m, # 기본물리수학 · 기본물리수학 2018년 2학기 homework 6 2018.11.29 -- 유재준...

기본물리수학 2018년 2학기 HOMEWORK 6 2018.11.29 -- 유재준

Due date: 2018.12.19 (Wed)

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552 Chapter 9 Differential Equations

9.2.7 The differential equation

P(x, y) dx + Q(x,y) dy = 0

is exact. Construct a solution

ϕ(x, y) =! x

x0

P(x, y) dx +! y

y0

Q(x0, y) dy = constant.

9.2.8 The differential equation

P(x, y) dx + Q(x,y) dy = 0

is exact. If

ϕ(x, y) =! x

x0

P(x, y) dx +! y

y0

Q(x0, y) dy,

show that

∂ϕ

∂x= P(x, y),

∂ϕ

∂y= Q(x,y).

Hence ϕ(x, y) = constant is a solution of the original differential equation.

9.2.9 Prove that Eq. (9.26) is exact in the sense of Eq. (9.21), provided that α(x) satisfiesEq. (9.28).

9.2.10 A certain differential equation has the form

f (x)dx + g(x)h(y) dy = 0,

with none of the functions f (x), g(x), h(y) identically zero. Show that a necessary andsufficient condition for this equation to be exact is that g(x) = constant.

9.2.11 Show that

y(x) = exp"−! x

p(t) dt

#$! x

exp"! s

p(t) dt

#q(s) ds + C

%

is a solution of

dy

dx+ p(x)y(x) = q(x)

by differentiating the expression for y(x) and substituting into the differential equation.

9.2.12 The motion of a body falling in a resisting medium may be described by

mdv

dt= mg − bv

when the retarding force is proportional to the velocity, v. Find the velocity. Evaluatethe constant of integration by demanding that v(0) = 0.

9.2 First-Order Differential Equations 553

9.2.13 Radioactive nuclei decay according to the law

dN

dt=−λN,

N being the concentration of a given nuclide and λ, the particular decay constant. In aradioactive series of n different nuclides, starting with N1,

dN1

dt= −λ1N1,

dN2

dt= λ1N1 − λ2N2, and so on.

Find N2(t) for the conditions N1(0) = N0 and N2(0) = 0.

9.2.14 The rate of evaporation from a particular spherical drop of liquid (constant density) isproportional to its surface area. Assuming this to be the sole mechanism of mass loss,find the radius of the drop as a function of time.

9.2.15 In the linear homogeneous differential equation

dv

dt=−av

the variables are separable. When the variables are separated, the equation is exact.Solve this differential equation subject to v(0) = v0 by the following three methods:

(a) Separating variables and integrating.(b) Treating the separated variable equation as exact.(c) Using the result for a linear homogeneous differential equation.

ANS. v(t) = v0e−at .

9.2.16 Bernoulli’s equation,

dy

dx+ f (x)y = g(x)yn,

is nonlinear for n ̸= 0 or 1. Show that the substitution u = y1−n reduces Bernoulli’sequation to a linear equation. (See Section 18.4.)

ANS.du

dx+ (1− n)f (x)u = (1− n)g(x).

9.2.17 Solve the linear, first-order equation, Eq. (9.25), by assuming y(x) = u(x)v(x), wherev(x) is a solution of the corresponding homogeneous equation [q(x) = 0]. This is themethod of variation of parameters due to Lagrange. We apply it to second-order equa-tions in Exercise 9.6.25.

9.2.18 (a) Solve Example 9.2.1 for an initial velocity vi = 60 mi/h, when the parachute opens.Find v(t). (b) For a skydiver in free fall use the friction coefficient b = 0.25 kg/m andmass m = 70 kg. What is the limiting velocity in this case?


