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We introduce a quantum mechanical time-of-arrival operator using a quantum clock. This new operator is Hermitian, physically interpretable, and resolves issues with prior methods for measuring event times.

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Area of Science:

  • Quantum Mechanics
  • Quantum Measurement Theory

Background:

  • Defining a time-of-arrival operator in quantum mechanics is challenging due to fundamental issues.
  • Previous proposals often lack a clear physical interpretation or violate key quantum principles.

Purpose of the Study:

  • To propose a novel, Hermitian time-of-arrival operator in quantum mechanics.
  • To develop a method for measuring event times that is consistent with the Born rule and has physical meaning.
  • To generalize the measurement procedure for arbitrary events, not just particle arrival.

Main Methods:

  • Conditioning quantum mechanical states on a precisely defined quantum clock.
  • Developing a Hermitian operator for time of arrival.
  • Applying the Born rule to derive probability distributions for time measurements.

Main Results:

  • A Hermitian time-of-arrival operator is successfully formulated.
  • The operator's probability distribution adheres to the Born rule, providing clear physical interpretation.
  • The methodology is shown to be applicable to measuring the time of arbitrary quantum events.

Conclusions:

  • The proposed quantum clock-conditioned approach offers a robust solution for the time-of-arrival problem in quantum mechanics.
  • This framework provides a physically meaningful and mathematically sound method for temporal measurements in quantum systems.
  • The generalized procedure enhances the scope of quantum temporal measurements beyond particle arrival.