Abstract
In this chapter, we shall describe several specific models of “small” quantum mechanical systems interacting with (“measured by”) the “large” quantal system, whose elementary subsystems interact by short range interactions. The macroscopic/classical variables of the infinite systems will change now due to the interaction with the “small system” in the limit of infinite time only, with very slow convergence to the changed value of the “pointer position”. A model of a large but finite “measuring apparatus”, in which the convergence to the final “measurement result” is, on the other hand, very quick, is proposed as well.
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Notes
- 1.
In the case of some different choices of (locally perturbed stationary) initial states in this model, the subsequent time evolutions of the chain could be different: e.g., an initial segment could move quasiperiodically and the infinite rest of the chain will converge to a macroscopically different state.
- 2.
We shall use here the Dirac bra-ket notation for convenience.
- 3.
We consider here, for the sake of simplicity, the measured system after the measurement as a part of the apparatus, what makes no difference for observing results of measurements via various macrostates—the macroobservables of the compound system are identical with those of the measuring apparatus alone. See however the Sect. 7.3.7 below.
- 4.
In accordance with that, the notation in this section will be changed slightly with respect to the notation in the Sect. 7.3.5.
- 5.
We are speaking here about the states on the algebra generated by \(a_j, a^*_j\) with \(j>0\) only.
- 6.
Note that, due to time-reflection symmetry of all the systems considered here, quite analogical equations and the corresponding results could be obtained also for the function \(t\mapsto F(-t),\ t\ge 0\).
- 7.
To see this, calculate \(\sum _{n=0}^\infty (h_+*)^n(t)\) for \(h\equiv const\).
- 8.
This notation should not be confused with \(\mathcal{F}(F)_+:=\theta \cdot \mathcal{F}(F)\equiv (\hat{F})_+ \), differing by the place where the sign “+” occurs.
- 9.
We shall omit usually in the following the tensor-product symbol \(\otimes \), according our preceding conventions.
- 10.
The formulation and main features of the dynamics of this model were presented first time in [33]. The technical details are described in [39].
- 11.
An exception consists in possible introduction ‘by hand’ by a theoretician some ‘superselection rules’ representing a model of ‘macroscopic difference’ and forbidding interference between vectors from specific subspaces of \({\mathcal{H}_v}\), cf. e.g. [167].
- 12.
Another possibility is some, up to now not clearly specified basic change of QM, as it was most urgently proposed by Penrose in several his publications, e.g. in [236–238]; the main motivation for these reformulations of QM was some inclusion of the usually postulated “reduction of wave packet” [226], called by Penrose the “process R” , into the dynamics of general QM systems.
- 13.
Let us illustrate briefly this idea on a long but finite spin-1/2 chain of the length N with the \(C^*\)-algebra \(\mathfrak {A}\) of its observables generated by the spin creation-annihilation operators \(a_j, a^*_j\ (j=1,2,\dots N)\) acting on the finite dimensional Hilbert space \(\mathcal{{H}}_N:=(C^2)^N\): If we are able to use apparatuses detecting the observables of this chain occurring in an arbitrary of the \(C^*\)-subalgebras \(\mathfrak {B}\subset \mathfrak {A}\) generated by any of the fixed restricted set of operators \(a_{j_m}, a^*_{j_m}\ (m=1,2,\dots K\ll N,\ 0\le j_m\le N)\) only, then the states \(|\Psi \rangle , |\Phi \rangle \) from \(\mathcal{{H}}_N\) for which it holds \(\langle \Psi |B|\Phi \rangle \equiv 0\ \forall B\in \mathfrak {B}\) could be considered as ‘almost macroscopically different’, resp. ‘empirically disjoint’. This happens, e.g., if in the state \(|\Psi \rangle \) all the spins are ‘pointing up’, and in the state \(|\Phi \rangle \) all the spins are ‘pointing down’.
- 14.
The “second quantization” \(d\Gamma ({\varvec{h}})\) of the ‘one-Fermi-particle-operator’ \({\varvec{h}}\) is the linear operator acting in the Fermi Fock space \(\mathcal{{H}}_F:=\oplus _{n=0}^\infty P_-\otimes _1^n\mathfrak {h}\), where \(P_-\) is the antisymmetrization operator, such that \(d\Gamma ({\varvec{h}})P_-\otimes _{k=1}^n\psi _k:=P_-\sum _{j=1}^n \psi _1\otimes \psi _2\otimes \dots \otimes {\varvec{h}}\psi _j\otimes \dots \otimes \psi _n\) for all \(n\in {\mathbb Z}_+\).
- 15.
We will work here with pure states (resp. vector states) only. In fact, it is not necessary to use density matrices in an analysis of the process of measurement in QM, as shown, e.g. by Wigner in [339].
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Bóna, P. (2020). Some Models of “Quantum Measurement”. In: Classical Systems in Quantum Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-030-45070-0_7
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DOI: https://doi.org/10.1007/978-3-030-45070-0_7
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