Table 9.3 Solutions in Circular Cylindrical Coordinatesa

ψ =!

m,α

amαψmα

a. ∇2ψ + α2ψ = 0 ψmα ="

Jm(αρ)

Nm(αρ)

#"cosmϕ

sinmϕ

#"e−αz

eαz

#

b. ∇2ψ − α2ψ = 0 ψmα ="

Im(αρ)

Km(αρ)

#"cosmϕ

sinmϕ

#"cosαz

sinαz

#

c. ∇2ψ = 0 ψm ="

ρm

ρ−m

#"cosmϕ

sinmϕ

#

aReferences for the radial functions are Jm(αρ), Section 11.1;Nm(αρ), Section 11.3;Im(αρ) and Km(αρ), Section 11.5.

PDE is invariant under rotations that comprise the group SO(3). Its diagonal genera-tor is the orbital angular momentum operator Lz =−i ∂

∂ϕ , and its quadratic (Casimir)invariant is L2. Since both commute with H (see Section 4.3), we end up with threeseparate eigenvalue equations:

Hψ = Eψ, L2ψ = l(l + 1)ψ, Lzψ = mψ.

Upon replacing L2z in L2 by its eigenvalue m2, the L2 PDE becomes Legendre’s ODE,and similarlyHψ = Eψ becomes the radial ODE of the separation method in sphericalpolar coordinates.

• For cylindrical coordinates the PDE is invariant under rotations about the z-axis only,which form a subgroup of SO(3). This invariance yields the generator Lz = −i∂/∂ϕ

and separate azimuthal ODELzψ = mψ , as before. If the potential V is invariant undertranslations along the z-axis, then the generator −i∂/∂z gives the separate ODE in thez variable.

• In general (see Section 4.3), there are nmutually commuting generatorsHi with eigen-values mi of the (classical) Lie group G of rank n and the corresponding Casimir in-variants Ci with eigenvalues ci (Chapter 4), which yield the separate ODEs

Hiψ = miψ, Ciψ = ciψ

in addition to the (by now) radial ODE Hψ = Eψ .

Exercises

9.3.1 By letting the operator ∇2 + k2 act on the general form a1ψ1(x, y, z) + a2ψ2(x, y, z),show that it is linear, that is, that (∇2 + k2)(a1ψ1 + a2ψ2) = a1(∇

2 + k2)ψ1 +a2(∇

2 + k2)ψ2.

9.3.2 Show that the Helmholtz equation,

∇2ψ + k2ψ = 0,9.3 Separation of Variables 561

is still separable in circular cylindrical coordinates if k2 is generalized to k2 + f (ρ) +(1/ρ2)g(ϕ) + h(z).

9.3.3 Separate variables in the Helmholtz equation in spherical polar coordinates, splitting offthe radial dependence first. Show that your separated equations have the same form asEqs. (9.61), (9.64), and (9.65).

9.3.4 Verify that

∇2ψ(r, θ,ϕ) +!k2 + f (r) +

1r2

g(θ) +1

r2 sin2 θh(ϕ)

"ψ(r, θ,ϕ) = 0

is separable (in spherical polar coordinates). The functions f,g, and h are functionsonly of the variables indicated; k2 is a constant.

9.3.5 An atomic (quantum mechanical) particle is confined inside a rectangular box of sidesa,b, and c. The particle is described by a wave function ψ that satisfies the Schrödingerwave equation

−h̄2

2m∇2ψ = Eψ.

The wave function is required to vanish at each surface of the box (but not to be identi-cally zero). This condition imposes constraints on the separation constants and thereforeon the energy E. What is the smallest value of E for which such a solution can be ob-tained?

ANS. E =π2h̄2

2m

#1a2

+1b2

+1c2

$.

9.3.6 For a homogeneous spherical solid with constant thermal diffusivity, K , and no heatsources, the equation of heat conduction becomes

∂T (r, t)

∂t= K∇2T (r, t).

Assume a solution of the form

T = R(r)T (t)

and separate variables. Show that the radial equation may take on the standard form

r2d2R

dr2+ 2r

dR

dr+%α2r2 − n(n + 1)

&R = 0; n = integer.

The solutions of this equation are called spherical Bessel functions.

9.3.7 Separate variables in the thermal diffusion equation of Exercise 9.3.6 in circular cylin-drical coordinates. Assume that you can neglect end effects and take T = T (ρ, t).

