 z2 r J (z r/r ) + J (z r/r )= n,m xJ (z r/r ) (9.30) dr dr n n,m d r n n,m d r2 n n,m d ✓ ◆ d 2 2 with an eigenvalue = zn,m/rd in which zn,m is the mth zero of the nth Bessel function, J (z ) = 0. For each integer n 0, there are infinitely n n,m many zeros zn,m at which the Bessel function vanishes, and for each zero the frequency is ! =(zn,m/rd) /µ. The general solution to the wave equation (9.24) of the membrane thus is p 1 1 z r h(r, ✓, t)= c sin n,m (t t ) sin [n(✓ ✓ )] J z n,m r µ 0 0 n n,m r n=0 m=1  d r ✓ d ◆ X X (9.31) in which t0 and ✓0 can depend upon n and m. For each n and big m,the zero zn,m is near m⇡ +(n + 1)⇡/2+⇡/4. We learned in section 6.5 that in three dimensions Helmholtz’s equation V = ↵2 V separates in cylindrical coordinates (and in several other co- 4 ordinate systems). That is, the function V (⇢,, z)=B(⇢)()Z(z) satisfies the equation 1 1 V = (⇢V ) + V + ⇢V = ↵2 V (9.32) 4 ⇢ ,⇢ ,⇢ ⇢ , ,zz  if B(⇢) obeys Bessel’s equation d dB ⇢ ⇢ + (↵2 + k2)⇢2 n2 B = 0 (9.33) d⇢ d⇢ ✓ ◆ and and Z respectively satisfy d2 d2Z = n2() and = k2Z(z) (9.34) d2 dz2 or if B(⇢) obeys the Bessel equation d dB ⇢ ⇢ + (↵2 k2)⇢2 n2 B = 0 (9.35) d⇢ d⇢ ✓ ◆ and and Z satisfy d2 d2Z = n2() and = k2Z(z). (9.36) d2 dz2 In the first case (9.33 & 9.34), the solution V is

2 2 in kz Vk,n(⇢,, z)=Jn( ↵ + k ⇢)e± e± (9.37) p 9.1 Bessel functions of the ﬁrst kind 389 while in the second case (9.35 & 9.36), it is

2 2 in ikz V (⇢,, z)=J ( ↵ k ⇢)e± e± . (9.38) k,n n In both cases, n must be an integerp if the solution is to be single valued on the full range of from 0 to 2⇡. When ↵ = 0, the Helmholtz equation reduces to the Laplace equation V = 0 of electrostatics which the simpler functions 4 in kz in ikz Vk,n(⇢,, z)=Jn(k⇢)e± e± and Vk,n(⇢,, z)=Jn(ik⇢)e± e± (9.39) satisfy. ⌫ The product i J⌫(ik⇢) is real and is known as the modified Bessel function ⌫ I (k⇢) i J (ik⇢). (9.40) ⌫ ⌘ ⌫ It occurs in various solutions of the di↵usion equation D = ˙.The 4 function V (⇢,, z)=B(⇢)()Z(z) satisfies 1 1 V = (⇢V ) + V + ⇢V = ↵2 V (9.41) 4 ⇢ ,⇢ ,⇢ ⇢ , ,zz  if B(⇢) obeys Bessel’s equation d dB ⇢ ⇢ (↵2 k2)⇢2 + n2 B = 0 (9.42) d⇢ d⇢ ✓ ◆ and and Z respectively satisfy d2 d2Z = n2() and = k2Z(z) (9.43) d2 dz2 or if B(⇢) obeys the Bessel equation d dB ⇢ ⇢ (↵2 + k2)⇢2 + n2 B = 0 (9.44) d⇢ d⇢ ✓ ◆ and and Z satisfy d2 d2Z = n2() and = k2Z(z). (9.45) d2 dz2 In the first case (9.42 & 9.43), the solution V is

2 2 in kz V (⇢,, z)=I ( ↵ k ⇢)e± e± (9.46) k,n n while in the second case (9.44 & 9.45),p it is

