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- TL;DR Summary
- In Quantum Mechanics the standard interpretation is that |ψ|^2 is only a probability and not a physical thing and the electron is a point particle. I feel like it should be reinterpreted as a function that literally describes the smeared out electron the magnitude corresponding to the probability of measuring it.

Schrodinger’s original interpretation of the wavefunction was that it represented a smeared out charge density however this was replaced with Max Born’s probability interpretation. The issue was from what I understand that a charge density would repel and have self interactions as all the charge elements would repel. However I believe the correct interpretation is somewhere inbetween. I believe that -e|ψ|^2 for a single electron literally represents a smeared out electron period. If you could somehow observe an electron without interacting with it what you would see a wavelike cloud which is the wavefunction squared.

As for the issue of the charge density interacting with itself, it shouldnt be thought of as a classical charge density in the sense of infinitesimal charges spread out but rather one single charge smeared out over space. Essentially its one solid piece not many small elements of charge. Theres no self interaction as there is no second charge to interact with, its all one single charge, as the electron has no internal structure its just one entity, essentially pointlike. However that doesnt mean that it literally is a point just that it has some properties similar to a point charge. It can be thought of as the quantum mechanical analogue of a classical point object. However the magnitude of this electron density corresponds to the probability of where we measure the electron. Think of it this way where do you think the electron is most likely to be measured, In regions where there is more electron density or less electron density, so the born rule isnt totally wrong, just slightly off.

Thinking about the electron as a single point we cant measure only use probability doesnt work. For one why do we need a probabilistic theory. Some may say that measurement changes the system but thats only in practice. In theory we could model the electron classically as having a well defined position as a literal point particle, but why dont we. for example in a Hydrogen Atom, an electron cant be circling too fast to track because it would emit radiation falling in to the nucleus and technically we could still model it theoretically. Simply put the electron cant be moving around the nucleus at all as it would lose energy by radiation. As for a still point charge again it simply doesnt work as the nucleus exerts a force. We simply cant apply this notion to quantum objects.

We simply have to consider the electron as a single charge smeared out over space. for a multi electron system the total electron density is what the electrons actually look like and are doing. However the wavefunction is somewhat real in the sense that electric potential is real. Its merely a calculation tool to get the correct description as we only observe Electromagnetic fields not potentials. In a similar way, when we measure the particle many times the distribution of points is the probability density not the wavefunction. I dont consider wavefunction collapse to be real rather its related to decoherence.

This is just how I view quantum mechanics and how I think it should be interpreted. Its not perfect but I think its plausible.

As for the issue of the charge density interacting with itself, it shouldnt be thought of as a classical charge density in the sense of infinitesimal charges spread out but rather one single charge smeared out over space. Essentially its one solid piece not many small elements of charge. Theres no self interaction as there is no second charge to interact with, its all one single charge, as the electron has no internal structure its just one entity, essentially pointlike. However that doesnt mean that it literally is a point just that it has some properties similar to a point charge. It can be thought of as the quantum mechanical analogue of a classical point object. However the magnitude of this electron density corresponds to the probability of where we measure the electron. Think of it this way where do you think the electron is most likely to be measured, In regions where there is more electron density or less electron density, so the born rule isnt totally wrong, just slightly off.

Thinking about the electron as a single point we cant measure only use probability doesnt work. For one why do we need a probabilistic theory. Some may say that measurement changes the system but thats only in practice. In theory we could model the electron classically as having a well defined position as a literal point particle, but why dont we. for example in a Hydrogen Atom, an electron cant be circling too fast to track because it would emit radiation falling in to the nucleus and technically we could still model it theoretically. Simply put the electron cant be moving around the nucleus at all as it would lose energy by radiation. As for a still point charge again it simply doesnt work as the nucleus exerts a force. We simply cant apply this notion to quantum objects.

We simply have to consider the electron as a single charge smeared out over space. for a multi electron system the total electron density is what the electrons actually look like and are doing. However the wavefunction is somewhat real in the sense that electric potential is real. Its merely a calculation tool to get the correct description as we only observe Electromagnetic fields not potentials. In a similar way, when we measure the particle many times the distribution of points is the probability density not the wavefunction. I dont consider wavefunction collapse to be real rather its related to decoherence.

This is just how I view quantum mechanics and how I think it should be interpreted. Its not perfect but I think its plausible.