# Can an electron be in more than two places at once?

1. Jul 27, 2014

### jebii

Forgive me if this is a dumb question; from what I've been reading an electron can exist simultaneously in two places. Is this definite?
Why can/can't it exist in more forms?

2. Jul 27, 2014

### Staff: Mentor

No.

QM is a theory about observations. It's never observed to not be in one place and one place only - end of story.

What its doing, being etc etc when not observed the formalism is silent on. We have interpretations with various takes on the issue - but they are simply conjecture.

Thanks
Bill

Last edited: Jul 27, 2014
3. Jul 27, 2014

### akhmeteli

I suspect that a "yes" or "no" answer to your question might be somewhat misleading, as your question, while natural, may be somewhat messy.

The traditional mantra of standard quantum mechanics (QM) is that QM can say nothing about electron's position between measurements, and this mantra is relatively consistent. Therefore, according to this mantra, we can only say something about electron's position (or "existence" in some "place", in the wording of your question) at the time of measurement. However, strictly speaking, we can never say that an electron was at time t0 at point x0, as no position measurement can be infinitely precise due to the uncertainty principle: infinitely precise position measurement at a certain moment in time would require infinite energy. Moreover, as Landau and Peierls noted, due to special relativity, position measurement with accuracy better than the Compton wavelength makes little sense, as the energy required for such precise measurement can be sufficient for pair production, so you cannot know anymore if you have measured the position of the initial electron or the position of an electron of a pair created in the course of measurement. Therefore, if we stick to the mantra, we can never say even that an electron can exist (be observed) in one particular place, let alone two:-)

So let us try to go somewhat beyond the mantra. According to quantum electrodynamics (QED), we never observe a "naked" electron, but only a "physical" electron, i.e. the naked electron surrounded by photons and pairs. One of the features of the physical electron is its Coulomb field, which has infinite effective radius. However, as I said, the mantra is relatively consistent: we can detect the electron by its Coulomb field at a large distance, but the photon detected in such measurement will have large wavelength, so the position uncertainty will be high.

In this context, the recent experiments by Couder e.a. may be interesting (https://hekla.ipgp.fr/IMG/pdf/Couder-Fort_PRL_2006.pdf , they were discussed in several threads here). They observed trajectories of silicone oil droplets on the surface of silicone oil (for some reason, it is called "silicon oil" in the article, but I am not sure this is correct). Those trajectories provided surprisingly good emulation of the two-slit experiment in QM. This experiment can be described in terms of classical mechanics, and what happens is the droplet creates a wave on the surface of liquid and then interacts with its own wave, creating the interference pattern. While you can apply the mantra of QM to the droplets in Couder's experiment as well, and it will provide a relatively consistent picture, this would not look very good:-)

4. Jul 27, 2014

### haael

Everything depends on what you mean by "electron". If you think of it as "tiny charged ball", then you indeed will need to assume it is in 2 different places. But that's the least of your problems, there are in fact much more paradoxes. Take for example tunneling through a high potential wall, which is an experimentally proven phenomenon. What is the potential energy of the electron when it passes through the potential? It's higher than the initial electron energy, thus violating energy conservation. There are more such paradoxes.

All contradictions begin from the initial assumption that the electron is a classical object. First you think: "electron is a tiny hard classical ball", then you try to get in agreement with the fact that it is in fact a quantum object. And you get paradoxes.

You may try argue that electron is "generally classical object plus some minor quantum corrections", but that really leads nowhere. This is the basis for the wave-particle duality. In some popular explanations you may read that electron is a classical hard ball that can sometimes mystically transform into a wave, perform wave behaviors and reappear again as a classical ball. But it really doesn't explain anything.

The proper answer is: electron is not a classical object. It doesn't have a "position" in traditional sense. Position is a classical term that does not apply to quantum objects.

5. Jul 27, 2014

Staff Emeritus

6. Jul 28, 2014

### Staff: Mentor

Doesn't matter how you view it, it never is in two places at once. End of story.

That position measurements are never, in practice, 100% precise has nothing to do with the issue.

Also there is nothing in QM that says you cant measure position with 100% accuracy. The uncertainty principle does't say anything about it.

To be clear on this point it needs to be stated carefully and rigorously.

Take a large number of similarly prepared systems. Measure the position in half of them. They will always give a definite position, but will have a statistical distribution. Measure the momentum in the other half. It too will always give a definite momentum, but will also have a statistical distribution. The uncertainty principle is a statement about those distributions - but each measurement can, in principle, be exact.

Thanks
Bill

Last edited: Jul 28, 2014
7. Jul 28, 2014

### Ravi Mohan

I am little confused. In principles of Quantum Mechanics, Chapter 1 section 4, Paul Dirac writes

8. Jul 28, 2014

### Staff: Mentor

A lot of water has gone under the bridge since Dirac wrote that classic.

It's well known there are a number of mistakes and misconceptions in that book.

If you want to understand the modern view see Ballentine.

The above is one of that books misconceptions, and its not the only one.

The principle of superposition is simply a reflection of the fact pure states can be mapped to a vector space.

As to what a state is Gleason's Theorem has a lot to say about it:

Its simply what's required for the outcomes to be mapped to POVM's and non-contextuality.

Thanks
Bill

Last edited: Jul 28, 2014
9. Jul 28, 2014

### Ravi Mohan

I think, please correct me if I am wrong, the meaning of superposition depends on the interpretation we choose.

If we assume that $|\Psi\rangle$ provides a complete and exhaustive description of an individual system, then superposed states imply the existence of system in two or more states.

But if we assume that a $|\Psi\rangle$ describes the statistical properties of an ensemble of similarly prepared systems, then the superposition of states is simply a different description of the ensemble.

