When an object move in space, what decides where it moves to? For instance there's no rule saying you can't move from 0 to 3, but have to move from 0 to 1 simply because 1 is next to 0 in terms of distance.
I'm not entirely sure what your question is but I believe what you are asking about is the principle of stationary action: http://en.wikipedia.org/wiki/Principle_of_least_action
I'm looking more for a science proof has to do with the particle movement. The smallest distance is known as Planck Length as someone mentioned. (http://en.wikipedia.org/wiki/Planck_length) So if you move through Planck Length from point A to point B, what decides where you show up when you hit point B, because with Planck Length being the smallest distance it seems like you've skipped this distance and ended up in a new place. Does movement of any length and size comes down to this as a minimum?
There are no "proofs"; the physical interpretation of the Planck length still seems to depend on what theory you subscribe to. But I think it is important to note that under a common interpretation, it is defined as the shortest "measurable" length, not the shortest length.
I'm no expert on this but... It sounds like the OP is assuming particles can only occupy positions that are on a plank scale grid. Are there theories that suggest this might be the case?
Well for an object to be moving it must have some kinetic energy*. This describes the work needed to accelerate a body, the direction of travel will be dependant on how the body received this energy i.e. if some one pushed it, the direction of travel and where it ends up will be in the direction of the applied force i.e. the push. As for the second part of this question I'm not sure what you mean by selecting arbitrary numbers and saying there is no rule against moving from one to the other. If you are using them as a measure of displacement from a starting point, in meters say, then there is a general rule that you have to move 1, 2 meters before you can have moved 3, unless you found a suitable method of teleportation. Have I misunderstood the question? * http://en.wikipedia.org/wiki/Kinetic_energy
This brings in the question of the size of the smallest particle. Is a particle of 'zero' size really to be regarded as less than the Planck length? Whatever the answer is to the above, I think that Space must be assumed to be monotonic - at least when considering three dimensional space. But when the multidimensional space of String Theory is brought into it, this may not be right.
Or are you referring to the discrete nature of energies and confusing this with the inability of a particle to stay at a location with not being able to move through it? Let's say there is some minimum movement for a particle based on a discrete energy that causes it to move some distance from A to B, as it's semi stable able to occupy those locations space. That does not indicate that it did not pass through an infinite number of possible locations in between. My interpretation from quantum mechanics is that there are a number of discrete locations that could be inhabited by a particle but I believe this is the nature of the particle not the nature of space and motion itself. Although if someone wouldn't mind explaining the physical path followed by a particle during tunneling.
I understand the object is supposed to move in the direction of the applied force, I'm just not sure how the object ends up at that position after the displacement. To me it seems like every movement made is going through multiples of Planck Length, and since movement is possible, there must be a smooth transition for an object to travel from one location to the next. Like something is telling you, after you move away from point A, you will end up in point B. I guess my question would be what goes on between Planck Length that displaces you to the next position.
Are you aware that you're asking for what is going on for something that (i) is hypothetical and (ii) no one has observed/detected before? There's a distinct possibility that this "Planck Length" might not even exist the way it has been envisioned. Wouldn't such a discussion be a total waste of time if that were to happen? Zz.
When a paritlce moves from A -> B is its motion broken into finitely many small jumps as I think the OP is suggesting, or is the motion more accurately described as a smooth transition with no possible way to break it down into steps? Is there/Has there been a way to prove this theoretically one way or another?
Maybe it's easier to think of a probability density instead of a point particle. So then you can invision a continuous smooth shift in the most likely place for the particle to exist, because I guess you can assign an infinite number of points in space with a probability function.
The idea of the Planck length is that you could not use quantum mechanics (as it is now) to describe motion on a length of scale smaller than the Planck length. You would need some kind of theory of quantum gravity, or some other extension which they haven't thought up yet. But this doesn't necessarily mean that motion occurs in 'steps' of the Planck length.
For a single object in space, without any fixed point of reference, does displacement/distance have any meaning? So the question is what does motion mean to the object itself. BTW you may be interested in reading up on Zeno's paradox....
After some thought, I think force decides the direction the object is supposed to travel, but not the destination. Since force remain constant at all places, but the place you travel to is different. So does that mean the space you're in uses force to find the destination you travel to? And does bending space generate a force in return. Just some thought, hopefully I'm not off topic.
yeah, and no. I would agree with some of the stuff you're saying. (Talking about non-quantum physics now). Force is the instantaneous rate of change of momentum, so the force tells us how the motion is changing in that instant. I don't understand what you mean by "Since force remain constant at all places, but the place you travel to is different." Do you mean that if the force is constant, then the place you travel to can still be different? This is because force only tells us the second differential of position, so if we know the constant force, then there are still two (vector) unknowns, which correspond to initial position and initial velocity.
What you said is correct, I'm just wondering where the information of the destination is held. The idea about moving object with force is credited to BenG549 and the direction is through someone I met online. I think I'll stop here. There might be more questions in the future.
I think the OP means if you were to roll a single atom along a desk for example. You lay a (planck length measuring stick) on the desk next to it. Then when you roll the atom, it has to move in the form; 1 - 2 - 3 - 4 - 5 ect planck length at a time. What is the law that governs this movement? Why can the atom not jump a few spaces. This is what the OP is asking I think. Well if the atom did jump a few spaces, that would be teleportation. What law tells the atom it must move in this fashion? I do not know if such a law exists... I think it most certainly exists though otherwise atoms would teleport micro-distances all the time I would think.
Right but as soon as the atom moves from stationary it should make a jump since the atom's shape doesn't change. And it jumps from the center of a sphere by its radius according to the direction of the force. Then again there might be other explanation if the atom does not "jump".