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Exact Energy-Time Uncertainty Relation for Arrival Time by Absorption · Around

2011

Exact Energy-Time Uncertainty Relation for Arrival Time by Absorption

Kiukas, Jukka, Ruschhaupt, Andreas, Schmidt, Piet O. et al.

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We prove an uncertainty relation for energy and arrival time, where the arrival of a particle at a detector is modeled by an absorbing term added to the Hamiltonian.

In this well-known scheme the probability for the particle's arrival at the counter is identified with the loss of normalization for an initial wave packet.
Under the sole assumption that the absorbing term vanishes on the initial wave function, we show that $\Delta T \Delta E \geq \sqrt p \hbar/2$ and $<T> \Delta E\geq 1.37\sqrt p\hbar$, where $<T>e$ denotes the mean arrival time, and $p$ is the probability for the particle to be eventually absorbed.
Nearly minimal uncertainty can be achieved in a two-level system, and we propose a trapped ion experiment to realize this situation.

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J. G. Muga, A. Ruschhaupt, and A. del Campo, eds., Time in quantum mechanics – Vol.2 , Lect. Notes Phys., Vol. 789 (Springer, 2009)
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H. G. Barros, A. Stute, T. E. Northup, C. Russo, P. O. Schmidt, and R. Blatt, New J. Phys. 11

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P. Busch, “The time-energy uncertainty relation,” in [ 18 ]

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I. L. Egusquiza, J. G. Muga, and A. D. Baute, “‘Standard’ quantum mechanical approach to times of arrival,” in [ 18 ]

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A. Ruschhaupt, J. G. Muga, and G. C. Hegerfeldt, “Detector models for the quantum time of arrival,” in [ 19 ]

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2011

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