Feynman Paths

StevieTNZ

If, at one particular spot a particle can from (point A), all the paths from that spot to point B cancel each other out, does that mean the particle cannot come from point A?

Or likewise, if only some paths from point A to point B cancel out, and paths from point C to point B cancel the rest of the paths from point A to B, the particle can't come from point A?

lugita15

Your question is a bit unclear, but this may help: if the amplitudes for all the paths going from A to B cancel each other out, then the total amplitude to go from A to B is zero, and thus there is zero probability that the particle will go to B if it starts out at A. (By the way, zero probability doesn't mean it will never happen, at least for continuous states; it means it will almost surely not happen.)

And one thing you should understand is that you can't just talk about the amplitude that the particle will be measured in point B. In quantum mechanics, you need both an initial state and a final state, and the question asked is "Given a particle in this initial state, what is the amplitude for it to be detected in that final state.'

StevieTNZ

Yup - that answers my question. Thanks!

StevieTNZ

I guess that explains why even though certain areas on the double slit screen are hit with electrons even when there it is predicted they don't hit there

EDIT: from a Maths lecturer at my university "This is all inevitable. If you have an infinite number of possible disjoint outcomes and only a total probability of one to share around, then plenty of possible events must receive probability zero. It may seem counterintuitive, but the mathematics of infinity often is."
But can't you just assign each outcome a probability such 0.00000001 (of course with a lot of more zeros). 0.0000001 isn't 0.

lugita15

I'm not sure what you're talking about, but if you're talking about a situation where electrons are systematically hitting some place, then definitionally they have some nonzero probability of hitting there. If you had zero probability, then it would literally occur infinitely rarely. Of course, in the real world the amplitudes almost never cancel out perfectly, so you almost never get zero probabilities for anything.

StevieTNZ

Yeah, I was wondering about the perfect cancellation, as doesn't everything have a finite (but not a zero probability) of being anywhere in the universe?

lugita15

Yes, in the real world probabilities all tend to be nonzero, just because there are so many interactions that are affecting the amplitude. But in simple example like a spherically symmetric potential we can get lots of regions where particles of zero probability of appearing.