# The electron as a smeared charge density

• I
• physwiz222
I do think that Schrodinger's original interpretation of the wavefunction, which is that it represents a smeared out charge density, was replaced with Max Born's probability interpretation.

#### physwiz222

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.

PeroK
physwiz222 said:
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.
I'm more sure how to react to your post as I feel that, in general, you say nothing that's novel, but at the same time are close to stating a personal theory, which we do not allow at PF.

physwiz222 said:
I believe that -e|ψ|^2 for a single electron literally represents a smeared out electron period.
Some calculations, such as the Coulomb integral in atomic physics, can be interpreted as a spread out charge density. The problem is in saying that it is nothing more than that.

physwiz222 said:
If you could somehow observe an electron without interacting with it what you would see a wavelike cloud which is the wavefunction squared.
You have a fundamental problem here in that this is a contradiction in itself in quantum mechanics. Observing is interacting.

physwiz222 said:
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.

I won't comment in detail the rest, but I think that you are trying too much to put classical notions on quantum phenomena. It can be useful to do this: it can help in understanding some phenomena, but one shouldn't think that it is a representation of reality.

Remember also that at the end of the day, when we observe the location of an electron, it is always at a single point in space.

DrClaude said:
I'm more sure how to react to your post as I feel that, in general, you say nothing that's novel, but at the same time are close to stating a personal theory, which we do not allow at PF.

Some calculations, such as the Coulomb integral in atomic physics, can be interpreted as a spread out charge density. The problem is in saying that it is nothing more than that.

You have a fundamental problem here in that this is a contradiction in itself in quantum mechanics. Observing is interacting.

I won't comment in detail the rest, but I think that you are trying too much to put classical notions on quantum phenomena. It can be useful to do this: it can help in understanding some phenomena, but one shouldn't think that it is a representation of reality.

Remember also that at the end of the day, when we observe the location of an electron, it is always at a single point in space.
A few things By observing without interacting I meant that if someone could see whats happening at the atomic scale without interacting. It was only hypothetical saying that the electron is literally a spread out charge density but not necessarily in a classical sense. It’s not a contradiction as it wouldnt happen in practice.

Its spread out and I said pointLIKE. Pointlike doesnt necessarily mean that its literally a point but that it has no internal structure in that there is no “Building block of the electron.” But it is spread out however but instead of being made of many small bits of charge density its one single charge smeared out, so theres no self interaction as theres no second charge to interact with its one single charge.

I am not making a new theory Quantum Mechanics IS correct period we dont need any new theory. Just saying that maybe the standard interpretation should change a bit maybe.

As for measuring whole electrons I am not sure about that yet but I believe it has to do with decoherence and entanglement, the latter I forgot to mention.

I disagree with @DrClaude , I think this isn't almost a personal theory. It is one.
As personal theories go, the motivation "I feel that", is not very good. The universe doesn't care how it makes you feel.
It is known from experiment that the electron has nb substucture down to the scale of 10-19 m or smaller. That blows your personal theory out of the water.

PeterDonis and vanhees71
I disagree with @DrClaude , I think this isn't almost a personal theory. It is one.
As personal theories go, the motivation "I feel that", is not very good. The universe doesn't care how it makes you feel.
It is known from experiment that the electron has nb substucture down to the scale of 10-19 m or smaller. That blows your personal theory out of the water.
Well I did say it has no internal structure. I meant that even though its a smeared out charge it technically isnt made of anything. It literally says in what I wrote.

I disagree with @DrClaude , I think this isn't almost a personal theory. It is one.
As personal theories go, the motivation "I feel that", is not very good. The universe doesn't care how it makes you feel.
It is known from experiment that the electron has nb substucture down to the scale of 10-19 m or smaller. That blows your personal theory out of the water.
Also those experiments count as a “measurement.” Meaning it would have had to interact with the system thus we only detect single electrons.

physwiz222 said:
Also those experiments count as a “measurement.” Meaning it would have had to interact with the system thus we only detect single electrons.
Your ideas in general don't fit well with QM. You are making the beginner's error that QM is wave mechanics in position space.

You can Fourier transform the wave function into momentum space and what does "literally smeared out momentum" mean?

Likewise, if you consider the electron spin, this does not fit the smeared out model.