2 2 in ikz Vk,n(⇢,, z)=In( ↵ + k ⇢)e± e± . (9.47) p 390 Bessel Functions In both cases, n must be an integer if the solution is to be single valued on the full range of from 0 to 2⇡. Example 9.2 (Charge near a Membrane). We will use ⇢ to denote the density of free charges—those that are free to move in or out of a dielectric medium, as opposed to those that are part of the medium, bound in it by molecular forces. The time-independent Maxwell equations are Gauss’s law D = ⇢ for the divergence of the electric displacement D, and the static r· form E = 0 of Faraday’s law which implies that the electric field E is r⇥ the gradient of an electrostatic potential E = V . r Across an interface between two dielectrics with normal vector nˆ,the tangential electric field is continuous, nˆ E = nˆ E , while the normal ⇥ 2 ⇥ 1 component of the electric displacement jumps by the surface density of free charge, nˆ (D D )=. In a linear dielectric, the electric displacement D · 2 1 is 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 of ⇥ nanometers, the membrane is flat. We will take it to be a slab extending to infinity in the x and y directions. If the interface between the lipid bilayer and the extracellular salty water is at z = 0, then the cytosol extends thousands of nm down from z = t = 5 nm. We will ignore the phosphate head groups and set the permittivity ✏` of the lipid bilayer to twice that of the vacuum ✏ 2✏ ; the permittivity of the extracellular water and that of the ` ⇡ 0 cytosol are ✏ ✏ 80✏ . w ⇡ c ⇡ 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 cylindrically in kz symmetric about the z-axis, so V = V (⇢, z). The functions Jn(k⇢) e e± form a complete set of solutions (9.39) of Laplace’s equation, but due to the kz symmetry, we only need the n = 0 functions J0(k⇢) e± . Since there are no free charges in the lipid bilayer or in the cytosol, we may express the potential in the lipid bilayer V` and in the cytosol Vc as

1 kz kz V`(⇢, z)= dk J0(k⇢) m(k) e + f(k) e 0 Z h i (9.48) 1 kz Vc(⇢, z)= dk J0(k⇢) d(k) e . Z0 The Green’s function (3.112) for Poisson’s equation G(x)=(3)(x)in 4 cylindrical coordinates is (5.147)

1 1 1 dk k z G(x)= = = J0(k⇢) e | |. (9.49) 4⇡ x 4⇡ ⇢2 + z2 0 4⇡ | | Z p 9.1 Bessel functions of the ﬁrst kind 391 Thus we may expand the potential in the salty water as

1 q k z h kz V (⇢, z)= dk J (k⇢) e | | + u(k) e . (9.50) w 0 4⇡✏ Z0  w Using nˆ E = nˆ E and nˆ (D D )=,suppressingk, and setting ⇥ 2 ⇥ 1 · 2 1 qe kh/4⇡✏ and y = e2kt, we get four equations ⌘ w m + f u = and ✏`m ✏`f + ✏wu = ✏w (9.51) ✏ m ✏ yf ✏ d = 0 and m + yf d =0. ` ` c In terms of the abbreviations ✏w` =(✏w + ✏`) /2 and ✏c` =(✏c + ✏`) /2 as well as p =(✏ ✏ )/(✏ + ✏ ) and p =(✏ ✏ )/(✏ + ✏ ), the solutions are w ` w ` 0 c ` c ` p p /y ✏ 1 u(k)= 0 and m(k)= w 1 pp /y ✏ 1 pp /y 0 w` 0 (9.52) ✏ p /y ✏ ✏ 1 f(k)= w 0 and d(k)= w ` . ✏ 1 pp /y ✏ ✏ 1 pp /y w` 0 w` c` 0 Inserting these solutions into the Bessel expansions (9.48) for the potentials, expanding their denominators

1 1 n 2nkt = (pp0) e (9.53) 1 pp /y 0 0 X and using the integral (9.49), we ﬁnd that the potential Vw in the extracellular water of a charge q at (0, 0,h) in the water is

