Abstract
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 φ(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.