18.369 Problem Set 2
losed under omposition (i.e. that the omposition of two operations always gives another oper-
Due Friday, 29 February 2008.
ation in the group) by giving the multipliation
table of the group (whose rows and olumns are
Problem 1: Projetion operators (easy)
group members and whose entries give their omposition).
The representation-theory handout gives a formula
(b) Find the harater table of
for the projetion operator from a state onto its omponent that transforms as a partiular representation. Prove the orretness of this formula (using the
irreduible representation of
Problem 2: Symmetries of a eld in a
square metal box
(d) Sketh possible
square of air (ε
We solved the ase of
luding
(e) If
and
saw solutions orresponding to ve dierent represen-
(you will need the
Ez = 0
E
degenerate
90
Problem 4: Cylindrial symmetry
enumerated in lass.
Suppose
that
we
have
a
ylindrial
metalli
waveguidethat is, a perfet metalli tube with ra-
dental degeneraies.)
R,
dius
whih is uniform in the
z diretion.
= 1).
The
interior of the tube is simply air (ε
Problem 3: Symmetries of a eld in a
triangular metal box
(a) This struture has ontinuous rotational symmetry around the
Consider the two-dimensional solutions in a triangu-
L.
(non-aidental)
matries.
(Like in lass, you will get some reduible ai-
lar perfet-metal box with side
are any
the triangular struture is not symmetri under
90◦ rotations). Hint: use your representation
eigen-
at the metal walls.
C4v
there
rotation of the other for a degenerate pair, but
(b) Sketh and lassify these solutions aording to
the representations of
of eah polarization orrespond
get the other orthogonal eigenfuntion(s) (e.g.
◦
problem from problem set 1), with the boundary
ondition that
ω = 0)
in the square ase we ould get one from the
(a) Solve for the eigenmodes of the other polariza-
E = Ez (x, y)ẑ
solutions that
modes, show how given one of the modes we an
tations of the symmetry group (C4v ).
tion:
Hz
and
to?
= 1)
H = Hz (x, y)ẑ
for eah
representation should the lowest-ω mode (ex-
surrounded by perfetly onduting walls (in whih
E = 0).
ω 6= 0 Ez
D
C3v .
would transform as these representations. What
In lass, we onsidered a two-dimensional (xy ) prob-
L×L
using the rules
() Give unitary representation matries
Great Orthogonality Theorem).
lem of light in an
C3v ,
from the representation-theory handout.
z
axis, alled the
C∞
1
group.
Find the irreduible representations of this group
Don't try to solve
(there are innitely many beause it is an innite
this analytially; instead, you will use symmetry to
group).
sketh out what the possible solutions will look like
for both
Ez
and
Hz
(b) For simpliity, onsider the (Hermitian) salar
ω2
2
wave equation −∇ ψ =
c2 ψ with ψ|r=R = 0.
Show that, when we look for solutions ψ that
polarizations.
(a) List the symmetry operations in the spae group
transform like one of the representations of the
(hoose the origin at the enter of the triangle so
that the spae group is symmorphi), and break
1 It
ditionally alled
C3v ).
also has an innite set of mirror planes ontaining the
z axis, but let's ignore these for now. If they are inluded, the
group is alled C∞v .
them into onjugay lasses. (This group is traVerify that the group is
1
C∞
eikz
group from above, and have
z
dependene
(from the translational symmetry), then we
obtain a Bessel equation (Google it if you've
forgotten Mr.
Bessel).
Write the solutions in
terms of Bessel funtions, assuming that you are
given their zeros
n = 1, 2, . . .,
xm,n
Jm
where
Jm (xm,n ) = 0
(i.e.
for
is the Bessel funtion of
the rst kind...if you Google for Bessel funtion
zeros you an nd them tabulated). Sketh the
dispersion relation
ω(k)
for a few bands.
() From the general orthogonality of Hermitianoperator eigenfuntions, derive/prove an orthogonality integral for the Bessel funtions.
(No,
just looking one up on Wikipedia doesn't ount.)
Problem 5: Conservation Laws
Suppose
that
J(x)e−iωt into
queny ω , and
we
introdue
a
nonzero
urrent
Maxwell's equations at a given frewe want to nd the resulting timeE(x)e−iωt (i.e. we are only
harmoni eletri eld
looking for elds that arise from the urrent, with
E→0
as
|x| → ∞
if
J
is loalized).
(a) Show that this results in a linear equation of the
 |Ei = |bi, where  is some linear oper|bi is some known right-hand side in
terms of the urrent density J.
form
ator and
(b) Prove that, if
J
transforms as some irreduible
representation of the spae group then
|Ei (= E,
whih you an assume is a unique solution) does
also.
(This is the analogue of the onservation
in time that we showed in lass, exept that now
we are proving it in the frequeny domain. You
ould prove it by Fourier-transforming the theorem from lass, I suppose, but do not do so
instead, prove it diretly from the linear equation
here.)
() Formally,
|Ei = Â−1 |bi,
where
Â−1
is related
to the Green's funtion of the system.
happens if
ω
What
is one of the eigenfrequenies?
2