Chapter 7
Some Models of “Quantum
Measurement”
7.1 Introductory Notes
7.1.1 The interactions in the models of large quantal systems described in Chap. 6
were of specific long-range type. All the elementary subsystems (“particles” or
“spins”) mutually interacted with each other ‘in the same way’ as if all the subsystems were not distinguishable from each other, i.e. the multi-particle interaction
was invariant with respect to permutations of the particles independent of their positions in the lattice , as specified by (6.1.1). Such interactions led in infinite limit of
the number N of the subsystems to the dynamics of “mean-field type”, i.e. to such a
dynamics that each individual subsystem moved as if it was immersed in an external
(in general time dependent) field produced by the whole collection of the infinite
number of all the subsystems and independent of any changes of the state of any
of these subsystems. The resulting dynamics was such that macroscopic (classical)
parameters of the infinite system were varying in time according to the dynamics of
some classical mechanical Hamiltonian system.
In this chapter, we shall describe several specific models of large quantal systems
whose elementary subsystems interact by short range interactions. The macroscopic,
or “classical”, variables of the infinite systems will change now just in the limit
t → ∞, because the short range interaction results in finite velocity of spreading of
local changes across the infinite system, hence in finite times only local variables
corresponding to changes of finite subsystems are changed.
7.1.2 We shall briefly describe here a few quantum-mechanical model systems
describing interactions of a ‘microscopic system’ with a ‘macroscopic system’ leading to a ‘macroscopic change’ in the second system. This means that such systems
describe schemes modeling dynamics of processes like ‘quantum measurement’ as a
process ascribing a classical probability distribution of ‘measurement results’ (given
by macroscopically distinct states of the ‘macroscopic system’ which plays the role of
the ‘measuring apparatus’) to the corresponding (according to the ‘measured observable’) quantum-mechanical linear decomposition of the wave function of the ‘microscopic system’. Construction of these models was inspired mainly by the classical
© Springer Nature Switzerland AG 2020, corrected publication 2020
P. Bóna, Classical Systems in Quantum Mechanics,
https://doi.org/10.1007/978-3-030-45070-0_7
167
Some Models of “Quantum
Measurement”
7.1 Introductory Notes
7.1.1 The interactions in the models of large quantal systems described in Chap. 6
were of specific long-range type. All the elementary subsystems (“particles” or
“spins”) mutually interacted with each other ‘in the same way’ as if all the subsystems were not distinguishable from each other, i.e. the multi-particle interaction
was invariant with respect to permutations of the particles independent of their positions in the lattice , as specified by (6.1.1). Such interactions led in infinite limit of
the number N of the subsystems to the dynamics of “mean-field type”, i.e. to such a
dynamics that each individual subsystem moved as if it was immersed in an external
(in general time dependent) field produced by the whole collection of the infinite
number of all the subsystems and independent of any changes of the state of any
of these subsystems. The resulting dynamics was such that macroscopic (classical)
parameters of the infinite system were varying in time according to the dynamics of
some classical mechanical Hamiltonian system.
In this chapter, we shall describe several specific models of large quantal systems
whose elementary subsystems interact by short range interactions. The macroscopic,
or “classical”, variables of the infinite systems will change now just in the limit
t → ∞, because the short range interaction results in finite velocity of spreading of
local changes across the infinite system, hence in finite times only local variables
corresponding to changes of finite subsystems are changed.
7.1.2 We shall briefly describe here a few quantum-mechanical model systems
describing interactions of a ‘microscopic system’ with a ‘macroscopic system’ leading to a ‘macroscopic change’ in the second system. This means that such systems
describe schemes modeling dynamics of processes like ‘quantum measurement’ as a
process ascribing a classical probability distribution of ‘measurement results’ (given
by macroscopically distinct states of the ‘macroscopic system’ which plays the role of
the ‘measuring apparatus’) to the corresponding (according to the ‘measured observable’) quantum-mechanical linear decomposition of the wave function of the ‘microscopic system’. Construction of these models was inspired mainly by the classical
© Springer Nature Switzerland AG 2020, corrected publication 2020
P. Bóna, Classical Systems in Quantum Mechanics,
https://doi.org/10.1007/978-3-030-45070-0_7
167