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Register a new userArrival time from the general theory of quantum time distributions
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We apply the recently developed general theory of quantum time distributions arXiv:2010.07575 to find the distribution of arrival times at the detector. Even though the Hamiltonian in the absence of detector is hermitian, the time evolution of the system before detection involves dealing with a non-hermitian operator obtained from the projection of the hermitian Hamiltonian onto the region in front of the detector. Such a formalism eventually gives rise to a simple and physically sensible analytical expression for the arrival time distribution, for arbitrary wave packet moving in one spatial dimension with negligible distortion.
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Via the proper-time eigenstates (event states) instead of the proper-mass eigenstates (particle states), free-motion time-of-arrival theory for massive spin-1/2 particles is developed at the level of quantum field theory. The approach is based on a position-momentum dual formalism. Within the framework of field quantization, the total time-of-arrival is the sum of the single event-of-arrival contributions, and contains zero-point quantum fluctuations because the clocks under consideration follow the laws of quantum mechanics.
We develop a general theory of the time distribution of quantum events, applicable to a large class of problems such as arrival time, dwell time and tunneling time. A stopwatch ticks until an awaited event is detected, at which time the stopwatch stops. The awaited event is represented by a projection operator $pi$, while the ideal stopwatch is modeled as a series of projective measurements at which the quantum state gets projected with either $bar{pi}=1-pi$ (when the awaited event does not happen) or $pi$ (when the awaited event eventually happens). In the approximation in which the time $delta t$ between the subsequent measurements is sufficiently small (but not zero!), we find a fairly simple general formula for the time distribution ${cal P}(t)$, representing the probability density that the awaited event will be detected at time $t$.
The analysis of the model quantum clocks proposed by Aharonov et al. [Phys. Rev. A 57 (1998) 4130 - quant-ph/9709031] requires considering evanescent components, previously ignored. We also clarify the meaning of the operational time of arrival distribution which had been investigated.
The concept of time as used in various applications and interpretations of quantum theory is briefly reviewed.
Repeated measurements of a quantum particle to check its presence in a region of space was proposed long ago [G. R. Allcock, Ann. Phys. {bf 53}, 286 (1969)] as a natural way to determine the distribution of times of arrival at the orthogonal subspace, but the method was discarded because of the quantum Zeno effect: in the limit of very frequent measurements the wave function is reflected and remains in the original subspace. We show that by normalizing the small bits of arriving (removed) norm, an ideal time distribution emerges in correspondence with a classical local-kinetic-energy distribution.
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