Mathematical Methods for Physics and Engineering - Matematica.NET

13.4 EXERCISES(a) Find the Fourier transform off(γ, p, t) ={e −γt sin pt t > 0,0 t 0) and p are constant parameters.(b) The current I(t) flowing through a certain system is related to the appliedvoltage V (t) by the equationI(t) =∫ ∞−∞K(t − u)V (u) du,whereK(τ) =a 1 f(γ 1 ,p 1 ,τ)+a 2 f(γ 2 ,p 2 ,τ).The function f(γ, p, t) is as given in (a) and all the a i ,γ i (> 0) and p i are fixedparameters. By considering the Fourier transform of I(t), find the relationshipthat must hold between a 1 and a 2 if the total net charge Q passed throughthe system (over a very long time) is to be zero for an arbitrary appliedvoltage.13.14 Prove the equality∫ ∞e −2at sin 2 at dt = 1 ∫ ∞a 20π 0 4a 4 + ω dω. 413.15 A linear amplifier produces an output that is the convolution of its input and itsresponse function. The Fourier transform of the response function for a particularamplifier isiω˜K(ω) = √ .2π(α + iω)2Determine the time variation of its output g(t) when its input is the Heavisidestep function. (Consider the Fourier transform of a decaying exponential functionand the result of exercise 13.2(b).)13.16 In quantum mechanics, two equal-mass particles having momenta p j = k j andenergies E j = ω j and represented by plane wavefunctions φ j =exp[i(k j·r j −ω j t)],j =1, 2, interact through a potential V = V (|r 1 − r 2 |). In first-order perturbationtheory the probability of scattering to a state with momenta and energies p ′ j ,E′ jis determined by the modulus squared of the quantity∫∫∫M = ψf ∗ Vψ i dr 1 dr 2 dt.The initial state, ψ i ,isφ 1 φ 2 and the final state, ψ f ,isφ ′ 1 φ′ 2 .(a) By writing r 1 + r 2 =2R and r 1 − r 2 = r and assuming that dr 1 dr 2 = dR dr,show that M can be written as the product of three one-dimensional integrals.(b) From two of the integrals deduce energy and momentum conservation in theform of δ-functions.(c) Show that M is proportional to the Fourier transform of V ,i.e.toṼ (k)where 2k =(p 2 − p 1 ) − (p ′ 2 − p′ 1 ) or, alternatively, k = p′ 1 − p 1.13.17 For some ion–atom scattering processes, the potential V of the previous exercisemay be approximated by V = |r 1 − r 2 | −1 exp(−µ|r 1 − r 2 |). Show, using the resultof the worked example in subsection 13.1.10, that the probability that the ionwill scatter from, say, p 1 to p ′ 1 is proportional to (µ2 + k 2 ) −2 ,wherek = |k| and kis as given in part (c) of that exercise.463