Qucstion 1
The
set
of
functions
Po
=
1,
P =
I,
2 =
*;
are
orthogonal
Anothcr
sct
of
functions
that
are
also
ort
hogonal
with
nit
wcight
and
span
tlhe
same
spare
are
Fo
=
c',
F = , F
=5r-3.
a.
IVritc
the
normalizcd
forms
ofP
and
E
for
i
==
1,2.3.
1|
with
nit
weight
on
the
range
-1
< <
+I.
3
b.
Find
the
unitary
matrix
U
that
transforms
from
thc
normalized
P,
basis
to
the
nor
Imalized
E
basis.
3
C.
Find
the
nitary
matrix
V that
trans•orns
from
the
normalizcd
E,
basis
to
the
nor
malizcd
P
basis.
[1])
d.
Expand
f(r)
=
5r?
-3r
+ I
in
terns
of
the
nomnalizcd
versions
of
both
bases,
and
verify
tlhat
the
transformnation matrix U coverts
the
P-basis
cxpansion
of
f()
into its
F-basis cxpansion.
(2+2-+1]
Qucstion 2
Thc
ON
set
of
functions
(over
a
suitablc
rangc
of
z,
y,
)
given
by
B
={lo)
=
Crc-.
Jo) = Cye-r,
Jos)
= Czc-r}
spans
a
threc
}spans
a threc dimemsional voctor
space ol functions.
V.
a.
ind
the
matrix reprcscnt
ation,
M,
of
the
operator
Lr
=
-i(y
-)in
the
B;
basis.
(3)
b.
IVhat
are
the
cigenvalucs
of
M
and
the
corresponding
cigenvectors?
(2
C.
Writc
the
cxplicit
form
of
M'
which
is
thc
matrix M writton
in
a dilcrent
ON
basis,
B =
{lo)lo,)lo3)}
using
the
unitary
givcn
below
(The
matrix
elements
of
U
are
Uij
i=
(0,,
o;))
(2):
æ =
0
1//2 -i/V2
0
1/V2
i/2
d.
Calculate
the cxpcctation
valucs
of
the matrix
MM
w.rt.
lo),
o,),lo).
3|
c.
Calculatc
the
expectation
valucs
of
the
matrix
M
w.r.t.
lo),).l).
(3
(1)
Qucstion 3
A,
B.C
arc
finitc-dimensional Hormitian
matrices
such
that
C=A+B
and
A,
B)
=
0.
The
ON
cigonvectors
of
A
are
u)u2)....
u,).
a.
Writc
down
thc
mitary
transformation
to
digonalize
Cin
terms
of
u;),i=
1,2,....
n.
b.
Consider
tlhe
natrix D =
A'
-
AB
+ A
BA
+ B +
B,
cvaluatc the conmntator
|D,C1.
(3|
c.
Lct.
A and B
be
positive-dcfnite matrices
and
and
that
for
B
be
A
9o
such
that A +
Ap
= 1
with
the
same
corresponling
cige
vcctor.
All
other
cigenval1es
of
A and B arc
less
thap
1/2.
Find
the
cxpcct
ation
value
(uC)
when
)
is
a
unit-nornalizcd
vector
3
the
largcst
cigenvalue
of
A
be
Aa
