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Register a new userFlow of time during energy measurements and the resulting time-energy uncertainty relations
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Uncertainty relations play a crucial role in quantum mechanics. A well-defined method exists for deriving such uncertainties for pairs of observables. It does not include, however, an important family of fundamental relations: the time-energy uncertainty relations. As a result, different approaches have been used for obtaining them in diversified scenarios. The one of interest here revolves around the idea of the existence or inexistence of a minimum duration for an energy measurement with a certain precision. In our study, we use the Page and Wooters timeless framework to investigate how energy measurements modify the relative flow of time between internal and external clocks. This provides a unified framework for discussing the topic, recovering previous results and leading to new ones. We also show that the evolution of the external clock with respect to the internal one is non-unitary.
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The Heisenberg and Mandelstam-Tamm time-energy uncertainty relations are analyzed. The conlusion resulting from this analysis is that within the Quantum Mechanics of Schr{o}dinger and von Neumann, the status of these relations can not be considered as the same as the status of the position-momentum uncertainty relations, which are rigorous. The conclusion is that the time--energy uncertainty relations can not be considered as universally valid.
Using the Mandelstam-Tamm method we derive time-energy uncertainty relations for neutrino oscillations. We demonstrate that the small energy uncertainty of antineutrinos in a recently considered experiment with recoilless resonant (Mossbauer) production and absorption of tritium antineutrinos is in conflict with the energy uncertainty which, according to the time-energy uncertainty relation, is necessary for neutrino oscillations to happen. A Mossbauer neutrino experiment could provide a unique possibility to test the applicability of the time-energy uncertainty relation to neutrino oscillations and to reveal the true nature of neutrino oscillations.
We prove the uncertainty relation $sigma_T , sigma_E geq hbar/2$ between the time $T$ of detection of a quantum particle on the surface $partial Omega$ of a region $Omegasubset mathbb{R}^3$ containing the particles initial wave function, using the absorbing boundary rule for detection time, and the energy $E$ of the initial wave function. Here, $sigma$ denotes the standard deviation of the probability distribution associated with a quantum observable and a wave function. Since $T$ is associated with a POVM rather than a self-adjoint operator, the relation is not an instance of the standard version of the uncertainty relation due to Robertson and Schrodinger. We also prove that if there is nonzero probability that the particle never reaches $partial Omega$ (in which case we write $T=infty$), and if $sigma_T$ denotes the standard deviation conditional on the event $T<infty$, then $sigma_T , sigma_E geq (hbar/2) sqrt{mathrm{Prob}(T<infty)}$.
We present a unified view of the Berry phase of a quantum system and its entanglement with surroundings. The former reflects the nonseparability between a system and a classical environment as the latter for a quantum environment, and the concept of geometric time-energy uncertainty can be adopted as a signature of the nonseparability. Based on this viewpoint, we study their relationship in the quantum-classical transition of the environment, with the aid of a spin-half particle (qubit) model exposed to a quantum-classical hybrid field. In the quantum-classical transition, the Berry phase has a similar connection with the time-energy uncertainty as the case with only a classical field, whereas the geometric phase for the mixed state of the qubit exhibits a complementary relationship with the entanglement. Namely, for a fixed time-energy uncertainty, the entanglement is gradually replaced by the mixed geometric phase as the quantum field vanishes. And the mixed geometric phase becomes the Berry phase in the classical limit. The same results can be draw out from a displaced harmonic oscillator model.
One manifestation of quantum resonances is a large sojourn time, or autocorrelation, for states which are initially localized. We elaborate on Lavines time-energy uncertainty principle and give an estimate on the sojourn time. For the case of perturbed embedded eigenstates the bound is explicit and involves Fermis Golden Rule. It is valid for a very general class of systems. We illustrate the theory by applications to resonances for time dependent systems including the AC Stark effect as well as multistate systems.
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