1
Ph 12b
Homework Assignment No. 3
Due: 5pm, Thursday, 28 January 2010
1. “A watched quantum state never moves.”
Consider a simple model
of an atom with two energy levels — the ground state

g
a
has energy
E
g
and the excited state

e
a
has energy
E
e
> E
g
, where
ω
= (
E
e

E
g
)
/
¯
h
;
the Hamiltonian of this system is
ˆ
H
=
E
g

g
aA
g

+
E
e

e
aA
e

.
An experimentalist is equipped to perform a measurement that projects
the state of the atom onto the orthonormal basis

+
a
=
1
√
2
(

g
a
+

e
a
)
,
a
=
1
√
2
(

g
a 
e
a
)
,
and to prepare the atom in the state

+
a
.
a
)
Suppose that the state

+
a
is prepared at time 0 and that the
measurement projecting onto
{
+
a
,
a}
is performed at time
t
.
Find the probability
P
t
(+) of the + measurement outcome and
the probability
P
t
(

) of the

measurement outcome.
b
) Suppose that the measurement projecting onto
{
+
a
,
a}
is per
formed twice in succession. The state

+
a
is prepared at time
0, the ±rst measurement is performed at time
t
, and the second
measurement is performed at time 2
t
. Find the probability of
a + outcome and the probability of a

outcome in the
second
measurement.
c
)
Now suppose that
N
measurements, equally spaced in time, are
performed in succession. The state

+
a
is prepared at time 0, the
±rst measurement is performed at time
t
, the second measurement
at time 2
t
, and so on, with the
N
th measurement performed at
time
Nt
. Find the probability
P
(+
N
) that the + outcome occurs
in
every one
of the
N
measurements.
d
)
For the same situation as in part (
c
), denote the total elapsed time
by
T
=
Nt
, so that the time interval between the measurements
is
t
=
T/N
. Show that
P
(+
N
) can be expressed as
P
(+
N
) = 1

f
(
ωT
)
/N
+
O
(1
/N
2
)
,
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2
and fnd the Function
f
(
ωT
). Thus, taking the limit
N
→ ∞
with
ωT
fxed, we conclude that iF the atom is observed continuously
its state never evolves.
2. A better bomb test
. The bombtesting protocol explained in class
uses a beam splitter described by the unitary transFormation
ˆ
U
=
1
√
2
p
1

1
1
1
P
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 DUDKO
 mechanics, Probability, Energy, Work, bomb, beam splitter, port exit1

Click to edit the document details