170
7 Some Models of “Quantum Measurement”
It is motivating and orientating for researchers to believe in the existence of some
‘final truth’. It is an important psychological aspect of scientific progress. The faith
in our ‘reliably verified knowledge’ is perhaps necessary also for the success of our
practical life. But if a theory is completed (i.e. if it is in agreement with all available
‘trustworthy’ experimental results), it can be (and eventually should be) challenged
in science.
Any theory, as well as any concept appearing in our consciousness or/and used
in our communication is a human construction. Hence it is dependent on human
interests and activities, and these activities are perpetually evolving–sometimes even
substantially changing. Hence, also our attention and interests are changing. This
implies that the motives for our intellectual activity are perpetually developing. The
resulting our ‘pictures of the world’, either global, or various special, are correspondingly changing along with these other changes. And, people also look then on ‘the
same things’ by different ways and from different points of view than before.
The ‘models’ presented in this chapter are just very simple abstractions imitating certain features of mutual interactions of general classes of physical systems: microsystems described adequately by QM, and macroscopic systems (usually described by CM) consisting of a large number of microsystems. We tried to
be mathematically rigorous in proceeding from basic axioms of QM to definitions
of introduced concepts and constructions of the mathematical models, as well as to
description and obtaining the consequences of the used dynamics. This emphasis on
mathematical rigor was motivated by our desire to show clearly that the obtained
results are exact consequences of the currently generally accepted formal theory
of QM.
7.3 Quantum Domino
7.3.1 We shall describe here briefly (for more details we refer to [35, 36]) the
model of infinite spin chain which we call, due to the character of its time evolution,
Quantum Domino (QD). The 1/2 spins are ordered by the values of the index i ∈ Z
and the Hamiltonian produces a local nearest three body interaction. This interaction
can be described easily as follows: If the hamiltonian acts on the state with the i-th
spin “pointing up” and the (i + 2)-nd spin “pointing down”, then the (i + 1)-st spin
changes its orientation to the opposite one. The dynamics of the two sided infinite
spin-1/2 quantum chain has spin configurations “all spins pointing up”, and “all spins
pointing down” as stationary states, which are unstable: If we reverse the direction
of one of the spins in these states, the new state will develop in the limit t → ∞
into another stationary (and ‘macroscopically’ stable) state, in which all the spins
lying on one side of the reversed spin are also reversed, and all the spins lying on
the other side of that spin stay unchanged. Since this evolution leads to the change
of the value of a macroscopic observable of the chain, it can be used as a model for
‘quantum measurement’ of microscopic observables of a single spin of the chain.
We shall show in this section how such model works.
7 Some Models of “Quantum Measurement”
It is motivating and orientating for researchers to believe in the existence of some
‘final truth’. It is an important psychological aspect of scientific progress. The faith
in our ‘reliably verified knowledge’ is perhaps necessary also for the success of our
practical life. But if a theory is completed (i.e. if it is in agreement with all available
‘trustworthy’ experimental results), it can be (and eventually should be) challenged
in science.
Any theory, as well as any concept appearing in our consciousness or/and used
in our communication is a human construction. Hence it is dependent on human
interests and activities, and these activities are perpetually evolving–sometimes even
substantially changing. Hence, also our attention and interests are changing. This
implies that the motives for our intellectual activity are perpetually developing. The
resulting our ‘pictures of the world’, either global, or various special, are correspondingly changing along with these other changes. And, people also look then on ‘the
same things’ by different ways and from different points of view than before.
The ‘models’ presented in this chapter are just very simple abstractions imitating certain features of mutual interactions of general classes of physical systems: microsystems described adequately by QM, and macroscopic systems (usually described by CM) consisting of a large number of microsystems. We tried to
be mathematically rigorous in proceeding from basic axioms of QM to definitions
of introduced concepts and constructions of the mathematical models, as well as to
description and obtaining the consequences of the used dynamics. This emphasis on
mathematical rigor was motivated by our desire to show clearly that the obtained
results are exact consequences of the currently generally accepted formal theory
of QM.
7.3 Quantum Domino
7.3.1 We shall describe here briefly (for more details we refer to [35, 36]) the
model of infinite spin chain which we call, due to the character of its time evolution,
Quantum Domino (QD). The 1/2 spins are ordered by the values of the index i ∈ Z
and the Hamiltonian produces a local nearest three body interaction. This interaction
can be described easily as follows: If the hamiltonian acts on the state with the i-th
spin “pointing up” and the (i + 2)-nd spin “pointing down”, then the (i + 1)-st spin
changes its orientation to the opposite one. The dynamics of the two sided infinite
spin-1/2 quantum chain has spin configurations “all spins pointing up”, and “all spins
pointing down” as stationary states, which are unstable: If we reverse the direction
of one of the spins in these states, the new state will develop in the limit t → ∞
into another stationary (and ‘macroscopically’ stable) state, in which all the spins
lying on one side of the reversed spin are also reversed, and all the spins lying on
the other side of that spin stay unchanged. Since this evolution leads to the change
of the value of a macroscopic observable of the chain, it can be used as a model for
‘quantum measurement’ of microscopic observables of a single spin of the chain.
We shall show in this section how such model works.