2 n 1 q 1 p 1 p0 1 p (pp0) Vw(⇢, z)= + 4⇡✏ r 2 2 2 2 w " ⇢ +(z + h) n=1 ⇢ +(z +2 nt + h) # X (9.54) p p in which r = ⇢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 bilayer p ` is n n n+1 q 1 (pp0) p p0 V`(⇢, z)= 4⇡✏w` " ⇢2 +(z 2nt h)2 ⇢2 +(z + 2(n + 1)t + h)2 # n=0 X (9.55) p p and that in the cytosol is

n q✏` 1 (pp0) Vc(⇢, z)= . (9.56) 4⇡✏ ✏ 2 2 w` c` n=0 ⇢ +(z 2nt h) X These potentials are the same as thosep of example 4.19, but this derivation is much simpler and less error prone than the method of images. 392 Bessel Functions Since p =(✏ ✏ )/(✏ +✏ ) > 0, the principal image charge pq at (0, 0, w ` w ` h) has the same sign as the charge q and so contributes a positive term proportional to pq2 to the energy. So a lipid slab repels a nearby charge in water 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 groups which are electric dipoles. These electric dipoles cause the cell membrane to weakly 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 in i(kz !t) in i(kz !t) E e e and B e e (9.57) in which E and B depend upon ⇢

E = E⇢⇢ˆ + Eˆ + Ezzˆ and B = B⇢⇢ˆ + Bˆ + Bzzˆ (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 wave guide the parallel components of E and the normal component of B must vanish 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 denote derivatives as in @(⇢E )/@⇢ (⇢E ) and so forth. In this notation, the ⌘ ,⇢ vacuum forms E = B˙ and B = E˙ /c2 of the Faraday and r ⇥ r ⇥ Maxwell-Ampère laws give us (exercise 9.15) the field equations E /⇢ ikE = i!B inB /⇢ ikB = i!E /c2 z, ⇢ z ⇢ ikE E = i!B ikB B = i!E /c2 (9.60) ⇢ z,⇢ ⇢ z,⇢ [(⇢E ) inE ] /⇢ = i!B [(⇢B ) inB ] /⇢ = i!E /c2. ,⇢ ⇢ z ,⇢ ⇢ z Solving them for the ⇢ and components of E and B in terms of their z components (exercise 9.16), we find ikE + n!B /⇢ nkE /⇢ + i!B E = z,⇢ z E = z z,⇢ ⇢ k2 !2/c2 k2 !2/c2 (9.61) ikB n!E /c2⇢ nkB /⇢ i!E /c2 B = z,⇢ z B = z z,⇢ . ⇢ k2 !2/c2 k2 !2/c2 The fields Ez and Bz obey the wave equations (11.90, exercise 6.6) E = E¨ /c2 = !2E /c2 and B = B¨ /c2 = !2B /c2. (9.62) 4 z z z 4 z z z 9.1 Bessel functions of the first kind 393 Because their z-dependence (9.57) is periodic, they are (exercise 9.17) linear combinations of J ( !2/c2 k2 ⇢)einei(kz !t). n Modes with B = 0 are transverse magnetic or TM modes. For them z p the boundary conditions (9.59) will be satisfied if !2/c2 k2 r is a zero z of J .Sothefrequency! (k) of the n, m TM mode is n,m n n,m p

2 2 2 !n,m(k)=c k + zn,m/r . (9.63) q Since ﬁrst zero of a Bessel function is z 2.4048, the minimum frequency 0,1 ⇡ ! (0) = cz /r 2.4048 c/r occurs for n = 0 and k = 0. If the radius of 0,1 0,1 ⇡ the wave-guide is r =1cm,then!0,1(0)/2⇡ is about 11 GHz, which is a microwave 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

