Chapter 1
Introduction
1.1 Motivation and Summary
1.1.1 Successful communication and manipulation with ‘objects’ requires construction of some adequate theoretical models (≈ theories) of some classes of ‘objects’,
resp. ‘phenomena’. Different phenomena might be described by different theoretical
schemes. These schemes should be, however, mutually consistent in the sense giving
the same results for phenomena lying in the common domain of applicability of different theories. If one of the theories is considered to be ‘more general’ then a second
one, then the whole domain of applicability of the second theory has to be contained
in the domain of the first one. This is the case of quantum mechanics (QM), which is
believed to be a ‘covering theory’ of the more special classical mechanics (CM)—to
the extent of measurement precision of apparatuses determining of ‘classical systems’. Hence we can ask how to describe phenomena belonging to the domain of
applicability of CM in the framework of QM.
1.1.2 Any single phenomenon, which is unambiguously and reproducibly determined by a specification of an empirical situation is, however, expressible in terms of
parameters (resp. variables) occurring in CM: coordinates of positions and velocities
of points distinguished and measured by ‘macroscopic bodies’ and various correlations between these variables. Hence also any experimentally realizable situation
described in QM (which need not be a consequence of laws of CM, e.g. observation
of spectra of atoms) is expressible in terms of CM (e.g. preparation of sources of
radiation and measurement of positions of spectral lines displayed on screens). Quantal (:= quantum mechanical) phenomena are not only observed on a ‘background’
and ‘from the point of view’ of quantities describing states of macroscopic bodies
(resp. of such parameters, the behaviour of which is adequately described by laws
of CM), but also specific theoretical models for description of such phenomena in
the framework of QM are constructed under strong influence of existing models in
CM (e.g. the quantal models of atoms compared to classical planetary motions, or,
more generally, some systems of canonically conjugated observables in the sense of
the Hamiltonian CM correspond isomorphica1ly to a subset of quantal observables).
© 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_1
1
Introduction
1.1 Motivation and Summary
1.1.1 Successful communication and manipulation with ‘objects’ requires construction of some adequate theoretical models (≈ theories) of some classes of ‘objects’,
resp. ‘phenomena’. Different phenomena might be described by different theoretical
schemes. These schemes should be, however, mutually consistent in the sense giving
the same results for phenomena lying in the common domain of applicability of different theories. If one of the theories is considered to be ‘more general’ then a second
one, then the whole domain of applicability of the second theory has to be contained
in the domain of the first one. This is the case of quantum mechanics (QM), which is
believed to be a ‘covering theory’ of the more special classical mechanics (CM)—to
the extent of measurement precision of apparatuses determining of ‘classical systems’. Hence we can ask how to describe phenomena belonging to the domain of
applicability of CM in the framework of QM.
1.1.2 Any single phenomenon, which is unambiguously and reproducibly determined by a specification of an empirical situation is, however, expressible in terms of
parameters (resp. variables) occurring in CM: coordinates of positions and velocities
of points distinguished and measured by ‘macroscopic bodies’ and various correlations between these variables. Hence also any experimentally realizable situation
described in QM (which need not be a consequence of laws of CM, e.g. observation
of spectra of atoms) is expressible in terms of CM (e.g. preparation of sources of
radiation and measurement of positions of spectral lines displayed on screens). Quantal (:= quantum mechanical) phenomena are not only observed on a ‘background’
and ‘from the point of view’ of quantities describing states of macroscopic bodies
(resp. of such parameters, the behaviour of which is adequately described by laws
of CM), but also specific theoretical models for description of such phenomena in
the framework of QM are constructed under strong influence of existing models in
CM (e.g. the quantal models of atoms compared to classical planetary motions, or,
more generally, some systems of canonically conjugated observables in the sense of
the Hamiltonian CM correspond isomorphica1ly to a subset of quantal observables).
© 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_1
1