Space-time symmetric extension of non-relativistic quantum mechanics

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.