1 1 1 zn,m ⇢ in Ez(⇢,, z, t)= cn,m(k) Jn e exp i [kz !n,m(k)t] dk. 0 r n=0 m=1 Z ⇣ ⌘ X X (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) if !2/c2 k2 r 2 2 2 is a zero zn,m0 of Jn0 . Their frequencies are !n,m(k)=c k +pzn,m0 /r .Since first zero of a first derivative of a Bessel function is z0 q 1.8412, the mini- 1,1 ⇡ mum frequency ! (0) = cz /r 1.8412 c/r occurs for n = 1 and k = 0. If 1,1 10 ,1 ⇡ the radius of the wave-guide is r =1cm,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, perfectly conducting cylindrical cavity of radius r and height h are like those of a cylindrical 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 if !2/c2 ⇡2`2/h2 r is a zero z of J ,thentheTEﬁeldsE = 0 and n,m0 n0 z p in i!t Bz = Jn(zn,m0 ⇢/r) e sin(⇡`z/h) e (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 2 2 2 2 2 the resonant TE modes then are !n,m,` = c zn,m0 /r + ⇡ ` /h . q 394 Bessel Functions

The TM modes are Bz = 0 and

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

2 2 2 2 2 with resonant frequencies !n,m,` = c zn,m/r + ⇡ ` /h . q

9.2 Spherical Bessel functions of the ﬁrst kind

If in Bessel’s equation (9.4), one sets n = ` +1/2 and j` = ⇡/2xJ`+1/2, then one may show (exercise 9.22) that p 2 2 x j00(x)+2xj0 (x)+[x `(` + 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 V = k2V in 4 spherical coordinates

2 2 r V (r Rk,0 `)0 (sin ✓ ⇥`0 ,m)0 00 2 2 4 = + + 2 = k r . (9.69) V Rk,` sin ✓ ⇥`,m sin ✓ Thus if ()=eim so that = m2 , and if ⇥ satisﬁes the m m00 m `,m associated Legendre equation (8.92)

2 2 sin ✓ sin ✓ ⇥0 0 +[`(` + 1) sin ✓ m ]⇥ = 0 (9.70) `,m `,m then the product V(r, ✓,)=Rk,`(r)⇥`,m(✓)m() will obey (9.69) because in view of (9.68) the radial function Rk,`(r)=j`(kr) satisﬁes 2 2 2 (r R0 )0 +[k r `(` + 1)]R =0. (9.71) k,` k,` In terms of the spherical harmonic Y`,m(✓,)=⇥`,m(✓)m(), the solution is V (r, ✓,)=j`(kr) Y`,m(✓,). Rayleigh’s formula gives the spherical Bessel function ⇡ j (x) J (x) (9.72) ` ⌘ 2x `+1/2 r as the `th derivative of sin x/x

1 d ` sin x j (x)=( 1)` x` (9.73) ` x dx x ✓ ◆ ✓ ◆ (Lord Rayleigh (John William Strutt) 1842–1919). In particular, j0(x)= 9.2 Spherical Bessel functions of the ﬁrst kind 395 sin x/x and j (x)=sinx/x2 cos x/x. Rayleigh’s formula leads to the re- 1 cursion relation (exercise 9.23) ` j (x)= j (x) j0 (x) (9.74) `+1 x ` ` with which one can show (exercise 9.24) that the spherical Bessel functions as 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) 2 2 2 (r j0 )0 + `(` + 1)j = k r j (9.75) ` ` ` 2 2 with eigenvalue k and weight function ⇢ = r .Ifj`(z`,n) = 0, then the functions j`(kr)=j`(z`,nr/a) vanish at r = a and form an orthogonal basis

a a3 j (z r/a) j (z r/a) r2 dr = j2 (z ) (9.76) ` `,n ` `,m 2 `+1 `,n n,m Z0 for a self-adjoint system on the interval [0,a]. Moreover, since as n !1 the eigenvalues k2 = z2 /a2 [(n + `/2)⇡]2/a2 , the eigenfunctions `,n `,n ⇡ !1 j`(z`,nr/a) also are complete in the mean (section 6.37) as shown by the expansion of the delta function ↵ 2 ↵ 2r r0 1 j`(z`,nr/a) j`(z`,mr0/a) (r r0)= (9.77) a3 j2 (z ) n=1 `+1 `,n X for 0 ↵ 2. This formula lets us expand any twice-di↵erentiable function   f(r) that vanishes at r = a as

