Physics 7701: Problem Set #11
Here are some hints, suggestions, and comments on the assignment.
Recent changes to this page:
 23Nov2013  original version.
 Hollow sphere.
You can prove equivalence simply by deriving these two expressions for
the potential by separate methods.
 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!).
 This is a straight expansion in spherical harmonics. Project the
coefficients as usual.
 Point charge potentials.
 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.
 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.
 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).
 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.
 Multipole expansion example.
 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.
 Using the standard formula for the scalar potential given a charge density
in terms of an integral with the free Green function
1/xx'. 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 doable!
 Quantum mechanical spherical well.

You don't need to rederive 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?)
 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).

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!).
 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/xx' + 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/xx' 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).
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Last modified: 08:03 am, December 02, 2013.
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