I asked a related question to this here:
It was closed as a duplicate, but unfortunately the answers in related links didn't help me answer my question. I will try again and hope to do better.
In classical mechanics, if I have an object, it has a definite position $x$. We don't have to guess, we know where it is. But quantum mechanics doesn't seem to work that way.
Consider a particle with a probability density $\rho (x,t)$ for some given $(x,t)$. As I understand it, $\rho$ represents the probability of finding a particle at position $x$ given a measurement at time $t$. We still might not find it! But if we perform this procedure on an ensemble of identically prepared systems, then we should on average expect to find the particle at $x$. I hope I am understanding this correctly.
Now, consider a standard die or coin. There are a set of possible outcomes. For instance $\lbrace H,T\rbrace$ for a coin. Each possible outcome has a probability. And these probabilities are fixed. So the analog for a die's probability space is the position space for a particle of possible positions at a time $t$. But does the physical reality of a particle's probability density only have meaning at the point of measurement?
In other words, consider now a set of $n$ 6-side dice. We will imagine that as I move through time, my die $i$ may morph into a die $j$ and have a different probability distribution over the same set of outcomes. In other words, we have differently bias dice, differently "loaded". Suppose that at time $t_i$ if I roll it, I have a the probabilities given die $i$. But if I roll at some later time $t_j$, then I will have different die $j$ and thus a different set of probabilities. Same set of outcomes, but different probabilities. One could imagine that a particle is like this. If I measure it at time $t_i$ I get a probability distribution $\rho_i$ but if I measure at a later time $t_j$, then it could be a different distribution $\rho_j$. But the set of possible $\rho$ is known. So if we had a function $f$ that would tell us given $t_i < t_j$ and given $\rho_i$, then $f(\rho_i)=\rho_j$, then the probability densities would have meaning beyond the single moment $t_i$. Then they would give us information about future probability densities. So although we never know where the particle is, we know where it might be. Is this how I should interpret the probability densities? Or are we saying no such $f$ exists. That is, are we saying that given $(x,t_i)$ and $\rho(x,t_i)$, there is no way of knowing what the probability density is at a later time $t_j$. If that's what we are saying, then here's my question:
If we cannot infer a later probability density $\rho_j$ given a current one $\rho_i$, how do we describe the time evolution of the system in a way that gives us information about where the particle might be in the future?