Derive the integral expression for spherical bessel functions in
terms of plane waves at the same wavenumber.
The addition theorems:
are derived someplace, for both this special case and for the general
case. Find at least one such place (for
), copy the
derivation (with acknowledgement), and hand it in. If you work in a
group, see how many places you can find it and compare. LEARN what you
can from the process, that is, read the text accompanying the
derivation(s) you find and try to understand it. Work it out. For
extra credit, find in the literature the original paper that derives
the general addition theorem. Hints: JMP, Danos and Maximon. Study
Derive the Green's function for the Helmholtz equation in
free space (zero boundary conditions at infinity). Do not use the
addition theorem, since you do not (in principle) know its form yet
and so do not know that it is a Neumann or Hankel function.
Naturally, you can follow Wyld or Jackson or Arfken, but acknowledge
your source and show the entire derivation.
Make a neat sheet with Everything You Never Wanted To Know About
Spherical Bessel/Neumann/Hankel Functions but were Afraid Not To Ask on it.
Don't hand it in, this will be your guide through life (for at least a few
weeks). Do NOT simply photocopy my notes. Do it by hand. Pool your sheets
with those of your friends -- put together everything to make a ``best''
sheet and then photocopy it. I use the term ``sheet'' loosely. I expect it
will fill several (it did in my notes).
Using the addition theorem derived above (in the form of the
Green's function) and the asymptotic relations on your worksheet,
derive the static result for the vector potential A we
previously obtained for the near field zone (my equation 66). Find
the lowest order correction to this expression. This will, of
course, involve finding more out about spherical waves than I have so
far told you!
item Using the same addition theorem and the other asymptotic relations,
derive an expression for the v.p. A in the far zone. Is there a
correspondance of some sort with our previous result (Jackson 9.9)?
is equivalent to
Any vector quantity can be decomposed in a symmetric and an
antisymmetric piece. Prove that, in the case of the
above, the current term can be decomposed into
Evaluate the antisymmetric piece. Show (from the magnetic dipole
vector potential) that
Remark upon the similarities and differences between this result and the
electric dipole result.
Next start to evaluate the integral of the symmetric piece. Show
that you get:
The steps involved are:
integrate by parts (working to obtain divergences of
times whatever from the
continuity equation (for a harmonic source).
rearranging and recombining.
Don't forget the boundary condition at infinity!
Homemade tables, part II. What you did for spherical bessel
functions, do for spherical harmonics. In particular, derive the
commutation rules for the raising and lowering operators from the
cartesian commutation relations for L. From the commutation rules
derive the (normalized) action of