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Advanced Quantum Theory
Amath 473, 673 / Phys 454
Problem Set 5
This assignment is due
at the beginning of class
on Monday Dec 6th, 2010.
1.
Unitary Evolution vs Collapse Process.
In this problem you will prove that the collapse postulate can not be derived from unitary evolution acting on
(any) Hilbert space. To do this assume an initial system state

ψ
i
S
∈ H
S
, an initial measurement apparatus
state

ready
i
A
∈ H
A
and an initial environment state

χ
i
E
∈ H
E
(describing any other degrees of freedom of
possible relevance). For simplicity, assume that all initial states are pure and that the system state is a twolevel
system
H
S
=
C
2
, e.g., a spin, with orthonormal basis states

up
i
S
and

down
i
S
. A measurement device that
is ideal performs perfectly
faithful measurements
, which means that distinct input basis states produce
perfectly
distinguishable
ﬁnal detector states, i.e., a macroscopic pointer is moved to the ‘left’ if the particle is prepared in
the ‘up’ state and to the ‘right’ if the particle is prepared in the ‘down’ state.
Perfectly distinguishable
here means
that the detector states (i.e., the states associated with ‘left’ and ‘right’) are orthogonal.
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This note was uploaded on 01/17/2011 for the course AMATH 454 taught by Professor Joseph during the Fall '10 term at Waterloo.
 Fall '10
 joseph

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