  In quantum theory we refer to the probability of finding a particle between
positions $x$ and $x+dx$ at the instant $t$, although we have no capacity of
predicting exactly when the detection occurs. In this work, first we present an
extended non-relativistic quantum formalism where space and time play
equivalent roles. It leads to the probability of finding a particle between $x$
and $x+dx$ during [$t$,$t+dt$]. Then, we find a Schr\"odinger-like equation for
a "mirror" wave function $\phi(t,x)$ associated with the probability of
measuring the system between $t$ and $t+dt$, given that detection occurs at
$x$. In this framework, it is shown that energy measurements of a stationary
state display a non-zero dispersion, and that energy-time uncertainty arises
from first principles. We show that a central result on arrival time, obtained
through approaches that resort to {\it ad hoc} assumptions, is a natural,
built-in part of the formalism presented here.
