18.369 Problem Set 1
P
Due Friday, 15 February 2007.
are
n cn Hn in the eigenvetors Hn with some oeients cn . It is suient to show that there
Problem 1: Adjoints and operators
Show that for a
dimensional
linearly independent eigenvetors
(i) Show that every
O
(a) We dened the adjoint † of operators Ô by:
hH1 , ÔH2 i = hÔ† H1 , H2 i for all H1 and H2
in the vetor spae.
d
d×d
Hn :
Hermitian matrix
has at least one nonzero eigenvetor
H1
[... use the fundamental theorem of algebra:
every polynomial with degree
nite-
> 0
has at
least one (possibly omplex) root℄.
Hilbert spae, where H is a olumn
hn (n = 1, · · · , d), Ô is a square d × d
(1)
matrix, and hH
, H (2) i is the ordinary onP (1)∗ (2)
jugated dot produt
n hn hn , the above
(ii) Show
vetor
that
the
spae
{H | hH, H1 i = 0}
of
V1
orthogonal to
=
H1
is preserved (transformed into itself or a
Ô . From this, show that
(d − 1) × (d − 1) Hermitian
subset of itself ) by
adjoint denition orresponds to the onjugate-
we an form a
transpose for matries.
matrix whose eigenvetors (if any) give (via
a similarity transformation) the remaining
In the subsequent parts of this problem, you
may
not
Ô
assume that
(if any) eigenvetors of
is nite-dimensional
(iii) By indution, form an orthonormal basis of
(nor may you assume any spei formula for
d
the inner produt).
not
Ô
eigenvetors for the
d-dimensional
spae.
(b) Completeness is not automati for eigenvetors
(b) Show that if Ô is simply a number o, then
Ô† = o∗ . (This is
the same as the previous question, sine
Ô .
in general.
Give an example of a non-singular
non-Hermitian operator whose eigenvetors are
not omplete. (A 2 × 2 matrix is ne. This ase
here an at on innite-
dimensional spaes.)
is also alled defetive.)
() If a linear operator
Ô
the operator is alled
satises
unitary.
Ô† = Ô−1 ,
then
() Completeness of the eigenfuntions is not au-
Show that a uni-
tomati
tary operator preserves inner produts (that is,
if we apply
for
Hermitian
operators on
dimensional spaes either;
Ô to every element of a Hilbert spae,
innite-
they need to have
then their inner produts with one another are
some additional properties (e.g. ompatness)
u of a
(|u| = 1)
for this to be true. However, it is true of most op-
unhanged).
Show that the eigenvalues
unitary operator have unit magnitude
erators that we enounter in physial problems.
not
and that its eigenvetors an be hosen to be
If a partiular operator did
orthogonal to one another.
basis of eigenfuntions, what would this mean
have a omplete
about our ability to simulate the solutions on a
−1
exists),
(d) For a non-singular operator Ô (i.e. Ô
−1 †
† −1
show that (Ô
) = (Ô ) . (Thus, if Ô is Her−1
mitian then Ô
is also Hermitian.)
omputer in a nite omputational box (where,
when you disretize the problem, it turns approximately into a nite-dimensional problem)?
No rigorous arguments required here, just your
thoughts.
Problem 2: Completeness
(a) Prove that
dimensional
matrix) are
the eigenvetors
Hn
Hermitian operator
omplete :
that is,
of a
Ô
(a
that
nite- Problem 3: Maxwell eigenproblems
d × d
any d-
(a) In lass, we eliminated
dimensional vetor an be expanded as a sum
E
from Maxwell's equa-
tions to get an eigenproblem in
1
H
alone, of the
ω2
c2 H(x).
stead eliminate H, you
form
Θ̂H(x) =
eigenproblem in
ε = onstant.
Show that if you in-
annot
E
get a Hermitian
exept for the trivial ase
Instead, show that you get a
eralized Hermitian eigenproblem :
ω2
c2 B̂E(x), where
are Hermitian operators.
the form
B̂
(b) For
ÂE(x) =
any
generalized
gen-
an equation of
both Â
and
Hermitian
eigenproblem
where
denite (i.e.
hE, B̂Ei > 0
for
show that the eigenvalues
B̂ is positive
1
all E(x) 6= 0 ),
ÂE = λB̂E) are real and
E1 and E2 satisfy a
orthogonality. Show that B̂ for
(i.e., the solutions of
that dierent eigenfuntions
modied kind of
the
E
eigenproblem above was indeed positive
denite.
() Show that
both the E and H formulations lead to
generalized Hermitian eigenproblems with real
if we allow magneti materials
µ
require
(d)
or
ω ).
µ
and
media.
ǫ
µ(x) 6= 1
real, positive, and independent of
are only ordinary numbers for
More generally, they are
ω
(but
H
isotropi
3×3
matri-
es (tehnially, rank 2 tensors)thus, in an
anisotropi medium,
by putting an applied eld
in one diretion, you an get dipole moment in
dierent diretion in the material.
Show what
onditions these matries must satisfy for us to
still obtain a generalized Hermitian eigenproblem in
1 Here,
E
(or
H)
when we say
with real eigen-frequeny
E(x) 6= 0
ω.
we mean it in the sense of
generalized funtions; loosely, we ignore isolated points where
E
is nonzero, as long as suh points have zero integral, sine
suh isolated values are not physially observable.
Gelfand and Shilov,
Generalized Funtions.
See e.g.
2