If you have $10 in a bank account, are those individual dollars or do they all just form an account balance, which is "just" a number.

Electrons in a coherent state are the same. They don't have individual identity. There is just the wave function occupying some area in space.

A similar example is if you hit a key on a piano. This will not only produce the sine wave of the base frequency, but also a number of other frequencies, including overtones and other frequencies that provide timbre.

But these other frequencies are in reality only mathematical artifacts of doing a Fourier transform of the sound pressure from a time domain to a frequency domain.

What is real (if we ignore the molecular level and below) is that a pressure wave is propagated through the air. The individual frequencies of the wave cannot be found anywhere particular in the wave.

Likewise, the exitation of the electron field associated with some atom in a given state will, at every point, represent the combined contribution of ALL electrons.

These electrons are stacked on top of each other the same way the frequencies in a piano tone is or the way the dollars in a bank balance is. They only exist as part of a whole, not individually.

it brings to the table that electrons in an atom are in fact not different things. The negative part and the positive part of the sine functions are still part of the same function

> And even then, you are simply getting a measurement of a thing.

This is exactly where the Many World Interpretation and the Copenhagen Interpretation diverges.

If you send a proton with enough violence through the core of a lead atom, for instance, any given observer will see patterns that tend to be interpreted as interactions between nuclei (or quarks, depending on the situation). We can draw Feynmann diagrams for this and even some calculations.

For other particles, such as electron-electron interactions, these can even be quite clean and give the impression that the particle has a certain location at the time of interaction.

While your objection can still be made in this case, it's much weaker. Physicists tend to consider cases like these as an observation of an individual particle. (If they ignore the MWI).

Within the MWI, though, the assumption is that every possible interaction all occur at once. It's just that usually the outcomes are not mutually coherent. And when so, later interactions between the different paths the wavefunction is taking cannot interact with eachother.

In this case, your objection becomes true at a much deeper level.

In any case, we simply do not yet KNOW what exists at the deepest level. All we have is guesses.