Physics 7701: Problem Set #3

Recent changes to this page:

• 03-Sep-2013 --- Original version.
• 08-Sep-2013 --- Added some hints for the bonus problem.

1. Contour integrals
   1. As in the example in the class notes, choose the height of the rectangular contour so that the top edge (where z = x + i times height) is such that the integrand is a multiple of the original integrand. Should the height depend on a, b, both, or neither? The conditions on a and b are important to ensure that the right and left edges vanish as you take R to infinity; make sure you see how that happens. For the Mathematica evaluation, you want to tell it to assume that b > Re(a) > 0. You can do this with the Assumptions option to the Integrate command. In this case, Assumptions -> {b > Re[a] > 0} is what you want. (After Assumptions is a minus sign and a greater-than sign, which together make an arrow.)
   2. If you use the definition of a principal value integral we originally used in class (the average of the integrals with k - iε and k + iε), then this result follows quickly from your result from PS#2 and a very similar result with the opposite sign for ε. But you can also follow the texts or the lecture notes on treating principal value integrals. To get Mathematica to do a principal value integral, add the option PrincipalValue->True to the Integrate function.
2. Laguerre's equation
   Follow the examples in Arfken or Lea or Cahill for the Frobenius method (see the reading) or the lecture notes. Regular at the origin means that it doesn't blow up at x=0. We try a solution that is a Taylor expansion times x^p with p to be determined (Arfken uses the variable k instead of p). Substitute the series, doing the derivatives term by term, and equate coefficients of the each term. The lowest power of x determines p (you should find p=0 here), and then you get relations between coefficients of successively higher powers of x. (Note that it is only the equation with the lowest power of x that you solve for p. An equation for the next lowest power may determine whether a₁ is zero or a free parameter.) You should be able to identify the general term, but it is easiest to see if you try the first few individually. To show that when α is an integer you get a polynomial, you must show that the equations for the coefficients of the series are zero after a finite number. You should be able to check your general result against the first few Laguerre polynomials, which you will find in quantum mechanics books (or just Google it). Note that the Frobenius method only gives you one p value; we will discuss later how to find a second solution (which has a logarithm and so is not regular at the origin).
3. Bessel equation
   Again, follow the examples in Arfken or Lea or Cahill for the Frobenius method (see the reading) or the lecture 10 notes. The same comments as for Laguerre's equation apply here (in both cases you are expanding about 0). A difference here is that you will find two possible values of p, which you should consider separately. You should find that a₀ and a₁ are independent parameters (for one of the p values), which means you obtain two solutions. You should recognize the series that you get in the two cases (once you pull out an overall power of x). You can check it by noting that the solutions are Bessel functions of order 1/2, which you can find (for example) in Wikipedia under "Bessel function" (search for spherical bessel functions).
4. If you divide by x and then take the limit of x to infinity, the equation with constant coefficients will be revealed. Use e^sx as an ansatz and solve for s. Then introduce v(x) as discussed in the lecture notes (or in the texts) and solve the equation directly (you might try the substitution w = dv/dx). In the end you should find two independent solutions. Mathematica can tell you the general solution as well as the solution to the asymptotic equation (use DSolve as in the example notebooks on the mathematica page). Note: it is sufficient to find the solution for positive x.
5. Here "simplify" means to eliminate the appearence of x² in the differential equation. After finding the equation for v(x), it is Frobenius time again (actually you should find ordinary power series for the two solutions).
6. Langmuir waves
   
   1. Landau says the integration contour should pass under the pole --- this is equivalent to pushing the pole upward by iepsilon. Then use the magic identify in the integral.
   2. You want to integrate by parts to move the derivative from f(v) to (w-kv)^-1. Then you can Taylor expand the denominator and integrate term by term. Does the principle value matter?
   3. What would the sign of the imaginary part be in this case? Is it physical?