Toward a more symmetric relation between space and time in non-relativistic quantum mechanics
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.