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

TL;DR

This paper formulates quantum tunneling decay as a time-of-arrival measurement using POVMs, identifying conditions for exponential decay and analyzing long-time behavior of unstable quantum states.

Contribution

It introduces a POVM-based approach to quantum decay, clarifies conditions for exponential decay, and explores long-time decay regimes.

Findings

Exponential decay occurs under three specific mathematical conditions.

Decay behavior at long times can deviate from exponential.

The approach provides a clear probability interpretation for decay processes.

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.