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How can you talk about two electrons if they are identical (indistinguible)? Does it make sense to let an electron to have an identity by itself?

If they are on diferent places the place they are is a diference (they are not identical). The act of labeling, naming them make them distinct. If I say let A be an electron and B another one, then they are distingible by name...(they are not identical)

Edit: if electrons are indistinguible by principle, then you cannot name them properly. A tick after you name one of them it could be any other electron in the universe or not exist anymore. You could say let this be electron A, but a tick after, A means nothing. The most you can say is at some point in time there were A and that "electronism" would remain, but this "electronism" would emerge in diferent form as long as "electronism" holds.

I think that the indistinguibility property makes the interpretation of a particle who has an identity and live forever hard to follow. One need to create strange ideas like "it follows all the paths" or "there are several paralel universes"...

Eduard
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  • I am not sure I understand your question. You seem to be answering the title question in the text itself. – Steeven Aug 03 '18 at 19:53
  • My english is not very good. I don't see how can I say electron A and electron B in a statement if they are truely indistinguible. Naming them, providing a distinct label for them is logically inconsistent IMO. – Eduard Aug 03 '18 at 20:12
  • Can you kit have two exactly identical particles that are merely placed differently, or which possesses different energies or alike? How exactly is this inconsistent? – Steeven Aug 04 '18 at 11:46
  • If they are in diferent places they are not identical. Their position make them distinguishable. – Eduard Aug 04 '18 at 12:05
  • Well... In that case, with that defition of "identical", no two electrons in the universe are identical. – Steeven Aug 04 '18 at 12:47
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    Check out this other question. – DanielSank Aug 04 '18 at 21:50
  • I don't remember why I voted to close this question, but I don't think it is unclear. Apologies for my previous vote; I'm voting to reopen now. – AccidentalFourierTransform Aug 08 '18 at 00:37
  • @Eduard, there is still no question in the main text. Putting question there instead of the comments would encourage people to answer. – Ján Lalinský Aug 08 '18 at 12:48
  • @Eduard "Their position make them distinguishable"; that's not true. You are claiming that one is in $x_1$ and another in $x_2$, the deal is that you don't know which is which. I think you are confused with the notation and the way we write Physics using mathematics. The state $|x_1 x_2\rangle \pm |x_2 x_1\rangle$ (up to normalization) expresses correctly this indistinguishability. The labels, at this level, are "immaterial". – user2820579 Aug 09 '18 at 01:05
  • I agree. I was confused. Quantum state (position, momentum, etc...) is "the particle" from the "outside". My concern about identity is when thinking on the particle from "inside". – Eduard Aug 09 '18 at 09:21

3 Answers3

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We can do an experiment that distinguishes a box with two electrons in it from a box with just one electron in it. Just measure the electric field near the box and the field near the two electron box will be larger.

We cannot do an experiment that distinguishes between a box with electron A on the left and electron B on the right from a box with the electron A and B swapped. This is what indistinguishability means.

Luke Pritchett
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  • Not sure at all. You can make the box smaller. Still you can talk about two electrons inside a box and measure it? Each time I see nonintuitive statements about particles, I got the feeling there is only one single particle (or field) spread all over the experiment area. The experiment then only tries to avoid the external interactions, just to have "three electrons and 2 fotons" inside the box. – Eduard Aug 03 '18 at 20:04
  • Can the box where A and B lives be smaller than plank radius? – Eduard Aug 03 '18 at 20:20
  • I'm afraid I don't understand your these questions. I am simply trying to show you that we can tell the difference between two electrons and one electron experimentally, even without labeling. – Luke Pritchett Aug 03 '18 at 20:38
  • I'm sorry. Maybe later I would find a better way to explain my concern. If A and B are indistinguible by definition, how can you distinguish them to provide them a name from the beginning. I understand that yiu could trap them in two separate boxes, measure something, conclude there is one electron in one box an one in the other then you could name them you could name them "atom in 1st box", "atom in 2nd box", but then if you measure again even if the measures say one atom in one box, one atom in the other, they could be swapped or be a totally diferent one's. Make no sense to label them. – Eduard Aug 03 '18 at 21:07
  • I don't think I understand your question at all. Are you asking how to describe two-electron states? For example, I could say "I have a box with two electrons in it. There is an electron near the top left corner, and there is an electron near the bottom right corner." That is a complete description of a system with two electrons in it and I haven't labeled distinct electrons at all. – Luke Pritchett Aug 03 '18 at 21:12
  • "electron near the top left corner" is one label, "electron near the bottom right corner" the other label. – Eduard Aug 03 '18 at 21:18
  • You are distinguishing electrons by location isn't it? – Eduard Aug 03 '18 at 21:19
  • Then one "tick" after, maybe still there are an electron in the top left an another one in the bottom right but we cannot say if they are the same or not. So naming them was useless... – Eduard Aug 03 '18 at 21:22
  • No, I am not distinguishing the electrons. "Indistinguishability" means that if you switch the two electrons the description of the state does not change. That's all it means. Read my wording carefully and you will see that even if the electrons swapped places my state description would be exactly the same. – Luke Pritchett Aug 04 '18 at 00:24
  • It may help to read a thermodynamics or statistical mechanics textbook on the statistical effects of indistinguishability. It is mostly a math concept and trying to understand it deeply without seeing the math is probably a mistake. – Luke Pritchett Aug 04 '18 at 00:25
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This is actually a very deep question, good job!