9.3.8 The quantummechanical angular momentum operator is given by L=−i(r×∇). Showthat

L ·Lψ = l(l + 1)ψ

leads to the associated Legendre equation.Hint. Exercises 1.9.9 and 2.5.16 may be helpful.

9.3 Separation of Variables 561

is still separable in circular cylindrical coordinates if k2 is generalized to k2 + f (ρ) +(1/ρ2)g(ϕ) + h(z).

9.3.3 Separate variables in the Helmholtz equation in spherical polar coordinates, splitting offthe radial dependence first. Show that your separated equations have the same form asEqs. (9.61), (9.64), and (9.65).

9.3.4 Verify that

∇2ψ(r, θ,ϕ) +!k2 + f (r) +

1r2

g(θ) +1

r2 sin2 θh(ϕ)

"ψ(r, θ,ϕ) = 0

is separable (in spherical polar coordinates). The functions f,g, and h are functionsonly of the variables indicated; k2 is a constant.

9.3.5 An atomic (quantum mechanical) particle is confined inside a rectangular box of sidesa,b, and c. The particle is described by a wave function ψ that satisfies the Schrödingerwave equation

−h̄2

2m∇2ψ = Eψ.

The wave function is required to vanish at each surface of the box (but not to be identi-cally zero). This condition imposes constraints on the separation constants and thereforeon the energy E. What is the smallest value of E for which such a solution can be ob-tained?

ANS. E =π2h̄2

2m

#1a2

+1b2

+1c2

$.

9.3.6 For a homogeneous spherical solid with constant thermal diffusivity, K , and no heatsources, the equation of heat conduction becomes

∂T (r, t)

∂t= K∇2T (r, t).

Assume a solution of the form

T = R(r)T (t)

and separate variables. Show that the radial equation may take on the standard form

r2d2R

dr2+ 2r

dR

dr+%α2r2 − n(n + 1)

&R = 0; n = integer.

The solutions of this equation are called spherical Bessel functions.

9.3.7 Separate variables in the thermal diffusion equation of Exercise 9.3.6 in circular cylin-drical coordinates. Assume that you can neglect end effects and take T = T (ρ, t).

9.3.8 The quantummechanical angular momentum operator is given by L=−i(r×∇). Showthat

L ·Lψ = l(l + 1)ψ

leads to the associated Legendre equation.Hint. Exercises 1.9.9 and 2.5.16 may be helpful.


Table 9.4

Regular Irregularsingularity singularity

Equation x = x =

1. Hypergeometric 0,1,∞ –x(x − 1)y′′ + [(1+ a + b)x − c]y′ + aby = 0.

2. Legendrea −1,1,∞ –(1− x2)y′′ − 2xy′ + l(l + 1)y = 0.

3. Chebyshev −1,1,∞ –(1− x2)y′′ − xy′ + n2y = 0.

4. Confluent hypergeometric 0 ∞xy′′ + (c− x)y′ − ay = 0.

5. Bessel 0 ∞x2y′′ + xy′ + (x2 − n2)y = 0.

6. Laguerrea 0 ∞xy′′ + (1− x)y′ + ay = 0.

7. Simple harmonic oscillator – ∞y′′ +ω2y = 0.

8. Hermite – ∞y′′ − 2xy′ + 2αy = 0.

aThe associated equations have the same singular points.

the coefficients

2z− z

z2and

1− n2z2

z4.

Since the latter expression diverges as z4, point x =∞ is an irregular, or essential, singu-larity. !

The ordinary differential equations of Section 9.3, plus two others, the hypergeometricand the confluent hypergeometric, have singular points, as shown in Table 9.4.It will be seen that the first three equations in Table 9.4, hypergeometric, Legendre, and

Chebyshev, all have three regular singular points. The hypergeometric equation, with regu-lar singularities at 0, 1, and∞ is taken as the standard, the canonical form. The solutions ofthe other two may then be expressed in terms of its solutions, the hypergeometric functions.This is done in Chapter 13.In a similar manner, the confluent hypergeometric equation is taken as the canonical

form of a linear second-order differential equation with one regular and one irregular sin-gular point.

