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I realize that an electron can only ever be detected at a single location in space when it is observed. That is, post-detection, an electron can only ever be at a single position in space. But prior to the detection, when the probability distribution of finding this electron is spread over space, can we interpret this as the electron existing at multiple locations in space? Or is it that the electron exists at one definite position even prior to measuring it but it's just that we don't know where it is and hence have to represent it as a probability distribution?

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when the probability distribution of finding this electron is spread over space,

To have a probability distribution we have to have a solution to a quantum mechanical wave equations with boundary conditions that will give us the probability distribution to discuss about.

can we interpret this as the electron existing at multiple locations in space

We can then interpret the probability distribution just as we do for classical probability distributions. When we throw dice and the probability is 1/6 to get an ace, it means that the ace exists, that is all. The electron exists within the boundary conditions, which we can check by measurements.

Here are is a bubble chamber a pi mu e decay, and we name the tracks by the particles. The main interaction, of interest in studies is at the vertex. If there were no liquid hydrogen for the charged particles to have thousands of interactions of low momentum ionizing the hydrogen atoms, we would not have the tracks, but the pion would still be bending in the magnetic field, interacting with it . We just would not know where it was not because of the quantum probability but because of not enough measurements.

A free electron in a beam of electrons has a track predicted by the fields , and the energy momentum it starts with at creation. The only QM indeterminacy is by representing it with a wave packet ,ie. a spread in its energy momentum vector, or time space vector,within the errors of the way it was created, as it travels down the beam pipe.

It all depends on the boundary conditions for a specific quantum mechanical problem.

So it is not simple

pimue

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An electron existing at different positions in space would raise more interpretation questions than those it is intended to reduce.

Here a partial list of difficulties.

If a full electron coexisted with its clones at different positions, we would go into troubles with conservation laws:

  1. charge and spin conservation;
  2. conservation of energy: the coexistence of many copies of the original electron should imply their mutual (repulsive) interaction. Suddenly, after a position measurement, all the clones would disappear, and also such interaction energy would disappear. Notice that all known calculations of energy based on the presence of one electron at the time agree very well with the measurements of energy. Moreover, the number of clones would not be finite, thus implying infinite repulsive energy between the clones.

It would be possible to imagine the coexistence of partial clones of the original electron. That is the presence at different points of the space of a fraction of the electron. Also, in such a case, there would be the presence of a self-energy modifying the energy between two measurements of position. Such a self-energy would have observable effects in atomic spectroscopy, but they have never been observed.

Therefore, we are left with only one electron. The only allowed statement about its position in each state can only be a probability distribution of measuring one observable of the one electron.

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