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arXiv:quant-ph/0202156 · Physics Letters A

Time problem in quantum mechanics and weak measurements

Julius Ruseckas, B. Kaulakys

Published Sep 1, 2001• 5 citations
arXivHTMLPDF
DOI
10.1016/s0375-9601(01)00500-x

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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.

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Parks introduced a formulation of time dependent weak values in 2008, which is the formalism we use in this paper. In this paper we extend notions from time dependent weak values to show that Hamiltonians associated with weak value measurements can be shown to exhibit even or odd symmetric properties. They exhibit PT or anti-PT symmetry, respectively. These symmetries are manifested during the measurement process as pointer translations, which have vanishing imaginary or vanishing real parts. The consequence of this that one can characterize some of the aspects of these symmetries of the time dependent Hamiltonians that arise from time dependent weak values. This allows one to generalize some work on non-Hermitian variables in quantum mechanics (QM) due to Bender related to weak values and weak measurement. We also speculate how these symmetries might apply to distinguishing between the various two-time interpretations of QM in the Conclusions of this paper.

arXiv:2111.10035•Nov 19, 2021•0 citations
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.
Quantum physics provides us with a formalism with which we can try to understand the microscopic world. However, the underpinning concepts of the basic principles of quantum mechanics itself are surprisingly ill understood. Even the ubiquitous concept of the wavefunction collapsing post measurement, lives outside the formal theory of quantum mechanics and must be taken as a postulate. The Schr\"odinger equation, which governs the evolution of quantum systems, does not describe the system during the measurement process. This is referred to as the measurement problem, and has been debated since the inception of quantum theory. Many different interpretations of quantum mechanics try to elucidate what happens during wavefunction collapse, such as the famous many-worlds interpretation. However, these interpretations either give rise to the same physical outcome or simply are not testable, and so cannot be physically differentiated from one another. We attempt to gain insight into the measurement problem through experiments involving sequential measurements, both in the weak and strong regime. We find surprising outcomes such as the persistence of time-ordering asymmetry in the weak measurement limit, and the survival of coherence in the strong measurement limit.
Parks introduced a formulation of time dependent weak values in 2008, which is the formalism we use in this paper. In this paper we extend notions from time dependent weak values to show that Hamiltonians associated with weak value measurements can be shown to exhibit even or odd symmetric properties. They exhibit PT or anti-PT symmetry, respectively. These symmetries are manifested during the measurement process as pointer translations, which have vanishing imaginary or vanishing real parts. The consequence of this that one can characterize some of the aspects of these symmetries of the time dependent Hamiltonians that arise from time dependent weak values. This allows one to generalize some work on non-Hermitian variables in quantum mechanics (QM) due to Bender related to weak values and weak measurement. We also speculate how these symmetries might apply to distinguishing between the various two-time interpretations of QM in the Conclusions of this paper.