Physics 834: Problem Set #10

Here are some hints, suggestions, and comments on the assignment. Remember to keep track of the amount of time you spend doing the (entire) assignment and record this number on your problem solution.

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  1. Quantum mechanical spherical well.
    1. You don't need to re-derive the radial part of the Laplacian (although it won't hurt :), but show how it can be transformed with the given potential to be the equation for spherical Bessel functions. What is the difference in the equation between when r is less than R and greater than R? How does this change the solution to the equation? Be sure to note the general solution in each region and why one of the two terms in each must have zero coefficient. (If the wave function is to be normalizable, what are the conditions at the origin and as r goes to infinity?)
    2. This is completely analogous to solving the square well problem in one dimension. Use the matching conditions to determine possible eigenvalues. But it won't be in the form asked for: use the recursion relations (which you can find in either text --- be careful that signs are different for the two types of spherical Bessel functions) to show that you get Equation (3).
    3. The l=0 and l=1 representations of the spherical Bessel functions, including the modified Bessel functions, can be referenced directly in some cases in Mathematica, but in all cases can be defined in terms of the ordinary Bessel functions. I recommend doing this. Look up "Bessel function" in the "Documentation Center" under Help and use the formulas given in Arfken or Lea. Since α=10 is given, this is just a numerical problem, so you can use FindRoot. But always plot the function first, so you know what root you are trying to find (don't assume there is only one bound state!).
  2. Damped oscillator Green's function by division-of-region.
    1. To apply the division-of-region method, we need to solve the homogeneous equation in each region (t < t' and t' < t) with the relevant boundary conditions and then match the solutions at t = t'. This is supposed to be a physical Green's function, so it is the causal response at time t to an impulse force at time t'. Before the force is applied, what do you expect the response to be? The response is damped and the driving impulse is only given at t', so what do you expect for the response at large times? We'll find this same Green's function by a transform method in class (following section C.3 in Lea, so look here), so you should know where you are heading!
    2. Here we just plug in the force into the usual integral of the Green's function (like the examples). You are encouraged to do this with Mathematica, including making a plot with sample values of the parameters. Don't forget to include Assumptions and any HeavisideTheta functions!
  3. Neumann Green's function for one-d Helmholtz equation. (This was formerly given for the Poisson equation, which had problems.) You should be able to follow the lecture 18 notes, just changing the boundary conditions for each of the methods. Look at the "Green's functions, Part I" notebook on the Mathematica examples page for a guide to making the plot. An easy check of your two Green's functions is just to evaluate them directly for some test x and x' values and see if you get the same result. Don't forget the constant term in the expansion method!
  4. Green's function for the diffusion problem.
    1. What are the boundary conditions at t=0 and as t goes to infinity?
    2. Plot your answer at various different times to see if it behaves as expected (a spreading gaussian).
  5. Dirichlet Green's function for Poisson's equation. Section C.7 is relevant here. What orientations of the hemispheres do the two choices of angles represent? Use the easier one to do the third part.

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Physics 834: Assignment #10 hints.
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