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

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

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.