Exercises

9.4.1 Show that Legendre’s equation has regular singularities at x =−1, 1, and∞.

9.4.2 Show that Laguerre’s equation, like the Bessel equation, has a regular singularity atx = 0 and an irregular singularity at x =∞.


Table 9.4


Equation x = x =





5. Bessel 0 ∞x2y′′ + xy′ + (x2 − n2)y = 0.



8. Hermite – ∞y′′ − 2xy′ + 2αy = 0.


the coefficients

2z− z

z2and

1− n2z2

z4.





Exercises




Table 9.4


Equation x = x =





5. Bessel 0 ∞x2y′′ + xy′ + (x2 − n2)y = 0.



8. Hermite – ∞y′′ − 2xy′ + 2αy = 0.


the coefficients

2z− z

z2and

1− n2z2

z4.





Exercises




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The usefulness of the series solution in terms of what the solution is (that is, numbers)depends on the rapidity of convergence of the series and the availability of the coefficients.Many ODEs will not yield nice, simple recurrence relations for the coefficients. In general,the available series will probably be useful for |x| (or |x − x0|) very small. Computerscan be used to determine additional series coefficients using a symbolic language, suchas Mathematica,12 Maple,13 or Reduce.14 Often, however, for numerical work a directnumerical integration will be preferred.

Exercises

9.5.1 Uniqueness theorem. The function y(x) satisfies a second-order, linear, homogeneousdifferential equation. At x = x0, y(x) = y0 and dy/dx = y′0. Show that y(x) is unique,in that no other solution of this differential equation passes through the points (x0, y0)with a slope of y′0.Hint. Assume a second solution satisfying these conditions and compare the Taylorseries expansions.

9.5.2 A series solution of Eq. (9.80) is attempted, expanding about the point x = x0. If x0 isan ordinary point, show that the indicial equation has roots k = 0, 1.

9.5.3 In the development of a series solution of the simple harmonic oscillator (SHO) equa-tion, the second series coefficient a1 was neglected except to set it equal to zero. Fromthe coefficient of the next-to-the-lowest power of x, xk−1, develop a second indicial-type equation.

(a) (SHO equation with k = 0). Show that a1, may be assigned any finite value (in-cluding zero).

(b) (SHO equation with k = 1). Show that a1 must be set equal to zero.

9.5.4 Analyze the series solutions of the following differential equations to see when a1 maybe set equal to zero without irrevocably losing anything and when a1 must be set equalto zero.(a) Legendre, (b) Chebyshev, (c) Bessel, (d) Hermite.

ANS. (a) Legendre, (b) Chebyshev, and (d) Hermite: For k = 0, a1may be set equal to zero; for k = 1, a1 must be set equalto zero.

(c) Bessel: a1 must be set equal to zero (except fork = ±n =− 1

2 ).

9.5.5 Solve the Legendre equation!1− x2

"y′′ − 2xy′ + n(n + 1)y = 0

by direct series substitution.

12S. Wolfram, Mathematica, A System for Doing Mathematics by Computer, New York: Addison Wesley (1991).13A. Heck, Introduction to Maple, New York: Springer (1993).14G. Rayna, Reduce Software for Algebraic Computation, New York: Springer (1987).

9.5 Series Solutions — Frobenius’ Method 575

(a) Verify that the indicial equation is

k(k − 1) = 0.

(b) Using k = 0, obtain a series of even powers of x (a 1 = 0).

y even = a 0

!1−

n(n + 1)2!

x2 +n(n− 2)(n + 1)(n + 3)

4!x4 + · · ·

",

where

a j+2 =j (j + 1)− n(n + 1)

(j + 1)(j + 2)a j .