↵ 1 f(r)=r an j`(z`,nr/a) (9.78) n=1 X where a 2 2 ↵ a = r0 j (z r0/a) f(r0) dr0. (9.79) n a3 j2 (z ) ` `,m `+1 `,n Z0 On an inﬁnite interval, the delta-function formulas are for positive values of k and k0 2kk0 1 2 (k k0)= j (kr) j (k0r) r dr (9.80) ⇡ ` ` Z0 and for r, r0 > 0

2rr0 1 2 (r r0)= j (kr) j (kr0) k dk. (9.81) ⇡ ` ` Z0 396 Bessel Functions

If we write the spherical Bessel function j0(x) as the integral 1 sin z 1 izx j0(z)= = e dx (9.82) z 2 1 Z and use Rayleigh’s formula (9.73), we may ﬁnd an integral for j`(z)

` ` 1 ` ` 1 d sin z ` ` 1 d 1 izx j`(z)=( 1) z =( 1) z e dx z dz z z dz 2 1 ✓ ◆ ✓ ◆ ✓ ◆ Z ` 1 2 ` ` 1 2 ` ` z (1 x ) izx ( i) (1 x ) d izx = ` e dx = ` ` e dx 2 1 2 `! 2 1 2 `! dx Z Z ` 1 ` 2 ` ` 1 ( i) izx d (x 1) ( i) izx = e ` ` dx = P`(x) e dx (9.83) 2 1 dx 2 `! 2 1 Z Z (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 1 ` 1 ikr cos ✓ i j`(kr)= P`(cos ✓)e d cos ✓ (9.84) 2 1 Z and the Fourier-Legendre expansion (8.31) give

1 2` +1 1 ikr cos ✓ ikr cos ✓0 e = P`(cos ✓) P`(cos ✓0) e d cos ✓0 2 1 X`=0 Z (9.85) 1 ` = (2` + 1) P`(cos ✓) i j`(kr). X`=0 If ✓, and ✓0,0 are the polar angles of the vectors r and k,thenbyusing the addition theorem (8.122) we get the plane-wave expansion

ik r 1 ` e · = 4⇡i j`(kr) Y`,m(✓,) Y`⇤,m(✓0,0). (9.86) X`=0

The series expansion (9.1) for Jn and the deﬁnition (9.72) of j` give us for small ⇢ 1 the approximation | |⌧ ` !(2⇢)` ⇢` j (⇢) = . (9.87) ` ⇡ (2` + 1)! (2` + 1)!! 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⇢ d sin ⇢ cos ⇢ sin ⇢ j (⇢)= = + (9.88) 1 d⇢ ⇢ ⇢ ⇢2 ✓ ◆ 9.2 Spherical Bessel functions of the ﬁrst kind 397

