Exact Energy-Time Uncertainty Relation for Arrival Time by Absorption

Abstract

We prove an uncertainty relation for energy and arrival time, where the arrival of a particle at a detector is modeled by an absorbing term added to the Hamiltonian. In this well-known scheme the probability for the particle's arrival at the counter is identified with the loss of normalization for an initial wave packet. Under the sole assumption that the absorbing term vanishes on the initial wave function, we show that $\Delta T\Delta E\ge \sqrt{p}\hslash /2$ and $<T>\Delta E\ge 1.37\sqrt{p}\hslash$, where $<T>e$ denotes the mean arrival time, and $p$ is the probability for the particle to be eventually absorbed. Nearly minimal uncertainty can be achieved in a two-level system, and we propose a trapped ion experiment to realize this situation.

8 pages, 2 figures

References in corpus (2)

• Heisenberg's Uncertainty Principle
• The uncertainty principle determines the non-locality of quantum mechanics

Cited by in corpus (8)

• Recurrence for discrete time unitary evolutions
• Uncertainty Relations for Angular Momentum
• Nonexponential quantum decay under environmental decoherence
• Time-of-arrival probabilities for general particle detectors
• Equivalence of matrix product ensembles of trajectories in open quantum systems
• Does the Heisenberg uncertainty principle apply along the time dimension?
• Absorbing detectors meet scattering theory
• Energy-Time Uncertainty Relation for Absorbing Boundaries