ECE 604 LINEAR SYSTEMS
Problem Set #3
Issued: Thursday, October 1, 2009
Due: Thursday, October 8, 2009
Problem 1:
a) Consider the homogeneous system:
!
x
t
( )
=
0
e
t
e
!
t
!
1
"
#
$
$
%
&
’
’
x
t
( )
x
0
( )
=
1
1
"
#
$
%
&
’
(1)
Define
P
t
( )
=
e
!
t
1
!
1
e
t
"
#
$
$
%
&
’
’
Let
z
t
( )
=
P
t
( )
x
t
( )
. Find a differential equation (including initial conditions)
for
z
t
( )
for
t
!
0
.
b) Compute the transition matrix for (1).
Problem 2:
One of the simplest problems in quantum mechanics is the one
dimensional
potential well problem
. The well potential is shown in Figure 1. Here, the wave
function
!
satisfies
d
2
dx
2
!
x
( )
"
#
2
!
x
( )
=
0
x
>
a
d
2
dx
2
!
x
( )
+
"
2
!
x
( )
=
0
x
#
a
Assume that
!
and
!
are real, and that
!
and
d
!
dx
are continuous at
x
= ±
a
.
Determine conditions on
!
and
!
such that there exists a solution
!
which goes
to zero at
±
!
. Sketch
!
x
( )
as a function of
x
.
Note:
!
is the response of a homogeneous linear system in each of the three
regions. Match the initial conditions of the three responses using the continuity
assumptions. (Although the system for
x
! "
a
is noncausal, you can replace
t
by
!
"
to get a system that is causal in
!
. After finding the solution, you can back
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 Spring '10
 Derivative

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