The experimental realization of successive non-demolition measurements on single microscopic systems brings up the question of ergodicity in Quantum Mechanics (QM). We investigate whether time averages over one realization of a single system are related to QM averages over an ensemble of similarly prepared systems. We adopt a generalization of von Neumann model of measurement, coupling the system to $N$ "probes" --with a strength that is at our disposal-- and detecting the latter. The model parallels the procedure followed in experiments on Quantum Electrodynamic cavities. The modification of the probability of the observable eigenvalues due to the coupling to the probes can be computed analytically and the results compare qualitatively well with those obtained numerically by the experimental groups. We find that the problem is not ergodic, except in the case of an eigenstate of the observable being studied.The experimental realization of successive non-demolition measurements on single microscopic systems brings up the question of ergodicity in Quantum Mechanics (QM). We investigate whether time averages over one realization of a single system are related to QM averages over an ensemble of similarly prepared systems. We adopt a generalization of von Neumann model of measurement, coupling the system to $N$ "probes" --with a strength that is at our disposal-- and detecting the latter. The model parallels the procedure followed in experiments on Quantum Electrodynamic cavities. The modification of the probability of the observable eigenvalues due to the coupling to the probes can be computed analytically and the results compare qualitatively well with those obtained numerically by the experimental groups. We find that the problem is not ergodic, except in the case of an eigenstate of the observable being studied.

arXiv:1312.6907Dec 25, 20130 citations

On the Measurement Problem of Quantum Mechanics

Jeffrey Bub

Springer eBooksJan 1, 198911 citations

Measurement and ergodicity in quantum mechanics

M. Bauer, Pier A. Mello

The experimental realization of successive non-demolition measurements on single microscopic systems brings up the question of ergodicity in quantum mechanics (QM). We investigate whether time averages over one realization of a single system are related to QM averages over an ensemble of similarly prepared systems. We adopt a generalization of the von Neumann model of measurement, coupling the system to N ‘probes’, with a strength that is at our disposal, and detecting the latter. The model parallels the procedure followed in experiments on quantum electrodynamic cavities. The modification of the probability of the observable eigenvalues due to the coupling to the probes can be computed analytically and the results compare qualitatively well with those obtained numerically by the experimental groups. We find that the problem is not ergodic, except in the case of an eigenstate of the observable being studied.The experimental realization of successive non-demolition measurements on single microscopic systems brings up the question of ergodicity in quantum mechanics (QM). We investigate whether time averages over one realization of a single system are related to QM averages over an ensemble of similarly prepared systems. We adopt a generalization of the von Neumann model of measurement, coupling the system to N ‘probes’, with a strength that is at our disposal, and detecting the latter. The model parallels the procedure followed in experiments on quantum electrodynamic cavities. The modification of the probability of the observable eigenvalues due to the coupling to the probes can be computed analytically and the results compare qualitatively well with those obtained numerically by the experimental groups. We find that the problem is not ergodic, except in the case of an eigenstate of the observable being studied.

Journal of Physics A Mathematical and TheoreticalApr 14, 20151 citation

Time problem in quantum mechanics and its analysis by the concept of weak measurements

Julius Ruseckas, Bronislovas Kaulakys

The model of weak measurements is applied to various problems, related to the time problem in quantum mechanics.The review and generalization of the theoretical analysis of the time problem in quantum mechanics based on the concept of weak measurements are presented.A question of the time interval the system spends in the specified state, when the final state of the system is given, is raised.Using the concept of weak measurements the expression for such time is obtained.The results are applied to the tunnelling problem.A procedure for the calculation of the asymptotic tunnelling and reflection times is proposed.Examples for δ-form and rectangular barrier illustrate the obtained results.Using the concept of weak measurements the arrival time probability distribution is defined by analogy with the classical mechanics.The proposed procedure is suitable to the free particles and to particles subjected to an external potential, as well.It is shown that such an approach imposes an inherent limitation to the accuracy of the arrival time definition.The model of weak measurements is applied to various problems, related to the time problem in quantum mechanics.The review and generalization of the theoretical analysis of the time problem in quantum mechanics based on the concept of weak measurements are presented.A question of the time interval the system spends in the specified state, when the final state of the system is given, is raised.Using the concept of weak measurements the expression for such time is obtained.The results are applied to the tunnelling problem.A procedure for the calculation of the asymptotic tunnelling and reflection times is proposed.Examples for δ-form and rectangular barrier illustrate the obtained results.Using the concept of weak measurements the arrival time probability distribution is defined by analogy with the classical mechanics.The proposed procedure is suitable to the free particles and to particles subjected to an external potential, as well.It is shown that such an approach imposes an inherent limitation to the accuracy of the arrival time definition.

arXiv:quant-ph/0409006Lithuanian Journal of PhysicsJan 1, 20040 citations

N

"probes" --with a strength that is at our disposal-- and detecting the latter. The model parallels the procedure followed in experiments on Quantum Electrodynamic cavities. The modification of the probability of the observable eigenvalues due to the coupling to the probes can be computed analytically and the results compare qualitatively well with those obtained numerically by the experimental groups. We find that the problem is not ergodic, except in the case of an eigenstate of the observable being studied.