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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?

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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.

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