Hi, I'm sorry if I've hit the wrong subforum. Although physics is not my strong side this question has been puzzling me for some time now and from I've read so far - there's no clear answer yet. Here's the question: If we take an atom and mark it's position in space at time X and then it's position at time X+ε then how did the atom physically moved from the first to the second position? I know that I'm asking quiet a general question, so if the above is too huge to be explained(summarized), I'll be happy even with a list of fields that I should look into to find out more about the current explanations.
Short answer - applies to anything, not just atoms: Newton's laws. If it is moving, it will keep moving in the same direction and speed as long as no force is applied. If a force is applied it will change speed and/or direction accordingly. If it is not moving and no force is applied, it will stay where it is. In extreme situations, relativity and quantum theory may apply, but the above description is basic.
If quantum mechanics (QM) is not the issue, then, as mathman stated, it's is governed by Newton's Laws. If you wish to consider the QM effects, then "marking" its position is a non-trivial act that will affects the path of the atom.
Joven, welcome to PF. You have asked a good question. Maybe a little too good, because the answer is that nobody knows. The problem is known as the "source of inertia" problem. Type that into google and that will get you some interesting ideas.
According to QM, you can't mark the position of an atom at a singular point. All you can do is specify a wavefunction, with a corresponding expectation value (Interpeted as the "centre" of the wavefunction). So the QM-correct way to pose the question is "what causes a wavefunction to change over time". Wavefunctions can be "moved" without applying force by using a moving reference frame (or equivalently, having the atom moving at an initial constant velocity). Recall that relativity tells us that no reference frame is preferable to any other. Accelerating wavefunctions (or an accelerating reference frame) requires the application of force. Claude.