How to visualize an electron existing in two different places at the same time?

Question

Let's consider a hypothetical situation where there are two electrons. The first electron is in superposition, simultaneously existing in two different locations. Let the locations be A and B. The other electron is in superposition, simultaneously existing in locations C and D.

If the first electron exists in location A and the second electron exists in location C both these electrons are close enough to interact with each other, which gives a resultant outcome. But if the first electron was to exist at location B and the second electron was to exist at the location D they don't interact with each other. For the other two possibilities, the electrons don't interact with each other.

So if this was the case, wouldn't we have two possible outcomes? The first outcome where the electrons interact with each other and the second out come where the electrons didn't interact with each other.

Let the first outcome occur. This influences the environment in one way. If the second outcome occurs, it influences the environment another way. So we have two states for the environment:

1. A change occurs in the environment as a result of the two electrons interacting.

2. The change that occurs to the environment when both the electrons don't interact.

Now, we have two alternating environments rather than alternating electron positions. As time goes, won't this slowly lead to the existence of two possible states of the universe coexisting? So when this happens, does it mean that I am posting this question and at the same time not posting this question here? How can this be resolved?

Answer 1

Let's first change the question slightly to get rid of the problem with identical particles: The initial state is that an electron is at A and B and a proton is at C and D, i.e. |i> = (|A>+|B>)(|C>+|D>) (normalization is not considered). This state can be written as |i> = |AC> + |AD> + ... meaning that "electron at A, proton at C" and, at the same time "electron at A, proton at D" and, ... As quantum mechanics is linear, you can evaluate what happens when the initial state were |AC> etc. The resulting final state |f> then is a superposition of these particular final states. All the particular final states are included as possibilities in the resulting final state.

If your environment is quantum, you will end up with a superposition of combined (system, environment) states (*). The system and the environment will be entangled.

If your environment is classical, it may be interpreted as a measurement. The type of measurement (= your environment) that you carry out will determine the possible outcomes and their respective probabilities.

Why should a quantum environment act differently as a classical environment? This is exactly the question that Schrödinger pointed out with his cat experiment. In the decoherence theory, the (*) state decoheres into a superposition of states with definite measurement outcomes (pointer states). How is this superposition resolved? In the many-worlds theory each of the definite outcomes will be realized in different "worlds". So in the end, there will be four "Sreram K" "instantiations". One sees as the result from the interaction of "electron at A and proton at C", the other the result from ... :-)

@CuriousOne: This is absolutely wrong. Quantum mechanics is not only applicable to emsembles but also to single quantum systems.

Edit: @CuriousOne: A single electron can be in a superposition of being at A and being at B. What does this mean? When you measure its location, you will find it either at A or at B. But before the measurement, it was not either at A or at B but in the superposition. Look at the double-slit experiment. When you send an electron through the double slit, it will end up at some point on the detector. When you pass many electrons independently, you will see an interference pattern (this is why one electron already can be at A and B. See https://en.wikipedia.org/wiki/Double-slit_experiment: Jönsson; Pozzi et al; Tonomura). Now measure through which slit the electron passes: A is "upper slit", B "in lower slit". Repeat the experiment. You will always find in passing either through A or through B. If you manipulate it in such a way that it definitely passes through A or through B, the interference pattern will vanish. This is the point: if the state is "superposition of A and B", the electron will be able to interfere. If the state is "either at A or B, I just don't know", it will not interfere.

Answer 2

There is no intuitive description of what it means for an electron to be in a superposition of two different locations. But what it definitely does not mean is that the electron is simultaneously in both positions.

The most important effect of superposition is the interference of superposed states. A quantum system in superposed states A and B is not both in state A and state B, it is in a state where A and B interfere. But there is more: whenever A and B are presented as the two interfering physical descriptions of the system, it is because we know from physics that A and B are the only acceptable descriptions of the system behavior: there is no C alternative to consider. In classical mechanics this would be because A and B are extrema of the system action (according to the principle of least action). Now Dirac then Feynman showed that the principle of least action itself originates from the quantum path integral: the paths of least action arise because of the interference of all imaginable paths. So we got it backward from the start: A and B are notable only because they themselves represent the superposition of all imaginable states of the system!

Back to our electron, what this means is that a more correct description of its quantum state is that it shows it busy exploring in parallel all imaginable ways to behave, all places, all motions, even arbitrarily strange ones with abrupt changes of direction, absolutely all of them, an infinity of them. By "exploring in parallel" I mean a very abstract interaction between all these behaviors. They all rotate in some abstract place according to their corresponding action, and when we combine them together taking these rotations into account as phase relationships governing an overall interference, we eventually get a probability distribution for measuring specific observables of the electron (such as its position or its momentum), not even its full state (which would be position and momentum together). The electron never has any classical state at all!

Now let's consider two electrons. According to the above discussion, it is only a semi-classical description to say that one in is A+B while the other is in C+D. We have to think they both are doing their thing in the path integral, where all imaginable situation is included. If positions A, B, C and D are privileged it is because of the environment they live in, which appears as boundary conditions in the path integral. Quantumly, the two-electron system just explores absolutely all behaviors compatible with the environment boundary conditions. You see here implied the fact that the environment is not changed by the system dynamics (in any of its superposed contributions). This was also implied in the above introduction of action, because the action needs boundary conditions to be defined. If that assumption is false, then we cannot represent the environment in terms of boundary conditions, so it is not really an environment anymore and we have to make it part of the quantum system; this is called entanglement. This is the point you rise in your question: a reactive environment is doomed to be entangled with the quantum system.

So if quantum systems spread by entanglement, how do we have a classical environment in the first place? This is still an open question. It is linked to the measurement problem because a measurement provides a boundary condition, but we do not know what a measurement exactly is.