Mathematical Methods for Physics and Engineering : A Comprehensive Guide

4.9 HINTS AND ANSWERS

find a closed-form expression forα, the Madelung constant for this (unrealistic)
lattice.
4.35 One of the factors contributing to the high relative permittivity of water to static
electric fields is the permanent electric dipole moment,p, of the water molecule.
In an external fieldEthe dipoles tend to line up with the field, but they do not
do so completely because of thermal agitation corresponding to the temperature,
T, of the water. A classical (non-quantum) calculation using the Boltzmann
distribution shows that the average polarisability per molecule,α,isgivenby

α=

p

E

(cothx−x−^1 ),

wherex=pE/(kT)andkis the Boltzmann constant.
At ordinary temperatures, even with high field strengths (10^4 Vm−^1 or more),
x1. By making suitable series expansions of the hyperbolic functions involved,
show thatα=p^2 /(3kT) to an accuracy of about one part in 15x−^2.
4.36 In quantum theory, a certain method (the Born approximation) gives the (so-
called) amplitudef(θ) for the scattering of a particle of massmthrough an angle
θby a uniform potential well of depthV 0 and radiusb(i.e. the potential energy
of the particle is−V 0 within a sphere of radiusband zero elsewhere) as

f(θ)=

2 mV 0


^2 K^3

(sinKb−KbcosKb).

Here
is the Planck constant divided by 2π, the energy of the particle is
^2 k^2 /(2m)

andKis 2ksin(θ/2).

Use l’Hopital’s rule to evaluate the amplitude at low energies, i.e. whenˆ kand

henceKtend to zero, and so determine the low-energy total cross-section.

[ Note: the differential cross-section is given by|f(θ)|^2 and the total cross-

section by the integral of this over all solid angles, i.e. 2π

∫π

0 |f(θ)|

(^2) sinθdθ.]
4.9 Hints and answers
4.1 Write as 2(

∑ 1000

n=1n−

∑ 499

n=1n) = 751 500.
4.3 Divergent forr≤1; convergent forr≥2.
4.5 (a) ln(N+ 1), divergent; (b)^13 [1−(−2)n], oscillates infinitely; (c) Add^13 SNto the
SNseries; 163 [1−(−3)−N]+^34 N(−3)−N−^1 ,convergentto 163.
4.7 Write thenth term as the difference between two consecutive values of a partial-
fraction function ofn.Thesumequals^12 (1−N−^2 ).
4.9 Sum the geometric series withrth term exp[i(θ+rα)]. Its real part is

{cosθ−cos[(n+1)α+θ]−cos(θ−α)+cos(θ+nα)}/4sin^2 (α/2),
which can be reduced to the given answer.
4.11 (a)− 1 ≤x<1; (b) allxexceptx=(2n±1)π/2; (c)x<−1; (d)x<0; (e)
always divergent. Clearly divergent forx>−1. For−X=x<−1, consider

∑∞

k=1

∑Mk

n=Mk− 1 +1

1

(lnMk)X

,

where lnMk=kand note thatMk−Mk− 1 =e−^1 (e−1)Mk; hence show that the
series diverges.
4.13 (a) Absolutely convergent, comparewith exercise 4.10(b). (b) Oscillates finitely.
(c) Absolutely convergent for allx. (d) Absolutely convergent; use partial frac-
tions. (e) Oscillates infinitely.