This preview shows pages 1–3. Sign up to view the full content.
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, Energy, Work

Click to edit the document details