Anonymous1212144 said:
Let's say that I observed a free particle at a certain location. Is there any way I can calculate the probability of finding that same particle at another location when I look for it again?
Yes, although there are some caveats. One is that the probability of finding a particle at any particular point is always exactly zero; the best that we can do is say that the probability of finding the particle within a distance ##\epsilon## of that point is ##P##. When ##\epsilon## is very small and ##P## is close to unity, we say that the particle is found at that point (that is, ##P=1## and ##\epsilon=0##) but that's a simplification, and when we do the math to calculate the future behavior of the particle we'll need the real values. The second problem is that we also need to know the velocity of the particle, and that is subject to the same sort of uncertainty.
But with that said, the basic idea is:
1) Solve the time-independent Schrodinger's equation for a free particle. This will tell you what the possible wave functions are.
2) Select the wave function from #1 that is consistent with your initial observation.
3) Use the time-dependent Schrodinger equation to calculate how that wave function changes over time.
4) Use the solution to #3 above to calculate the probability of finding the particle at a distance ##\epsilon## from the point you're interested in at whatever later time ##t## you're interested in.
A college-level intro QM class will work through this, but it requires a fair amount of calculus and differential equations, not stuff that belongs in a B-level thread. However, if you want to dig further, you can try googling for "free particle gaussian".