The Spherical Bessel Function j1(ρ)forρ ≪ 1andρ ≫ 1 0.5 ) ρ

( 0 1 j

−0.5 0 2 4 6 8 10 12 ρ

The Spherical Bessel Function j2(ρ)forρ ≪ 1andρ ≫ 1 0.5 ) ρ

( 0 2 j

−0.5 0 2 4 6 8 10 12 ρ

Figure 9.3 Top: Plot of j1(⇢) (solid curve) and its approximations ⇢/3 for small ⇢ (9.87, dashes) and sin(⇢ ⇡/2)/⇢ for big ⇢ (9.89, dot-dash). Bottom: 2 Plot of j2(⇢) (solid curve) and its approximations ⇢ /15 for small ⇢ (9.87, dashed) and sin(⇢ ⇡)/⇢ for big ⇢ (9.89, dot-dash). The values of ⇢ at which j`(⇢) = 0 are the zeros or roots of j`; we use them to ﬁt boundary conditions. adds a factor of 1/⇢ when it acts on 1/⇢ but not when it acts on sin ⇢.Thus the dominant term is the one in which all the derivatives act on the sine, and so for large ⇢ 1, we have approximately | | 1 d ` sin ⇢ ( 1)` d` sin ⇢ sin (⇢ `⇡/2) j (⇢)=( 1)`⇢` = (9.89) ` ⇢ d⇢ ⇢ ⇡ ⇢ d⇢` ⇢ ✓ ◆ ✓ ◆ with an error that falls o↵as 1/⇢2. The quality of the approximation, which is exact for ` = 0, is illustrated for ` = 1 and 2 in Fig. 9.3. Example 9.5 (Partial Waves). Spherical Bessel functions occur in the wave functions of free particles with well-deﬁned angular momentum. 2 The hamiltonian H0 = p /2m for a free particle of mass m and the square L2 of the orbital angular-momentum operator are both invariant under ro- 398 Bessel Functions tations; thus they commute with the orbital angular momentum operator L. 2 Since the operators H0, L , and Lz commute with each other, simultaneous eigenstates k, `, m of these compatible operators (section 1.31) exist | i p2 (~ k)2 H k, `, m = k, `, m = k, `, m (9.90) 0 | i 2m | i 2m | i 2 2 L k, `, m = ~ `(` + 1) k, `, m and Lz k, `, m = ~ m k, `, m . | i | i | i | i By (9.68–9.71), their wave functions are products of spherical Bessel functions and spherical harmonics (8.111)

2 r k, `, m = r, ✓, k, `, m = kj`(kr) Y`,m(✓,). (9.91) h | i h | i r⇡ They satisfy the normalization condition

2kk0 1 2 k, `, m k0,`0,m0 = j (kr)j (k0r) r dr Y ⇤ (✓,)Y (✓,) d⌦ h | i ⇡ ` ` `,m `0,m0 Z0 Z = (k k0) (9.92) `,`0 m,m0 and the completeness relation

` 1 1= 1dk k, `, m k, `, m . (9.93) | ih | Z0 `=0 m= ` X X

Their inner products with an eigenstate k0 of a free particle of momentum | i p0 = ~k0 are i` k, `, m k0 = (k k0) Y ⇤ (✓0,0) (9.94) h | i k `,m 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)

ik r ` e 0· 1 1 = r k0 = dk r k, `, m k, `, m k0 (2⇡)3/2 h | i h | ih | i Z0 `=0 m= ` X X (9.95) 1 2 ` = i j`(kr) Y`,m(✓,) Y`⇤,m(✓0,0). r⇡ X`=0 The small kr approximation (9.87), the deﬁnition (9.91), and the normalization (8.114) of the spherical harmonics tell us that the probability that a 9.2 Spherical Bessel functions of the ﬁrst kind 399 particle with angular momentum ~` about the origin has r = r 1/k is | |⌧ 2 r r 2`+3 2k 2 2 2 2`+2 (4` + 6)(kr) P (r)= j` (kr0)r0 dr0 2 (kr) dr = 2 ⇡ 0 ⇡ ⇡[(2` + 1)!!] 0 ⇡[(2` + 3)!!] k Z Z (9.96) which is very small for big ` and tiny k. So a short-range potential can only a↵ect partial waves of low angular momentum. When physicists found that nuclei scattered low-energy hadrons into s-waves, they knew that the range 15 of the nuclear force was short, about 10 m. 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 that of a free particle (9.95) which by the big kr approximation (9.89) is

(0) sin(kr `⇡/2) 1 i(kr `⇡/2) i(kr `⇡/2) u (r)=j (kr) = e e . ` ` ⇡ kr 2ikr h (9.97)i (0) Thus at big r the radial wave function u`(r) di↵ers from u` (r) only by a phase shift `

sin(kr `⇡/2+`) 1 i(kr `⇡/2+ ) i(kr `⇡/2+ ) u (r) = e ` e ` . ` ⇡ kr 2ikr h (9.98)i The phase shifts determine the cross-section to be (Cohen-Tannoudji et al., 1977, chap. VIII)

