Physics 7701: Problem Set #6

Here are some hints, suggestions, and comments on the assignment.

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  1. Fourier transform practice. Mostly this is about doing integrals of certain canonical types, which is why you have to do these by hand. You also may find that checking with Mathematica is not so transparent in some cases.
    1. Whenever you see a cosine or sine together with something like a gaussian, you should immediately think about changing it to exponential form. Then you have a "complete the square" problem. Be sure to justify (briefly!) why you can move or exchange the contour of integration. You can assume that you know the integral of a plain gaussian function e-a x2. The inverse transform is another complete-the-square integral and, of course, you should get what you started with.
    2. Use derivatives: x e-ikx = i (d/dk)e-ikx. In your answer for F(k) are the apparent poles really there? (That is, are they removable? If so, you can use a principal value or just shift the poles and do the integrals. I shifted the poles to the upper half plane in the end, because this meant fewer terms to combine, but you can make a different choice.) Doing the inverse transform, be sure to distinguish x>0 and x<0 or x>1 and x<1 when relevant (e.g., do you close in the upper or lower half plane?). When you have double poles, you can use the standard formula or just expand the numerator and read off the residue from the 1/k term.
    3. Fourier transforms of f(t) in physics use the convention of e+iωt rather than e-iωt in defining F(ω). Use derivatives here as in the last problem. Be careful to distinguish contours for t>0 and t<0 when doing the inverse transform. The inverse transform has a double pole, but it is not at ω = 0, so be careful (use the formula).
    4. The integral for this Fourier transform should be getting familiar by now! Write your answer using theta functions of k and -k. The inverse transform involves simple integrals (not contour integration).
  2. Parseval's theorem. This should be pretty straightforward. The main purpose of the problem is to get you thinking about Parseval's theorem in the form of Eq.(1). You can use Mathematica to do the inverse transform integral.
  3. Spring-and-dashpot system. We've seen this differential equation before! Note that in this case, f(t) is not periodic (although it has a periodic component) because of the exponential descrease in amplitude. So we use Fourier transforms instead of Fourier series. The usual game is to transform both sides (using the properties of Fourier transforms to deal with the derivatives) to obtain an algebraic equation (rather than a differential equation) involving X(ω) and F(ω), which you can solve for X(ω) and then transform back to find x(t). Be careful of where the poles are; how does this affect verifying that x=0 for t<0. You should check your answer for the Fourier transforms with Mathematica, but you must also do them by hand. If your answer doesn't look like the Mathematica one, enter yours and subtract them and apply FullSimplify to see if you get zero. You may find it convenient to define a frequency Ω0 = (ω02 - α2)1/2.
  4. Radon problem. Note that this uses a cosine transform, because we know the value of the first derivative of density at t=0 (see Lea section 7.7.3).
    1. Is the diffusion rate finite? So how much do you expect at infinite y given finite time?
    2. Your answer for this part should just be an integral over k. You need to solve a first order differential equation in t that has a constant term (so the answer is not just an exponential). Check your answer by plugging it back into the differential equation!
    3. You can take the derivative of your answer from the last part. You should get an integral that is proportional to a delta function. To find the constant, do a cosine tranform of a delta function in y.
    4. Use your expression for ρ(y,t) in this part. Mathematica can express the integrals in terms of "known" functions (the error function in one case).
    5. If you have a differential equation for ρ (take the derivative of your answer for ρ(y,t) and give the integral to Mathematica!) and an initial condition in time, then integrate from 0 to t to get the desired equation.
  5. Charge form factor. The density ρ is a function of the scalar r only, so what coordinate system is best? Don't forget to do all of the three-dimensional integrals in that system. Choose your z-axis to simplify your life!
  6. Just another Fourier transform. Actually, this is not so interesting; I'm not sure why I picked it! :)
  7. Remember to allow for the functions being complex. You can use Parseval's Theorem multiple times.

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