Physics 7701: Problem Set #11

Recent changes to this page:

• 23-Nov-2013 --- original version.

1. Hollow sphere. You can prove equivalence simply by deriving these two expressions for the potential by separate methods.
1. One way to get this form is to use the master formula with an appropriate Dirichlet Green function (which we derived in a previous problem set!).
2. This is a straight expansion in spherical harmonics. Project the coefficients as usual.
2. Point charge potentials.
1. Remember that the Green's function is the potential due to a point charge at x' and that we already wrote an expansion for that. You can simplify quite a bit using the special values of the angles for the charges.
2. Which of the terms survive in the limit that "a" goes to zero? What is the vector direction of the dipole? Use this in your final answer.
3. You might want to go back to the expansion in part (a) and use image charges to get the potential with a grounded shell, then take the limit as in part (b).
3. Multipole theorem.
• One way to proceed is to note that you can work in Cartesian coordinates because the momentums $q_{lm}$ can be written as sums of moments that are monomials in x, y, and z. Then it is easy to formulate a shift in coordinates.
4. Multipole expansion example.
1. First figure out how to rewrite the charge density in terms of particular spherical harmonics. Then you'll be able to use the orthonormality integrals for spherical harmonics to evaluate the angular integrals.
2. Using the standard formula for the scalar potential given a charge density in terms of an integral with the free Green function 1/|x-x'|. But you can use your result from the first part and the spherical harmonics expansion of the free Green function and suddenly all integrals are do-able!
5. 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 any of the texts --- 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!).
6. Dirichlet Green's function for Poisson's equation. Section C.7 of the Lea notes (posted on the webpage) is relevant here.
   • What orientations of the hemispheres do the two choices of angles represent? Use the easier one to do the third part.
   • You could find the Green's function by the division of region method, but it is probably easier to do this by writing G(x,x') = 1/|x-x'| + psi(x,x'), where Del^2 psi = 0 and psi is chosen to satisfy the boundary conditions on the hemisphere. We can use the general spherical harmonics expansion of 1/|x-x'| and the general expansion of psi(x,x') = sum_l,m A_lm r^l/a^l+1 Y_lm(theta,phi), where A_lm is a function of x'. Determine A_lm by imposing the boundary condition at r=a (for both parts a and b) and then imposing the boundary condition for the flat part.
   • You can check your answer by using the method of images with three image charges strategically placed to ensure that the potential is zero on both the curved and flat surfaces.
   • For part c), apply the master theorem (e.g., Equation C.37 in Lea).