Physics 7701: Problem Set #6
Here are some hints, suggestions, and comments on the assignment.
Recent changes to this page:
- 03-Oct-2013 --- original version.
- 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.
- 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.
- 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.
- 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).
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).
- 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
- 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
- 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
- Is the diffusion rate finite? So how much do you expect at
infinite y given finite time?
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!
- 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.
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).
- 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.
- 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!
- Just another Fourier transform.
Actually, this is not so interesting; I'm not sure why I picked
- Remember to allow for the functions being complex.
You can use Parseval's Theorem multiple times.
Physics 7701: Assignment #6 hints.
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