Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On Time in Quantum Mechanics   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.

22
The categories of this paper are: chao-dyn
5
The categories of this paper are: gr-qc
0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
24
The categories of this paper are: dg-ga
36
The categories of this paper are: cmp-lg
18
The categories of this paper are: econ
31
The categories of this paper are: ao-sci
12
The categories of this paper are: nucl-th
15
The categories of this paper are: nucl-ex
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
27
The categories of this paper are: comp-gas
29
The categories of this paper are: funct-an
9
The categories of this paper are: nlin