(c) Using k = 1, develop a series of odd powers of x (a 1 = 1).

y odd = a 1

!x −

(n− 1)(n + 2)3!

x3 +(n− 1)(n− 3)(n + 2)(n + 4)

5!x5 + · · ·

",

where

a j+2 =(j + 1)(j + 2)− n(n + 1)

(j + 2)(j + 3)a j .

(d) Show that both solutions, y even and y odd, diverge for x = ±1 if the series continueto infinity.

(e) Finally, show that by an appropriate choice of n, one series at a time may be con-verted into a polynomial, thereby avoiding the divergence catastrophe. In quantummechanics this restriction of n to integral values corresponds to quantization ofangular momentum.

9.5.6 Develop series solutions for Hermite’s differential equation

(a) y ′′ − 2xy ′ + 2αy = 0.

ANS. k(k − 1) = 0, indicial equation.

For k = 0,

a j+2 = 2a jj − α

(j + 1)(j + 2)(j even),

y even = a 0

!1+

2(−α)x2

2!+22(−α)(2− α)x4

4!+ · · ·

".

For k = 1,

a j+2 = 2a jj + 1− α

(j + 2)(j + 3)(j even),

y odd = a 1

!x +

2(1− α)x3

3!+22(1− α)(3− α)x5

5!+ · · ·

".

9.5 Series Solutions — Frobenius’ Method 577

Using the larger root of the indicial equation, develop a power-series solution aboutξ = 0. Evaluate the first three coefficients in terms of ao.

Indicial equation k2 −m2

4= 0,

u(ξ) = a0ξm/2!1−

α

m + 1ξ +

"α2

2(m + 1)(m + 2)−

E

4(m + 2)

#ξ2 + · · ·

$.

Note that the perturbation F does not appear until a3 is included.

9.5.13 For the special case of no azimuthal dependence, the quantum mechanical analysis ofthe hydrogen molecular ion leads to the equation

d

dη

"%1− η2

&du

dη

#+ αu + βη2u = 0.

Develop a power-series solution for u(η). Evaluate the first three nonvanishing coeffi-cients in terms of a0.

Indicial equation k(k − 1) = 0,

uk=1 = a0η

!1+

2− α

6η2 +

"(2− α)(12− α)

120−

β

20

#η4 + · · ·

$.

9.5.14 To a good approximation, the interaction of two nucleons may be described by amesonic potential

V =Ae−ax

x,

attractive for A negative. Develop a series solution of the resultant Schrödinger waveequation

h̄2

2md2ψ

dx 2+ (E − V )ψ = 0

through the first three nonvanishing coefficients.

ψ = a0'x + 1

2A′x 2 + 1

6( 12A

′2 −E′ − aA′)x 3 + · · ·

*,

where the prime indicates multiplication by 2m/h̄2.

9.5.15 Near the nucleus of a complex atom the potential energy of one electron is given by

V =Ze2

r

%1+ b1r + b2r

2&,

where the coefficients b1 and b2 arise from screening effects. For the case of zero angu-lar momentum show that the first three terms of the solution of the Schrödinger equationhave the same form as those of Exercise 9.5.14. By appropriate translation of coeffi-cients or parameters, write out the first three terms in a series expansion of the wavefunction.

9.6 A Second Solution 589

FIGURE 9.3 x and |x|.

has a regular solution Pn(x) and an irregular solution Qn(x). Show that the Wronskianof Pn and Qn is given by

Pn(x)Q′n(x)− P ′n(x)Qn(x) =

An

1− x2,

with An independent of x.

9.6.10 Show, by means of the Wronskian, that a linear, second-order, homogeneous ODE ofthe form

y′′(x) + P(x)y′(x) + Q(x)y(x) = 0

cannot have three independent solutions. (Assume a third solution and show that theWronskian vanishes for all x.)

9.6.11 Transform our linear, second-order ODE

y′′ + P(x)y′ + Q(x)y = 0

by the substitution

y = z exp!−12

" x

P (t) dt

#

and show that the resulting differential equation for z is

z′′ + q(x)z = 0,

where

q(x) = Q(x)− 12P

′(x)− 14P

2(x).

Note. This substitution can be derived by the technique of Exercise 9.6.24.

