Mathematical Methods for Physics and Engineering : A Comprehensive Guide

13.4 EXERCISES

(a) Find the Fourier transform of

f(γ, p, t)=

{

e−γtsinpt t > 0 ,

0 t< 0 ,

whereγ(>0) andpare constant parameters.

(b) The currentI(t) flowing through a certain system is related to the applied

voltageV(t) by the equation

I(t)=

∫∞

−∞

K(t−u)V(u)du,

where

K(τ)=a 1 f(γ 1 ,p 1 ,τ)+a 2 f(γ 2 ,p 2 ,τ).

The functionf(γ, p, t) is as given in (a) and all theai,γi(>0) andpiare fixed

parameters. By considering the Fourier transform ofI(t), find the relationship

that must hold betweena 1 anda 2 if the total net chargeQpassed through

the system (over a very long time) is to be zero for an arbitrary applied

voltage.

13.14 Prove the equality
∫∞

0

e−^2 atsin^2 at dt=

1

π

∫∞

0

a^2

4 a^4 +ω^4

dω.

13.15 A linear amplifier produces an output that is the convolution of its input and its
response function. The Fourier transform of the response function for a particular
amplifier is

K ̃(ω)=√ iω

2 π(α+iω)^2

.

Determine the time variation of its outputg(t) when its input is the Heaviside
step function. (Consider the Fourier transform of a decaying exponential function
and the result of exercise 13.2(b).)
13.16 In quantum mechanics, two equal-mass particles having momentapj=
kjand
energiesEj=
ωjand represented by plane wavefunctionsφj=exp[i(kj·rj−ωjt)],
j=1,2, interact through a potentialV=V(|r 1 −r 2 |). In first-order perturbation
theory the probability of scattering to a state with momenta and energiesp′j,E′j
is determined by the modulus squared of the quantity

M=

∫∫∫

ψ∗fVψidr 1 dr 2 dt.

The initial state,ψi,isφ 1 φ 2 and the final state,ψf,isφ′ 1 φ′ 2.

(a) By writingr 1 +r 2 =2Randr 1 −r 2 =rand assuming thatdr 1 dr 2 =dRdr,

show thatMcan be written as the product of three one-dimensional integrals.

(b) From two of the integrals deduce energy and momentum conservation in the

form ofδ-functions.

(c) Show thatMis proportional to the Fourier transform ofV,i.e.toV ̃(k)

where 2
k=(p 2 −p 1 )−(p′ 2 −p′ 1 ) or, alternatively,
k=p′ 1 −p 1.

13.17 For some ion–atom scattering processes, the potentialVof the previous exercise
may be approximated byV=|r 1 −r 2 |−^1 exp(−μ|r 1 −r 2 |). Show, using the result
of the worked example in subsection 13.1.10, that the probability that the ion
will scatter from, say,p 1 top′ 1 is proportional to (μ^2 +k^2 )−^2 ,wherek=|k|andk
is as given in part (c) of that exercise.