# Information loss of particles over time

Due to the nature of the Heisenberg uncertainty principle along with the Schrodinger equation, the position of a particle gains uncertainty / loses certainty over time, because its momentum is also uncertain.

If you know the wave function of a particle at time t0, and reevaluate it at a later time t1, have you gained or lost information now that you are less certain of the position of the particle? Or was there no change in information at all?

The Schrödinger equation is time-symmetric and deterministic, so the wave function at time $$t_1$$ has the same information as at time $$t_0$$. No information is lost or gained. However if you only have some partial information about the wave function, e.g. if you only know the standard deviation $$\Delta x$$, then arguably you will lose or gain information as $$\Delta x$$ increases or decreases over time.