The power of QM comes from abstracting the particle's properties into a state vector. Rather than seeing particles as physically smeared out classical objects in 3D space.

vanhees71 and DrClaude
physwiz222 said:
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.
Schrödinger's interpretation has some attractive aspects, and peer-reviewed articles in favor of it appear from time to time (e.g., Sebens, C.T. Found. Phys. 2021, 51, 75; Barut, A.O. Found. Phys. 1988, 18, 95–105). However, the interpretation has some difficulties. You mentioned some of them, but there are others. For example, the wave function of a free particle spreads with time, which seems to be in tension with the interpretation. Khrennikov (Beyond Quantum; Pan Stanford Publishing: Singapore, 2014) also offered the following critique of the interpretation: “Unfortunately, I was not able to find in Schrödinger’s papers any explanation of the impossibility to divide this cloud into a few smaller clouds, i.e., no attempt to explain the fundamental discreteness of the electric charge.”

So maybe my modification of Schrödinger's interpretation (Entropy 2022, 24, 261) might be of some interest for you. I consider e|\psi|^2 and similar charge density for the Klein-Gordon or Dirac equation as a smoothed charge density of a large collection containing N+1 electrons and N positrons, which seems to be compatible with the notion of vacuum polarization and to be immune to some difficulties of the Schrödinger's interpretation, such as wave function spreading. In particular, I prove (for one dimension so far) that any smooth charge density with an integer total charge can be approximated arbitrarily well by a collection of charges +e and -e (see the precise wording in Quantum Rep. 2022, 4, 486–508).

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physwiz222 said:
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.
The main problem (and the reason why Schrödinger gave up on his interpretation) is that this does not extend to the wave function of two electrons. Here ##\psi## has 6 coordinates but the the electron density has only three.

A much better interpretation is given by quantum field theory, where the density of an electron field can be defined as the expectation of the charge density operator. Apart from a constant factor, this happens to reduce to Schrödinger's interpretation in the special case where the quantum field is in a single electron state, but doesn't have the defect of Schrödinger's original proposal when states are more complex.

gentzen, Nugatory, vanhees71 and 2 others
A. Neumaier said:
The main problem (and the reason why Schrödinger gave up on his interpretation) is that this does not extend to the wave function of two electrons. Here ##\psi## has 6 coordinates but the the electron density has only three.

A much better interpretation is given by quantum field theory, where the density of an electron field can be defined as the expectation of the charge density operator. Apart from a constant factor, this happens to reduce to Schrödinger's interpretation in the special case where the quantum field is in a single electron state, but doesn't have the defect of Schrödinger's original proposal when states are more complex.
I literally said in the post that for a multi electron system the total electron density is the real thing that physically describes the system. Your interpretation is the same as mine

akhmeteli said:
Schrödinger's interpretation has some attractive aspects, and peer-reviewed articles in favor of it appear from time to time (e.g., Sebens, C.T. Found. Phys. 2021, 51, 75; Barut, A.O. Found. Phys. 1988, 18, 95–105). However, the interpretation has some difficulties. You mentioned some of them, but there are others. For example, the wave function of a free particle spreads with time, which seems to be in tension with the interpretation. Khrennikov (Beyond Quantum; Pan Stanford Publishing: Singapore, 2014) also offered the following critique of the interpretation: “Unfortunately, I was not able to find in Schrödinger’s papers any explanation of the impossibility to divide this cloud into a few smaller clouds, i.e., no attempt to explain the fundamental discreteness of the electric charge.”

So maybe my modification of Schrödinger's interpretation (Entropy 2022, 24, 261) might be of some interest for you. I consider e|\psi|^2 and similar charge density for the Klein-Gordon or Dirac equation as a smoothed charge density of a large collection containing N+1 electrons and N positrons, which seems to be compatible with the notion of vacuum polarization and to be immune to some difficulties of the Schrödinger's interpretation, such as wave function spreading. In particular, I prove (for one dimension so far) that any smooth charge density with an integer total charge can be approximated arbitrarily well by a collection of charges +e and -e (see the precise wording in Quantum Rep. 2022, 4, 486–508).

[1]
[2]
[3]
The discreteness of electric charge is the fact that the electron density cloud |Ψ|^2 isnt made of smaller charges its one single entity instead of being a collection of infinitesimal bits of charge. Technically the electron can be localized at 2 different spots but that doesnt mean that there are 2 clouds with charge e/2 now but rather that the cloud is most concentrated at 2 locations. Its one single entity still, theres no dividing the electron cloud into smaller clouds. It can be visualized as one single charge smeared out over space. The charge density spreads over time because of wave dispersion, as wavefunctions of different momenta travel differently. The charge density keep in mind can be thought of as a wave of electronness as well.

weirdoguy and PeroK
physwiz222 said:
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.
Does this interpretation make any testable predictions? Or is this another “we don’t allow unfalsifiable philosophy unless it’s a new interpretation of quantum mechanics” thread?

vanhees71