Time-of-arrival distributions for continuous quantum systems
arxiv(2024)
摘要
Using standard results from statistics, we show that for any continuous
quantum system (Gaussian or otherwise) and any observable A (position or
otherwise), the distribution π _a(t) of a time measurement
at a fixed state a can be inferred from the distribution ρ _t(
a) of a state measurement at a fixed time t via the transformation π _a( t) = |∂/∂ t∫_-∞^aρ _t( u) du |. This finding suggests that the
answer to the long-lasting time-of-arrival problem is in fact readily available
in the standard formalism, secretly hidden within the Born rule, and therefore
does not require the introduction of an ad-hoc time operator or a commitment to
a specific (e.g., Bohmian) ontology. The generality and versatility of the
result are illustrated by applications to the time-of-arrival at a given
location for a free particle in a superposed state and to the time required to
reach a given velocity for a free-falling quantum particle. Our approach also
offers a potentially promising new avenue toward the design of an experimental
protocol for the yet-to-be-performed observation of the phenomenon of quantum
backflow.
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