This question may well be predicated on a fundamental misunderstanding of quantum physics; if it is, maybe you could explain my error in very small words?
As I understand it, my atoms don’t so much occupy places as they occupy probability fields, or something: most of my electrons are right within the normal field you’d expect them to be in, but some of them at any given moment are swinging outside of the field. Someone once told me it’s not inconceivable for one of my electrons to be out near Mars right now.
I’m guessing that’s phenomenally unlikely, though. So here’s my question: at any given moment, what’s the farthest away from the bulk of my body that I’m likely to have an electron? A millimeter away? A centimeter? A meter? Ten meters? Less? More?
You gain and loose electrons all the time. If you rub a balloon on your hair, some of your electrons get wiped onto the balloon (this is why your hair might stand on-end when you do this – this creates a strong electric field between your hair and the balloon). Then if you let the balloon fly up into the sky, your electrons go with it, and can get far away from your body. Of course, then they aren’t your electrons anymore; they are the balloon’s electrons now! Also, this is not a quantum effect really. I’m mentioning this just so we are clear that you gain and loose electrons all the time, having nothing to do with quantum “fuzziness.”
So, you are asking,* due only to the quantum fuzziness*, how far away can an electron get away from the atom it is orbiting in your body. The answer is: usually much less than a millimeter. But sometimes that quantum fuzziness is what will decide whether or not an electron will jump over to the balloon mentioned in the last paragraph. This would happen whenever you had an electron that would come really really close to being swiped onto the balloon, and only needs the tiniest little kick to get all the way there. In this sense, quantum fuzziness has made many of your electrons go really far from your body, in the same way that a butterfly flapping its wings in the Congo can lead to a tornado in Mississippi 100 years later. But this is only in a pedantic sense. In the “intuitive” sense I think you are asking about, the quantum fuzziness only extends to very, very tiny distances less than a millimeter.
The proximity of the constituent electrons associated with your body need only be enough that they can interact with each other. That makes them a “part” of you.
The definition of “your” electron would be one that remains within the realm of co-influence with the rest of “your” electrons. If one goes to Mars, it is not yours anymore, unless there is a relevant persuasion on its part to return to your body, as opposed to random motion.
That’s for any individual electron, though, right? From my understanding, I’ve got literally dozens of electrons (hey, I did say literally), though, and at any given point, some are more than 3 nanometers away and some are less. If that’s correct, what’s the furthest from my body that any one of my dozens of electrons is right now?
jtur and iamnotbatman, thanks for your clarification of the question. Am I understanding you to say that once an electron exceeds this 3 nm distance, it’s not reasonable to call it my electron anymore?
Ah, I see. That’s a harder question to answer! It may very well come home even if it strays over a millimeter away, due to its electrical attraction to the positive ion it left behind. It depends on whether or not the electron is moving away from your body with enough energy to overcome its attraction to you! If the electron had zero kinetic energy but was suddenly displaced by a millimeter, it would probably find its way home!
This is not an easy question to answer. First, note that it’s a bit dicey to talk about electrons bound to a nucleus as occupying any definite place in space, or following definite orbits as in the familiar ‘solar system’ (Bohr) model of the atom. Rather, all that you can really say is that at a certain point in space, there is a certain probability to observe the electron there, if you look. These probabilities are given by the orbitals of the atom – you’ve probably seen funny pictures like this one before. What they tell you is the shape of the probability distribution (more accurately, the wave function; the probability is obtained by integrating over the absolute square of the wave function); at any point on their surfaces, the electron is equally likely to be found. Radially, i.e. at some distance from the atom, these go like r*exp(-r) in the simplest case (see here). As you can see, this never reaches 0 – there is always a non-vanishing probability to observe an electron at any distance from its atom (the probability gets really small really quick, though). The typical distance is around the Bohr radius, in the vicinity of half an Angström (10[sup]-10[/sup]m).
Now, you have on the order of 10[sup]28[/sup] atoms in your body. The likelihood of observing an electron in an interval between 35 and 36 times the Bohr radius away from its nucleus is ~10[sup]-28[/sup]. So, if you measured all your atoms at that distance, there’s a good chance you’d find an electron there at least once (and perhaps a couple of times; this is such a quick and dirty calculation that I’d be happy to be correct within an order of magnitude or two). This would translate to something around 2 nm, perhaps.
Of course, this calculation is heavily approximated. First of all, I’ve essentially assumed that you’re composed entirely out of hydrogen, and second, that all of your atoms are in the ground state, neither of which will be true. In particular, excited atoms can have a much larger radius – the radius scales with n², where n is the excitation level of the atom; you can produce extremely large atoms, so-called Rydberg atoms, which can have radii in the range of micrometres. I’ve also treated your atoms as a collection of systems in isolation, which is probably not a good approximation, so it’s dubious whether the quantum-mechanical calculation is really applicable at all.
So in short, yeah, it’s not an easy question to answer…
If you want to do quantum theory consistently with relativity, you have to appeal to quantum field theory. But even apparently superluminal features in quantum mechanics turn in general out to be truly apparent: for instance, there was a bit of a fuzz some years ago about quantum tunneling ‘faster than the speed of light’, but this essentially turned out to be just a confusion to which notion of the speed of a wave one should appeal to (since waves are not localized, you can assign different speeds to different ‘parts’: its outer edge may move ‘faster’ than its central point, if the wave – or wave packet – is dispersing while it moves; but you can’t transmit any information superluminally that way, so all is well).
Okay, yeah, I think that’s the question I’m wanting to ask (and I appreciate your patience with my ignorance on this topic). Half Man Half Wit, thanks for your explanation, which I think I was just barely able to follow, which is a reflection on me and not on you :).
If we define “my” electrons as ones that will continue (absent further interference) orbiting positive ions in my body, then, and if we allow for excited atoms (a term I’m taking pretty much on faith at this point–you could call them cuddly fuzzy snuggle-atoms and I’d believe you)–how far are we likely to find an electron as defined above?
Note that this question isn’t totally academic: I’m trying to build a teleportation device.
(Okay, in a role-playing game with a Tesla disciple. Still.)
Really, I think this is the best overall answer to the OP’s question itself. It’s not that electrons are flying around long distances away from their atoms. They aren’t ping-pong balls where we’re just not sure of the precise location. It’s that they are a wave function with an ever-diminishing influence as you move away from it.
For the “chance that an electron is really a millimeter away” idea to make any sense, there’d have to be an interaction that changes/collapses the wave function.
So “my electrons” are those with a wave field that is defined by interacting with “my atoms” - I would say that if we can define the electron as being even a millimeter away, we’re really talking about an interaction with the outside world that has made it cease to be “my electron.”
