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7 Some Models of “Quantum Measurement”
tionals ω on algebras of observable quantities” (which generalize the former two
classes of objects). The “observable quantities” (represented by operators, resp. elements of an algebra) correspond to experimental, or observational, arrangements of
empirical situations, in which the observer is able, after “installing” a specific state
ω of the observed system, to perceive and appreciate by his human senses some well
determined, in advance expected feelings (optical, auditory, acquired by touch or
in another way) of some specific perceptions that are clearly distinguishable from
others (e.g. when reading positions of a pointer, or hearing a characteristic sound
from a counter,…), so that they can be formalized into a form suitable for further
communication. A single observable A of a specific physical system appears in such
an empirical situation through a specific instance of a set of such clearly distinguishable phenomena, each of which can be (and, as a rule, is) denoted by a number
α j (∈ R) called the result of single measuring act in the state ω of a value of A
(not to be confused with “the value of A in ω ”—different single measuring acts
of the same observable on ω could lead to different results!). Many experiments on
microscopic systems performed in the history of microphysics have shown that we
are not able to prepare states of any microsystem in such a way that in a many times
repeated measurement of an observable on the same (prepared each time anew) state
ω one obtains the same measured value for each observable which can be chosen
for these repeated measurements. To state it briefly: For any state of any microsystem
there is some observable which does not have any specific value in that state. This
is reflected mathematically in, e.g., Heisenberg uncertainty relations. On the other
hand, to each value α j of the given observable A there exists (for observables with
discrete spectra) at least one state ω j such that the repeated measurements of A on
it give with certainty the same value α j . The problem arises because there is (with
certainty) some other observable B such that the repeated measurements of it on
the same state ω j give mutually different values β k = β l · · · ∈ R, i.e. the statistical
dispersion of the measured values of B in that ω j is nonzero. Sharp values (obtained
consistently in the identical, many times repeated measurements) β k of B can be
obtained in other states ω
k , for which, however, the measurements of some other
observables A, C, . . . would have nonzero dispersions.
The existing very successful mathematical model of QM provides solution of
this problem which consists in describing an arbitrarily chosen (but, by assumption,“pure”) state ω j as a linear superposition of some (again pure) states ω
k , ω
l . . . ,
i.e., if we express all the states in the form of vectors in a Hilbert space H v , in writing
the state in question as ψ j =
k c k ϕ k , where the correspondence with the values of
the observables A, B, . . . described now as linear operators on H v , is such that the
“state-vector” ψ j , corresponding to the state ω j , is an “eigenvector” of the operator
A (a common practice is to use the same symbol for the operator as for the physical
quantity represented by it): Aψ j = α j ψ j , and similarly the vectors ϕ k corresponding
to the states ω
k are the eigenvectors of the operator B : Bϕ k = β k ϕ k .
All this is, of course, very well known, and we have also briefly described it in our
Sect. 1.2. We recall it here to stress the unusual intuition required when dealing with
the phenomena described by the mathematical model of QM, in comparison with the
7 Some Models of “Quantum Measurement”
tionals ω on algebras of observable quantities” (which generalize the former two
classes of objects). The “observable quantities” (represented by operators, resp. elements of an algebra) correspond to experimental, or observational, arrangements of
empirical situations, in which the observer is able, after “installing” a specific state
ω of the observed system, to perceive and appreciate by his human senses some well
determined, in advance expected feelings (optical, auditory, acquired by touch or
in another way) of some specific perceptions that are clearly distinguishable from
others (e.g. when reading positions of a pointer, or hearing a characteristic sound
from a counter,…), so that they can be formalized into a form suitable for further
communication. A single observable A of a specific physical system appears in such
an empirical situation through a specific instance of a set of such clearly distinguishable phenomena, each of which can be (and, as a rule, is) denoted by a number
α j (∈ R) called the result of single measuring act in the state ω of a value of A
(not to be confused with “the value of A in ω ”—different single measuring acts
of the same observable on ω could lead to different results!). Many experiments on
microscopic systems performed in the history of microphysics have shown that we
are not able to prepare states of any microsystem in such a way that in a many times
repeated measurement of an observable on the same (prepared each time anew) state
ω one obtains the same measured value for each observable which can be chosen
for these repeated measurements. To state it briefly: For any state of any microsystem
there is some observable which does not have any specific value in that state. This
is reflected mathematically in, e.g., Heisenberg uncertainty relations. On the other
hand, to each value α j of the given observable A there exists (for observables with
discrete spectra) at least one state ω j such that the repeated measurements of A on
it give with certainty the same value α j . The problem arises because there is (with
certainty) some other observable B such that the repeated measurements of it on
the same state ω j give mutually different values β k = β l · · · ∈ R, i.e. the statistical
dispersion of the measured values of B in that ω j is nonzero. Sharp values (obtained
consistently in the identical, many times repeated measurements) β k of B can be
obtained in other states ω
k , for which, however, the measurements of some other
observables A, C, . . . would have nonzero dispersions.
The existing very successful mathematical model of QM provides solution of
this problem which consists in describing an arbitrarily chosen (but, by assumption,“pure”) state ω j as a linear superposition of some (again pure) states ω
k , ω
l . . . ,
i.e., if we express all the states in the form of vectors in a Hilbert space H v , in writing
the state in question as ψ j =
k c k ϕ k , where the correspondence with the values of
the observables A, B, . . . described now as linear operators on H v , is such that the
“state-vector” ψ j , corresponding to the state ω j , is an “eigenvector” of the operator
A (a common practice is to use the same symbol for the operator as for the physical
quantity represented by it): Aψ j = α j ψ j , and similarly the vectors ϕ k corresponding
to the states ω
k are the eigenvectors of the operator B : Bϕ k = β k ϕ k .
All this is, of course, very well known, and we have also briefly described it in our
Sect. 1.2. We recall it here to stress the unusual intuition required when dealing with
the phenomena described by the mathematical model of QM, in comparison with the