Time-of-arrival probabilities and quantum measurements: III Decay of unstable states

Abstract

We study the decay of unstable states by formulating quantum tunneling as a time-of-arrival problem: we determine the detection probability for particles at a detector located a distance L from the tunneling region. For this purpose, we use a Positive-Operator-Valued-Measure (POVM) for the time-of-arrival determined in quant-ph/0509020 [JMP 7, 122106 (2006)]. This only depends on the initial state, the Hamiltonian and the location of the detector. The POVM above provides a well-defined probability density and an unambiguous interpretation of all quantities involved. We demonstrate that the exponential decay only arises if three specific mathematical conditions are met. Their physical content is the following: (i) the decay time is much larger than any microscopic timescale, so that the fine details of the initial state can be ignored, (ii) there is no quantum coherence between the different `attempts' of the particle to traverse the barrier, and (iii) the transmission probability varies little within the momentum spread of the initial state. We also determine the long time limits of the decay probability and we identify regimes, in which the decays have no exponential phase.

arXiv comment27 pages, 2 figs--revised version, small changes, references added, version to appear in J. Math. Phys

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• Quantum temporal probabilities in tunneling systems
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• Decays of unstable quantum systems
• Characterizing Multiple Solutions to the Time-Energy Canonical Commutation Relation via Quantum Dynamics
• Characterizing Multiple Solutions to the Time - Energy Canonical Commutation Relation via Internal Symmetries
• Real-time particle-detection probabilities in accelerated macroscopic detectors
• Path of a tunneling particle
• Multi-Time Measurements in Hawking Radiation: Information at Higher-Order Correlations