Physics 834: Problem Set #4

Here are some hints, suggestions, and comments on the assignment. Remember that this problem set is not to be handed in.

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  1. If you divide by x and then take the limit of x to infinity, the equation with constant coefficients will be revealed. Use esx as an ansatz and solve for s. Then introduce v(x) as discussed in Lecture 7 (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).
  2. One solution here should be apparent by inspection because there are just derivatives of y with no term proportional to y. This should also come out from your analysis. I'm not actually sure how the other solution is supposed to work, since Frobenius seems to give identically zero if you use the recurrence relations! However, to get the point, just look at the recurrence relation for general m and m+1 and apply a convergence test for the series (try section 2.3.1 of Lea for possible tests). What kind of singularity doesn't converge for any value of x when expanded about x=0? Mathematica can tell you the general solution (use DSolve).
  3. Apply the standard Frobenius method, but now you need to expand the potential in a series as well. You only need a finite number of terms, so there's no need to try to determine a general expression: just work your way up until you have three nonzero terms. The condition on the wave function should lead you to a particular choice of the possible solutions.
  4. Here "simplify" means to eliminate the appearence of x2 in the differential equaiton. After finding the equation for v(x), it is Frobenius time again (actually you should find power series for the two solutions).
  5. A standard Frobenius application for a physics problem. We want the solution that is regular (finite) at the origin. The electric field shouldn't appear too early!
  6. Input for a numerical solution
    1. With the assumption on V(x), this is an equation you can solve by inspection. Is it sufficient to leave C0 arbitrary? Or do you need to argue for a particular value?
    2. Again, with the assumption you are told, you will have an equation to which you already know the answers. What does physics tell you about the two possible solutions?
    3. Remember that you only know V(x) on the mesh points, so if you use anything like derivatives, you will need finite difference approximations.

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