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arXiv:1307.0528

Complementarity and Classical Limit of Quantum Mechanics: Energy Measurement aspects

Adélcio C. Oliveira, Zolacir Trindade de Oliveira Junior, Nestor Santos Correia

Published Jul 1, 2013• 3 citations
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Abstract

In the present contribution we discuss the role of experimental limitations in the classical limit problem. We studied some simple models and found that Quantum Mechanics does not re-produce classical mechanical predictions, unless we consider the experimental limitations ruled by uncertainty principle. We have shown that the discrete nature of energy levels of integrable systems can be accessed by classical measurements. We have defined a precise limit for this procedure. It may be used as a tool to define the classical limit as far as the discrete spectra of integrable systems are concerned. If a diffusive environment is considered, we conclude that the "lifetime" of discreteness is approximately $1/κ$ ($κ$ is the diffusion constant), thus it was possible to relate the classical limit of a spectra with the action of an environment and experimental resolution.

DOI
10.48550/arxiv.1307.0528

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In the present contribution we discuss the role of experimental limitations in the classical limit problem. We studied some simple models and found that Quantum Mechanics does not re-produce classical mechanical predictions, unless we consider the experimental limitations ruled by uncertainty principle. We have shown that the discrete nature of energy levels of integrable systems can be accessed by classical measurements. We have defined a precise limit for this procedure. It may be used as a tool to define the classical limit as far as the discrete spectra of integrable systems are concerned. If a diffusive environment is considered, we conclude that the "lifetime" of discreteness is approximately 1/κ1/κ1/κ (κκκ is the diffusion constant), thus it was possible to relate the classical limit of a spectra with the action of an environment and experimental resolution.
In the present contribution we discuss the role of experimental limitations in the classical limit problem. We studied some simple models and found that Quantum Mechanics does not re-produce classical mechanical predictions, unless we consider the experimental limitations ruled by uncertainty principle. We have shown that the discrete nature of energy levels of integrable systems can be accessed by classical measurements. We have defined a precise limit for this procedure. It may be used as a tool to define the classical limit as far as the discrete spectra of integrable systems are concerned.
The classical limit of quantum mechanics is discussed for closed quantum systems in terms of observational aspects. Initially, the failure of the limit h->0 is explicitly demonstrated in a model of two quantum mechanically interacting oscillators by showing that neither quantum expectations reduce to Newtonian trajectories nor entanglement vanishes. This result suggests that the quantum-to-classical transition occurs only at an approximative level, which is regulated by the low accuracy of the measurements. In order to verify the consistence of these ideas we take into account the experimental resolution of physical measurements by introducing a discretized formulation for the quantum structure of wave functions. As a result, in the low-resolution limit the quasi-determinism is recovered and hence the quantum-to-classical transition is shown to occur adequately. Other puzzling problems, such as the classical limit of quantum superpositions and nonlocal correlations, are naturally address as well.
Although time measurements are routinely performed in laboratories, their theoretical description is still an open problem. Correspondingly, the status of the energy-time uncertainty relation is unsettled. In the first part of this work the necessity of positive operator valued measures (POVM) as descriptions of every quantum experiment is reviewed, as well as the suggestive role played by the probability current in time measurements. Furthermore, it is shown that no POVM exists, which approximately agrees with the probability current on a very natural set of wave functions; nevertheless, the choice of the set is crucial, and on more restrictive sets the probability current does provide a good arrival time prediction. Some ideas to experimentally detect quantum effects in time measurements are discussed. In the second part of the work the energy-time uncertainty relation is considered, in particular for a model of alpha decay for which the variance of the energy can be calculated explicitly, and the variance of time can be estimated. This estimate is tight for systems with long lifetimes, in which case the uncertainty relation is shown to be satisfied. Also the linewidth-lifetime relation is shown to hold, but contrary to the common expectation, it is found that the two relations behave independently, and therefore it is not possible to interpret one as a consequence of the other. To perform the mentioned analysis quantitative scattering estimates are necessary. To this end, bounds of the form ∥1Re−iHtψ∥22≤Ct−3\|1_Re^{-iHt}ψ\|_2^2 \leq C t^{-3}∥1R​e−iHtψ∥22​≤Ct−3 have been derived, where ψψψ denotes the initial state, HHH the Hamiltonian, RRR a positive constant, and CCC is explicitly known. As intermediate step, bounds on the derivatives of the SSS-matrix in the form $\|1_K S^{(n)}\|_\infty \leq C_{n,K} $ have been established, with n=1,2,3n=1,2,3n=1,2,3, and the constants Cn,KC_{n,K}Cn,K​ explicitly known.
This chapter focuses on the notions of classical instruments and of classical measurements. It reviews the theories, based on the concept of macroscopic properties, that is, on the essential complexity of the measuring apparatus. One of the most distinctive features of quantum mechanics is the fact that it often assigns precise structures—or discrete "choices" among possible different structures—to the physical systems it describes. Hisenberg's uncertainty relations bear on the dispersions of ensembles as a whole, whereas the limitations that serve to define "classical measurements" essentially concern our possibilities of actually performing measurements in different elements of the ensemble. Thus from the uncertainty relations alone one can certainly not derive the conclusion that nonclassical measurements are impossible. The chapter considers the restricted set of observables. It is convenient to use the word "macrostate" to designate the whole set of microstates on which all the classical observables forming a definite complete set have definite values.
The accuracies of measurements of almost all fundamental physical constants have increased by factors of about 10 000 during the past 60 years. Although some of the improvements are due to greater care, most are due to new techniques based on quantum mechanics. Although the Heisenberg Uncertainty Principle often limits measurement accuracies, in many cases the validity of quantum mechanics makes possible the vastly improved measurement accuracies. Seven quantum features that have a profound influence on the science of measurements are: (1) Existence of discrete quantum states of energy. (2) Energy conservation in transitions between two states. (3) Electromagnetic radiation of frequency v is quantized with energy hv per quantum. (4) The identity principle. (5) The Heisenberg Uncertainty Principle. (6) Addition of probability amplitudes (not probabilities). (7) Wave and coherent phase phenomena. Of these seven quantum features, only the Heisenberg Uncertainty Principle limits the accuracy of measurements, and its affect is often negligibly small. The other six features make possible much more accurate measurements of quantum systems than with almost all classical systems. These effects are discussed and illustrated.