L
Homework 9
Prob. 9.1
For a twolevel system with static couplingin Eq. (9.18),
(a)
Find
A
and
B
in terms of
θ
if
c
1
(0)
= 0 and
c
2
(0)
=1.
(5 pts)
(b)
Find
c
1
(t)
,
c
2
(t)
,
p
1
(t)
and
p
2
(t)
following the initial condition in (a).
(5 pts)
(c)
Find
A
and
B
in terms of
θ
if
c
1
(0)
=
c
2
(0)
=
2
/
1
.
(5 pts)
(d)
When is the first time
p
1
(t) = 0
following the initial condition in (c).
(10 pts)
Prob. 9.2
For a twolevel system with dynamic coupling, assume that the evolution equations
for the expansion coefficients
c
1
(t)
and
c
2
(t)
can be described by:
(
29
(
29
(
29
(
29
=

t
c
t
c
H
Ve
Ve
H
t
c
t
c
dt
d
i
t
i
t
i
2
1
2
1
2
1
ϖ
ϖ
where
V
is real.
(a)
If
(
29
(
29
(
29
t
H
i
t
b
t
c
i
i
~
exp

=
, find the time derivative of
b
i
(t)
.
(5 pts)
(b)
If
(
29
(
29
/
exp
1
1
t
i
b
t
b
λ
=
and
(
29
(
29
(
29
/
exp
2
2
t
i
b
t
b
ϖ
λ
+
=
with
b
1
and
b
2
as time
independent, find the matrix equation with
λ
being the eigenvalue of the matrix.
(5 pts)
(c)
Find the eigenvalues
λ
.
(10 pts)
(d)
If
c
1
(0)
= 1 and
c
2
(0)
=0, find the expression for
c
1
(t)
and
c
2
(t)
.
(10 pts)
Prob. 9.3
An infinite potential well of length
L
is shown in the figure.
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 '05
 TANG
 Derivative, pts, Rabi cycle

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