On Time in Quantum Mechanics

TL;DR

This paper explores the theoretical challenges of defining time in quantum mechanics, examines the energy-time uncertainty relation through a decay model, and derives bounds for scattering estimates, highlighting the independence of linewidth and lifetime relations.

Contribution

It provides new insights into the role of POVMs in time measurements, analyzes the energy-time uncertainty in alpha decay, and derives explicit bounds for scattering estimates and S-matrix derivatives.

Findings

No POVM closely matches the probability current for general wave functions.

The energy-time uncertainty relation holds for long-lived systems.

Linewidth-lifetime relation is independent, not derivable from the uncertainty relation.

Abstract

Although time measurements are routinely performed in laboratories, their theoretical description is still an open problem. Correspondingly, the status of the energy-time uncertainty relation is unsettled. In the first part of this work the necessity of positive operator valued measures (POVM) as descriptions of every quantum experiment is reviewed, as well as the suggestive role played by the probability current in time measurements. Furthermore, it is shown that no POVM exists, which approximately agrees with the probability current on a very natural set of wave functions; nevertheless, the choice of the set is crucial, and on more restrictive sets the probability current does provide a good arrival time prediction. Some ideas to experimentally detect quantum effects in time measurements are discussed. In the second part of the work the energy-time uncertainty relation is considered, in particular for a model of alpha decay for which the variance of the energy can be calculated explicitly, and the variance of time can be estimated. This estimate is tight for systems with long lifetimes, in which case the uncertainty relation is shown to be satisfied. Also the linewidth-lifetime relation is shown to hold, but contrary to the common expectation, it is found that the two relations behave independently, and therefore it is not possible to interpret one as a consequence of the other. To perform the mentioned analysis quantitative scattering estimates are necessary. To this end, bounds of the form $\|1_Re^{-iHt}\psi\|_2^2 \leq C t^{-3}$ have been derived, where $\psi$ denotes the initial state, $H$ the Hamiltonian, $R$ a positive constant, and $C$ is explicitly known. As intermediate step, bounds on the derivatives of the $S$-matrix in the form $\|1_K S^{(n)}\|_\infty \leq C_{n,K} $ have been established, with $n=1,2,3$, and the constants $C_{n,K}$ explicitly known.