4⇡ 1 = (2` + 1) sin2 . (9.99) k2 ` X`=0 If the potential V (r) is negligible for r>r, then for momenta k 1/r 0 ⌧ 0 the cross-section is 4⇡ sin2 /k2. ⇡ 0 Example 9.6 (Quantum dots). The active region of some quantum dots is a CdSe sphere whose radius a is less than 2 nm. Photons from a laser excite electron-hole pairs which fluoresce in nanoseconds. I will model a quantum dot simply as an electron trapped in a sphere of radius a. Its wave function (r, ✓,) satisfies Schrödinger’s equation ~2 + V = E (9.100) 2m 4 with the boundary condition (a, ✓,) = 0 and the potential V constant and negative for ra.Withk2 =2m(E 2 2 2 V )/~ = z`,n/a , the unnormalized eigenfunctions are (r, ✓,)=j (z r/a) Y (✓,) ✓(a r) (9.101) n,`,m ` `,n `,m 400 Bessel Functions in which the Heaviside function ✓(a r) makes vanish for r>a, and ` and m are integers with ` m ` because must be single valued for   all angles ✓ and . 2 The zeros z`,n of j`(x) fix the energy levels as En,`,m =(~z`,n/a) /2m+V . 2 For j0(x)=sinx/x, they are z0,n = n⇡.SoEn,0,0 =(~n⇡/a) /2m + V .If the 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 ~2 E = E E =(z2 ⇡2) . (9.102) n n,1,0 1,0,0 1,n 2ma2 Inserting factors of c2 and using ~c = 197 eV nm, and mc2 =0.511 MeV, we 2 find from the zero z1,2 =7.72525 that E2 =1.89 (nm/a) eV, which is red 2 light if a =1nm.Thenextzeroz1,3 = 10.90412 gives E3 =4.14 (nm/a) eV, which is in the visible if 1.2

9.3 Bessel Functions of the Second Kind In section 7.5, we derived integral representations (7.55 & 7.56) for the (1) (2) Hankel functions H (z) and H (z) for Re z>0. One may analytically continue them (Courant and Hilbert, 1955, chap. VII) to the upper half z-plane

(1) 1 i/2 1 iz cosh x x H (z)= e e dx Imz 0 (9.103) ⇡i Z1 and to the lower half z-plane

(2) 1 +i/2 1 iz cosh x x H (z)= e e dx Imz 0. (9.104) ⇡i  Z1 When both z = ⇢ and = ⌫ are real, the two Hankel functions are complex conjugates of each other

(1) (2) H⌫ (⇢)=H⌫ ⇤(⇢). (9.105) Hankel functions, called Bessel functions of the third kind, are linear combinations of Bessel functions of the first J(z) and second Y(z) kind (1) H (z)=J(z)+iY(z) (9.106) H(2)(z)=J (z) iY (z). 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 at z = 0 as illustrated in Fig. 9.4. When z = ix is imaginary, we get the modified Bessel functions

2m+↵ ↵ 1 1 x I (x)=i J (ix)= ↵ ↵ m!(m + ↵ + 1) 2 m=0 X ⇣ ⌘ (9.107) ⇡ ↵+1 (1) 1 x cosh t K (x)= i H (ix)= e cosh ↵t dt. ↵ 2 ↵ Z0 Some simple cases are

2 2 ⇡ z I 1/2(z)= cosh z, I1/2(z)= sinh z, and K1/2(z)= e . r⇡z r⇡z 2z r (9.108)

When do we need to use these functions? If we are representing functions that 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 representing is inﬁnite at that point, then we do need the Y⌫(⇢)’s. Example 9.7 (Coaxial Wave Guides). An ideal coaxial wave guide is perfectly conducting for ⇢r, and the waves occupy the region r0 <⇢