9.6.12 Use the result of Exercise 9.6.11 to show that the replacement of ϕ(r) by rϕ(r) may beexpected to eliminate the first derivative from the Laplacian in spherical polar coordi-nates. See also Exercise 2.5.18(b).


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9.6.13 By direct differentiation and substitution show that

y2(x) = y1(x)

! x exp[−" s

P (t) dt][y1(s)]2

ds

satisfies (like y1(x)) the ODE

y ′′2 (x) + P(x)y ′2(x) + Q(x)y2(x) = 0.

Note. The Leibniz formula for the derivative of an integral is

d

dα

! h(α)

g(α)f (x,α) dx =

! h(α)

g(α)

∂f (x,α)

∂αdx + f

#h(α),α

$dh(α)

dα− f

#g(α),α

$dg(α)

dα.

9.6.14 In the equation

y2(x) = y1(x)

! x exp[−" s

P (t) dt][y1(s)]2

ds

y1(x) satisfies

y ′′1 + P(x)y ′1 + Q(x)y1 = 0.

The function y2(x) is a linearly independent second solution of the same equation.Show that the inclusion of lower limits on the two integrals leads to nothing new, thatis, that it generates only an overall constant factor and a constant multiple of the knownsolution y1(x).

9.6.15 Given that one solution of

R′′ +1rR′ −

m2

r2R = 0

is R = rm, show that Eq. (9.127) predicts a second solution, R = r−m.

9.6.16 Using y1(x) =%∞

n=0(−1)nx2n+1/(2n + 1)! as a solution of the linear oscillator equa-tion, follow the analysis culminating in Eq. (9.142f) and show that c1 = 0 so that thesecond solution does not, in this case, contain a logarithmic term.

9.6.17 Show that when n is not an integer in Bessel’s ODE, Eq. (9.100), the second solutionof Bessel’s equation, obtained from Eq. (9.127), does not contain a logarithmic term.

9.6.18 (a) One solution of Hermite’s differential equation

y ′′ − 2xy ′ + 2αy = 0

for α = 0 is y1(x) = 1. Find a second solution, y2(x), using Eq. (9.127). Show thatyour second solution is equivalent to yodd (Exercise 9.5.6).

(b) Find a second solution for α = 1, where y1(x) = x, using Eq. (9.127). Show thatyour second solution is equivalent to y even (Exercise 9.5.6).

9.6.19 One solution of Laguerre’s differential equation

xy ′′ + (1− x)y ′ + ny = 0

for n = 0 is y1(x) = 1. Using Eq. (9.127), develop a second, linearly independent solu-tion. Exhibit the logarithmic term explicitly.

9.7 Nonhomogeneous Equation — Green’s Function 607

Exercises

9.7.1 Verify Eq. (9.168),!

(vL2u− uL2v) d τ2 =!

p(v∇2u− u∇2v) · d σ 2.

9.7.2 Show that the terms +k2 in the Helmholtz operator and −k2 in the modified Helmholtzoperator do not affect the behavior ofG(r1, r2) in the immediate vicinity of the singularpoint r1 = r2. Specifically, show that

lim|r1−r2|→0

!k2G(r1, r2) d τ2 = 1.

9.7.3 Show thatexp(ik|r1 − r2|)4π |r1 − r2|

satisfies the two appropriate criteria and therefore is a Green’s function for theHelmholtz equation.

9.7.4 (a) Find the Green’s function for the three-dimensional Helmholtz equation, Exer-cise 9.7.3, when the wave is a standing wave.

(b) How is this Green’s function related to the spherical Bessel functions?

9.7.5 The homogeneous Helmholtz equation

∇2ϕ + λ2ϕ = 0

has eigenvalues λ2i and eigenfunctions ϕi . Show that the corresponding Green’s functionthat satisfies

∇2G(r1, r2) + λ2G(r1, r2) =−δ(r1 − r2)

may be written as

G(r1, r2) =∞"

i=1

ϕi (r1)ϕi (r2)λ2i − λ2

.