My point is that axiomatically, QM is silent on the meaning of superposition. If we follow certain interpretation, then the answer to the OP's query could be affirmative.

Last edited: Jul 28, 2014
10. Jul 28, 2014

### Staff: Mentor

Precisely. Note carefully what I said in my reply. It was about the FORMALISM.

That does not follow - you are making the unwarranted assumption that complete and exhaustive description is in some sense real. Copenhagen for example assumes the state is a complete and exhaustive description - but it is subjective knowledge of the system similar to the Bayesian view of probability. But there are interpretations where such is the case.

Sure. Note the only difference between Copenhagen and Ensemble is their view of the state - but both are not in any sense real. Copenhagen is like the Bayesian view of probabilities - the Ensemble is like the frequentest view. Both are very minimal interpretations. Applied math guys like me would say, just like the Bayesian and frequentest view, in practice its not really that important, but strictly speaking yes they are interpretations. Since the axioms of QM mention probability all you can really infer is the Kolmogerov axioms. But again in applied math its not usual to draw any real distinction.

Again, I repeat - It was about the FORMALISM.

Thanks
Bill

Last edited: Jul 28, 2014
11. Jul 28, 2014

### Ravi Mohan

Yes sir. I read your reply. I was trying to put my views on why I think that Dirac's statement is correct and makes sense.

But now, as you have pointed that first interpretation (in my post) does not imply what Dirac has written, I am pondering over it.

To be sure, which interpretation does Dirac follow in Principles of Quantum Mechanics?

12. Jul 28, 2014

### Staff: Mentor

None really - it was written in the early days of QM before such things were understood - it was even before Copenhagen.

Its basically not fully worked out ideas like his comment on the principle of superposition.

We now know the principle of superposition is not the real foundation - it basically an extension of probability theory whose pure states allow continuous transformations:
http://arxiv.org/pdf/quantph/0101012.pdf

Thanks
Bill

13. Jul 28, 2014

### Staff: Mentor

Just reviewing my posts and I see I forgot to mention exactly what was Dirac's misconception.

Its not what he said was wrong, its what he didn't make clear - namely the state he is talking about does not necessarily exist in a real sense. When you read it you naturally assume, like the states of classical physics, its real and that's where the problem arises. You think that, for example, if you are considering a state whose observation will give say position A or B, and from the principle of superposition that can be considered a superposition of a state that when observed will definitely give A and one that will definitely give B, then literally it is two positions at once. But since a state in QM is like probabilities, not real in any usual sense, it doesn't have to be viewed that way. Its like if you toss a coin you know it will give a head or tail but while its spinning in the air you cant say its both a head and a tail at the same time - all you can say is its spinning. The same in QM - just because you know you will get A or B if you observe it you cant say it's both A and B when you are not observing it.

Thanks
Bill

14. Jul 28, 2014

### akhmeteli

I don't know. My impression was that, say, diagram (b) at https://www.llnl.gov/str/May06/Beiersdorfer.html is part and parcel of a physical electron, so you can detect two electrons with energetic enough photons where there is actually just one physical electron. Am I wrong?

15. Jul 28, 2014

### Staff: Mentor

That is absolutely incorrect. You will only ever detect one electron.

Also, diagram (b) is a Feynman diagram depicting an electron interacting with a pair of virtual particles when subjected to an electric or magnetic field. It does not represent a "piece" of an electron or a detection of more than one particle. (Virtual particles are very complicated and usually make zero sense until you delve into the math underlying them)

16. Jul 28, 2014

### akhmeteli

Any arguments?

17. Jul 28, 2014

### Ravi Mohan

While I understand your explanation about superposition of two states not implying the existence of the particle in those two states, I would like to direct attention towards the literature of gravity induced collapse models.

In the Schrodinger-Newton equation
$$i\hbar\frac{\partial\psi(\vec{r},t)}{\partial t} =\hat{H}_o\psi(\vec{r},t)+m\phi\psi(\vec{r},t), \nabla^2\phi=4\pi Gm|\psi|^2\label{poiss}.$$
the gravitational potential is computed using a suitable Green's function. The mass density of the particle is given by $\rho(\vec{r},t) = m|\psi(\vec{r},t)|^2$ which does support (or is based on) the viewpoint (a particle existing in two or more superposed states).

Last edited: Jul 28, 2014
18. Jul 29, 2014

### Born2bwire

I would like to echo bhobba here too.

Superposition is not saying that an electron is actually in two different places at once. It is saying that it can be in multiple states with an associated probability. But when you actually measure the system, that electron will always be found in one single state. If you setup a large number of identical systems and make the same measurement, then you will find that the statistics of how many times you measure the electron in state A, in state B, etc. to match up with the predictions of quantum mechanics.

19. Jul 29, 2014

### Staff: Mentor

No.

Drakkith is correct.

QM is weird enough without making it weirder than necessary.

At a technical level see the link I gave to Gleason's theorem to understand what's really happening.

If the link is unclear, happy to clarify what's going on. But basically QM is a theory about the probabilities of the outcomes of observations. The state is simply a device to help calculate those probabilities - that's the import of Gleason's beautiful theorem (note I gave the modern simplified version based on POVM's)

Thanks
Bill

20. Jul 29, 2014

### Staff: Mentor

As I said I am talking about the formalism. Interpretations have their own take - of which the above is simply one of many.

What we need is a way to experimentally distinguish between them - basically no-one has been able to figure out how to do that. In fact many interpretations have deliberately been cooked up so it is observationally exactly equivalent to the formalism - so doing that would seem rather difficult - likely impossible.

Thanks
Bill

Last edited: Jul 29, 2014
21. Jul 29, 2014

### Staff: Mentor

Multiple posts that were nothing but "noise" have been deleted.