Infinite Positions in QM

StevieTNZ

I'm not sure if that answers my question. Of course we cannot predict where a particle will be at a later time with certainty. We can, however, predict probabilities for every possible result from every future possible experiment conceivable based on a starting wave function.

questionpost

I don't think we can do that either, unless Douglas Adams invented the infinite improbability drive in his slot in the cemetery.

StevieTNZ

From point A to point B, a particle can take an infinite amount of paths to get to the next point. What paths are possible? Ones that don't currently have other systems in them?

questionpost

Particles don't "take paths" because if they did they would radiate their energy away. Instead, they merely have an undefined vector state, which is different than physically taking paths.

StevieTNZ

When you calculate the probability of a particle landing on a screen behind double slits, do we take into account all the paths possible from every point in the universe (would the particle evolved to spread everywhere before reaching the slits?) Of course, would it would need to be histories that involve the experiment, rather than a particle at point C (say where the screen is in another history) to point D (another point on the screen in another history) without there being the experiment happening? Or would you include every possible history that allows the particle to land at the point on the screen, even if the history doesn't involve a double slit experiment?

lugita15

In principle you have to consider every possible history in which the initial and final configurations of your system are the same, however exotic they may be. You can have the particle traveling a little bit, taking a brief detour to the Andromeda galaxy, the screen going to Mars, and then the particle and the screen both returning at exactly the right time so that the particle hits the screen. However, the contributions due to these exotic histories tend to cancel each other out to an extremely good approimation, so that you'll usually get excellent results if you just stick to more realistic paths.

StevieTNZ

What I mean is the initial position of the particle. Do we calculate the paths from every point in the universe, to the point on the screen?

lugita15

No, to calculate probabilities in quantum mechanics we need to specify both the initial condition and the final condition. A general question you answer in QM is, "Given a particle in the state state |ψ>, what is the probability that it will be measured in the state |χ>?" In this case, the specific question is, "Given a particle with position x0 at time t0, what is the probability that it will be measured with position at at time t?" To do that, you add up the amplitude for each history in which the particle started out at (x0,t0) and finished at (x,t), regardless of how exotic they are. Here (x0,t0) is the time and place the particle was launched, and (x,t) is the time and place it's detected at the screen.

StevieTNZ

I guess the basis of my question was based on that the Schrodinger equation has evolved a lot and when it predicts a particle after t=0 is everywhere at once, you have everything everywhere.

lugita15

Yes, you have everything having a probability of being anywhere. But of course, they can have more probability of being in certain regions rather than other regions. Your head is much more likely to stay attached to your body than to disappear and reappear on Venus.

StevieTNZ

Thank goodness for whoever set those probabilities into the laws of physics!

So I gather: you calculate paths from points to the same place (point B) if the experimental set-up is the same in each of those histories? (rather than calculate getting to point B without there being a double slit experiment)

lugita15

Both point A, the initial point, and point B, the final point, must be the same in all histories your summing over. And more generally, the initial configuration of your whole system and the final configuration of your whole system must be the same in all the histories you're summing over. This includes the photon source, the slits, the screen, etc.

StevieTNZ

Gotcha.