An expansion of this form is called a bilinear expansion. If the Green’s function isavailable in closed form, this provides a means of generating functions.

9.7.6 An electrostatic potential (mks units) is

ϕ(r) =Z

4πε0·e−ar

r.

Reconstruct the electrical charge distribution that will produce this potential. Note thatϕ(r) vanishes exponentially for large r , showing that the net charge is zero.

ANS. ρ(r) = Zδ(r)−Za2

4πe−ar

r.


has the spherical polar coordinate expansion

exp(−k|r1 − r2|)4π |r1 − r2|

= k

∞!

l=0il(kr<)kl(kr>)

l!

m=−l

Yml (θ1,ϕ1)Y

m∗l (θ2,ϕ2).

Note. The modified spherical Bessel functions il(kr) and kl(kr) are defined in Exer-cise 11.7.15.

9.7.12 From the spherical Green’s function of Exercise 9.7.10, derive the plane-wave expan-sion

eik·r =∞!

l=0il(2l + 1)jl(kr)Pl(cosγ ),

where γ is the angle included between k and r. This is the Rayleigh equation of Exer-cise 12.4.7.Hint. Take r2≫ r1 so that

|r1 − r2|→ r2 − r20 · r1 = r2 −k · r1

k.

Let r2→∞ and cancel a factor of eikr2/r2.

9.7.13 From the results of Exercises 9.7.10 and 9.7.12, show that

eix =∞!

l=0il(2l + 1)jl(x).

9.7.14 (a) From the circular cylindrical coordinate expansion of the Laplace Green’s function(Eq. (9.197)), show that

1(ρ2 + z2)1/2

=2π

" ∞

0K0(kρ) coskz dk.

This same result is obtained directly in Exercise 15.3.11.(b) As a special case of part (a) show that

" ∞

0K0(k) dk =

π

2.

9.7.15 Noting that

ψk(r) =1

(2π)3/2eik·r

is an eigenfunction of#∇2 + k2

$ψk(r) = 0

(Eq. (9.206)), show that the Green’s function of L= ∇2 may be expanded as

14π |r1 − r2|

=1

(2π)3

"eik·(r1−r2) d

3k

k2.


has the spherical polar coordinate expansion

exp(−k|r1 − r2|)4π |r1 − r2|

= k

∞!

l=0il(kr<)kl(kr>)

l!

m=−l

Yml (θ1,ϕ1)Y

m∗l (θ2,ϕ2).

Note. The modified spherical Bessel functions il(kr) and kl(kr) are defined in Exer-cise 11.7.15.

9.7.12 From the spherical Green’s function of Exercise 9.7.10, derive the plane-wave expan-sion

eik·r =∞!

l=0il(2l + 1)jl(kr)Pl(cosγ ),

where γ is the angle included between k and r. This is the Rayleigh equation of Exer-cise 12.4.7.Hint. Take r2≫ r1 so that

|r1 − r2|→ r2 − r20 · r1 = r2 −k · r1

k.

Let r2→∞ and cancel a factor of eikr2/r2.

9.7.13 From the results of Exercises 9.7.10 and 9.7.12, show that

eix =∞!

l=0il(2l + 1)jl(x).

9.7.14 (a) From the circular cylindrical coordinate expansion of the Laplace Green’s function(Eq. (9.197)), show that

1(ρ2 + z2)1/2

=2π

" ∞

0K0(kρ) coskz dk.

This same result is obtained directly in Exercise 15.3.11.(b) As a special case of part (a) show that

" ∞

0K0(k) dk =

π

2.

9.7.15 Noting that

ψk(r) =1

(2π)3/2eik·r

is an eigenfunction of#∇2 + k2

$ψk(r) = 0

(Eq. (9.206)), show that the Green’s function of L= ∇2 may be expanded as

14π |r1 − r2|

=1

(2π)3

"eik·(r1−r2) d

3k

k2.

m, # 기본물리수학 · 기본물리수학 2018년 2학기 homework 6 2018.11.29 -- 유재준...

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