In our tangible world, both position and time cannot be determined with arbitrary precision. Despite this evident experimental fact, in quantum theory, we routinely refer to the probability of measuring a particle between positions x and x+dx, exactly at the instant t, but never to the probability of detecting it during the time interval [t, t+dt], exactly at the position x. The latter question is clearly a well-posed one, and this space-time asymmetry has nothing to do with the lack of Lorentz covariance of the Schrödinger equation. In this talk, 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 Schrödinger-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. Several possible experimental consequences of this proposal will be discussed.