PHYS 455
Homework III
Assigned: November 22, 2011, Tuesday.
Due: November 30, 2011, Wednesday.
1.
Quantum Zeno Effect
is the quantum analog of the Zeno paradox. One of many alternative
versions of the paradox is summarized by the statement “A watched pot never boils”. It
is possible to claim that the quantum analog of the paradox is more convincing than the
classical ones. In this problem, we will see on a 2level system what the strange feature of
the paradox is.
Consider a twolevel system that we may call as a “pot”. We will use the 2
×
1 matrix
representation. Let us identify the two states in the standard basis as the “cold” state and
the “boiled” state.
φ
cold
=
•
1
0
‚
,
φ
boiled
=
•
0
1
‚
.
Let the time development operator for the pot be given as
U
(
t
1
+ Δ
t, t
1
) =
•
cos
ω
Δ
t

sin
ω
Δ
t
sin
ω
Δ
t
cos
ω
Δ
t
‚
.
This might me considered as describing a “heating” process as it transforms the “cold”
state
φ
cold
into the “boiled” state
φ
boiled
in a certain time.
But, of course, this is not
a perfect heating as the same process will convert the boiled state into the cold state
eventually. But, in a small 2level system, where there is not much “phase space”, these
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 Spring '11
 starg
 Tn, 2TB, POT

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