There are no mystical laws of nature associated with consciousness. Therefore the copy has no magical effect on the original you. Therefore it should appear the same to original you whether the copy is there or whether we just say the copy is there.
Which makes clear that the idea of a "body-jump" is wrong. No soul is leaping forth from the original you to go live in some copy.
So what does this probability of waking up as someone else mean? Well, we can think of your memories, including the memory of this probability, as a "mark" on minds. The probability really asks: given that you have the mark, what is the probability that you are the original? Copying you is just creating more minds with the mark, so if the only thing you know is that you have the mark, you have a 1/N chance of being the original.
The real problem seems to be uncertainty over the definition of "you" given multiple minds with the mark. To which I'd like to refer you to the official reference.
Of course there's no soul. Nor is there any spooky physical action at a distance whenever a copy is created.
What I mean by a body jump is the entirely subjective experience of the copy. Past the branching point, the original will experience a smooth continuation of the stream of consciousness, while the copy experiences a jarring disconnect if no interruption of consciousness occurred at the time of copy - which subjectively feels like a body jump even though it's not what actually happened.
Of course, if you're unconscious (sleeping, tranquilized, etc.) when the scan takes place, the waking experience is symmetric.
The subject of copying people and its effect on personal identity and probability anticipation has been raised and, I think, addressed adequately on Less Wrong.
Still, I'd like to bring up some more thought experiments.
Recently I had a dispute on an IRC channel. I argued that if some hypothetical machine made an exact copy of me, then I would anticipate a 50% probability of jumping into the new body. (I admit that it still feels a little counterintuitive to me, even though this is what I would rationally expect.) After all, they said, the mere fact the copy was created doesn't affect the original.
However, from an outside perspective, Maia1 would see Maia2 being created in front of her eyes, and Maia2 would see the same scene up to the moment of forking, at which point the field of view in front of her eyes would abruptly change to reflect the new location.
Here, it is obvious from both an inside and outside perspective which version has continuity of experience, and thus from a legal standpoint, I think, it would make sense to regard Maia1 as having the same legal identity as the original, and recognize the need to create new documents and records for Maia2 -- even if there is no physical difference.
Suppose, however, that the information was erased. For example, suppose a robot sedated and copied the original me, then dragged Maia1 and Maia2 to randomly chosen rooms, and erased its own memory. At this point, neither either of me, nor anyone else would be able to distinguish between the two. What would you do here from a legal standpoint? (I suppose if it actually came to this, the two of me would agree to arbitrarily designate one as the original by tossing an ordinary coin...)
And one more moment. What is this probability of subjective body-jump actually a probability of? We could set up various Sleeping Beauty-like thought experiments here. Supposing for the sake of argument that I'll live at most a natural human lifespan no matter which year I find myself in, imagine that I make a backup of my current state and ask a machine to restore a copy of me every 200 years. Does this imply that the moment the backup is made -- before I even issue the order, and from an outside perspective, way before any of this copying happens -- I should anticipate subjectively jumping into any given time in the future, and the probability of finding myself as any of them, including the original, tends towards zero the longer the copying machine survives?