If you think about what counting means, it is a sort of song and dance game, where the song is usually a sort of chant that we have just agreed upon, where the important thing is that each sound that you chant is different from every other sound in the sequence. Then the game has certain rules, where in time with the chant you are pointing at objects, and you must never point at the same object twice, and you must point at all of the objects, and then whatever the last word you chanted was, we say is the count of the thing.

You are right to intuit that this game cannot be played in this form with electrons: at least, not according to quantum mechanics. You can point at distinct states that the electrons are occupying, and you can count those, but you cannot count the actual electrons, because they are indistinguishable particles, and there is generally a non-zero probability that they switched places, and this probability actually has some measurable consequences, in that the electrons behave statistically according to Fermi-Dirac statistics.

With that said, the fact that there is no observable difference between any electrons, actually kind of helps us out a little bit. It means that we can speak very loosely about a state occupied by an electron, as “an electron,” and not worry about tracking the electron in that state to see if it changes places with another electron. The only reason we'd really care about them changing places, is if it made some sort of difference to some sort of measurement, and it cannot.

So when we count electrons, or try to label them, we are really labeling the state that they are in, and counting the states that they are in, and confirming that those states are occupied by an electron. But at a deep level indistinguishability does mean that we can't tell you that “it's for sure the same electron occupying this state at time $T+1$ as had been occupying it at time $T$.

Of course, the states occupied by electrons, do have real physical consequences. For example, you can measure the electric charge in a box, and count electrons that way. Or for another quick example, water contains these atoms of hydrogen and oxygen, which if the electrons were separated and not configured so that the electrons were in these shared states between the hydrogen and oxygen atoms, would be flammable. But they're not once they are water.

CR Drost
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  • Additionally avoiding electron identity is easier to visualize the double split experiment. It is not the "same" electron that goes thought the wall, folowing two paths. There is no path at all, if we cannot tell is the same point who follow it... – Eduard Aug 05 '18 at 17:47
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You can label billiards balls, ping pong balls, marbles and even BBs. But there is no way you can label electrons, even in principle. Think about 2 identical electrons in a head on collision. Before the collision they move in opposite directions along the same straight line. After the collision, 2 electrons move away from each other at some angle to the initial direction less than 180 degrees. How would you be able to tell which electron was which in the final state? The answer is you can't. So you have to consider the final state as a superposition of both possibilities. Exchange degeneracy.

Mephistopheles
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  • When I see the sentence "there is no way to label electrons, even in principle" I assume implies that you cannot label electrons. So a sentence like "let's A an electron and B another electron..." is wrong. The sentence is labeling electrons! – Eduard Aug 04 '18 at 21:54
  • I don't understand your comment Eduard. I haven't called electrons A or B or anything else. You did that. What can you do to distinguish identical electrons? You can't paint them or anything else I can think of. – Mephistopheles Aug 05 '18 at 12:31
  • The whole point is theory says you can't distinguish electrons. So refering to diferent electrons should be interpreted taking into account they dont have an identity. They dont "exist" at all. Only their properties seems to "exist" in the sense they are always there. – Eduard Aug 05 '18 at 14:46