Ez(r0,)=0,E(r0,)=0, and B⇢(r0,) = 0 (9.110) at ⇢ = r0.InTMmodeswithBz = 0, one may show (exercise 9.28) that the boundary conditions Ez(r0,) = 0 and Ez(r, ) = 0 can be satisﬁed if J (x) Y (vx) J (vx) Y (x) = 0 (9.111) n n n n in which v = r/r and x = !2/c2 k2 r . One can use the Matlab code 0 0 n = 0.; v = 10.; p f=@(x)besselj(n,x).*bessely(n,v*x)-besselj(n,v*x).*bessely(n,x) x=linspace(0,5,1000); 402 Bessel Functions

The Bessel Functions of the Second Kind Y 0(ρ), Y 1(ρ), Y 2(ρ) 0.5

−0.5

−1 0 2 4 6 8 10 12 ρ

The Bessel Functions of the Second Kind Y 3(ρ), Y 4(ρ), Y 5(ρ) 0.5

−0.5

−1 2 4 6 8 10 12 14 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 guesses in the code, one finds x1,1 =0.3941, x1,2 =0.7331, and x1,3 =1.0748. The 2 2 2 corresponding dispersion relations are !n,i(k)=c k + xn,i/r0. q 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 (⇢)= Y (⇢) (9.112) ` 2⇢ `+1/2 r and Rayleigh formulas express them as d ` cos ⇢ y (⇢)=( 1)`+1⇢` . (9.113) ` ⇢d⇢ ⇢ ✓ ◆ ✓ ◆ The term in which all the derivatives act on the cosine dominates at big ⇢ 1 d` cos ⇢ y (⇢) ( 1)`+1 = cos (⇢ `⇡/2) /⇢. (9.114) ` ⇡ ⇢ d⇢` 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 example 9.5, the potential of a hard sphere of radius r is V (r)= ✓(r r)in 0 1 0 which ✓(x)=(x + x )/2 x is Heaviside’s function. Since the point r =0is | | | | not 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 the constants c` and d`. Thus for ` = 0, Rayleigh’s formulas (9.73 & 9.113) and the boundary condition say that kr u (r )=c sin(kr ) d cos(kr )=0 0 0 0 0 0 0 0 or d /c = tan kr .Thes-wave then is u (kr)=c sin(kr kr )/(kr cos kr ), 0 0 0 0 0 0 0 which tells us that the tangent of the phase shift is tan (k)= kr .By 0 0 (9.99), the cross-section at low energy is 4⇡r2 or four times the classical ⇡ 0 value. Similarly, one ﬁnds (exercise 9.29) that the tangent of the p-wave phase shift is kr cos kr sin kr tan (k)= 0 0 0 . (9.117) 1 cos kr kr sin kr 0 0 0 For kr 1, we have (k) (kr )3/3; more generally, the `th phase shift 0 ⌧ 1 ⇡ 0 is (k) (kr )2`+1/ (2` + 1)[(2` 1)!!]2 for a potential of range r at ` ⇡ 0 0 low energy k 1/r . ⌧ 0 404 Bessel Functions

The Spherical Bessel Function y1(ρ)forρ ≪ 1andρ ≫ 1

0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 1 2 3 4 5 6 7 8 9 10 11 12 ρ

The Spherical Bessel Function y2(ρ)forρ ≪ 1andρ ≫ 1

0.2 0.1 0 −0.1 −0.2 −0.3 −0.4 −0.5 1 2 3 4 5 6 7 8 9 10 11 12 ρ

2 Figure 9.5 Top: Plot of y1(⇢) (solid curve) and its approximations 1/⇢ for small ⇢ (9.115, dot-dash) and cos(⇢ ⇡/2)/⇢ for big ⇢ (9.114, dashed). 3 Bottom: Plot of y2(⇢) (solid curve) and its approximations 3/⇢ for small ⇢ (9.115, dot-dash) and cos(⇢ ⇡)/⇢ for big ⇢ (9.114, dashed). The values of ⇢ at which y`(⇢) = 0 are the zeros or roots of y`;weusethemtoﬁt boundary conditions. All six plots run from ⇢ =1to⇢ = 12.