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

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10.1063/1.2839920

### 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 from the tunneling region. For this purpose, we use a positive-operator-valued measure (POVM) for the time-of-arrival determined by Anastopoulos and Savvidou [J. Math. Phys.47, 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.

© 2008 American Institute of Physics

Received 19 June 2007
Accepted 11 January 2008
Published online 15 February 2008

Acknowledgments: I would like to thank N. Savvidou and D. Ghikas for useful discussions on the issue.

Article outline:

I. INTRODUCTION

A. Our approach

B. Comparison to other approaches

C. Our results

II. SUMMARY OF THE FORMALISM

III. THE ORIGIN OF EXPONENTIAL DECAY

IV. EVALUATING THE DECAY PROBABILITY

V. THE REGIME FOR EXPONENTIAL DECAY

A. Derivation

B. The conditions for exponential decay

C. Special cases

VI. BEYOND EXPONENTIAL DECAY

A. The long-time limit(s)

B. Nonexponential decays

VII. COMPARISON WITH THE SURVIVAL PROBABILITY

VIII. CONCLUSIONS

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2008-02-15

2014-04-24

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