Are there experiments that show that the electron wave function can be point-like?

Question

I know that there are scattering experiments that show that electrons act like structureless particles up to extremely small scales. But in these experiments the electrons are moving, so their wave function cannot actually be considered point-like in the sense of a $\delta$ distribution.

Usually, if I imagine an electron, I think about a coulomb potential reaching up to but excluding $r\to 0$. Of course, as we all know, if the corresponding wave function was really shaped like $\delta$, this can only be an instantaneous event due to quantum mechanics and the completely undefined momentum in this case. So in the next moment, the single electron's wave function would already have dispersed to some extent, but this is not the focus of my question.

What I ask is, are there any experiments at all, where the electron's wave function reduces to a $\delta$ shape, and where we can check somehow that its coulomb potential is as nice as we imagine it? If we make a double slit diffraction experiment, the electron will only be localized to the size of the atom that captures it on the screen behind a double slit. If we get a single $-1e$ charged ion into a trap, it is the same situation.

Shouldn't electroweak interaction and the Higgs mechanism that gives the electron it's mass, give it also a structure on very small scales? I think if the electron was strictly Coulomb shaped, it was also strictly electromagnetic, and so there would not be the possibility for weak interaction, right? Or is this a too simple mental picture?

Answer 1

Scattering experiments are exactly the sort of experiments you describe.

Consider the experiments done at the Large Electron–Positron Collider where electrons were scattered off positrons. In the lab frame it is true that both the electrons and positrons are moving, but there are no absolute velocities so it is just as valid to analyse the results in a frame in which the electron is stationary. In this frame we have high energy positrons scattering off a stationary electron. Conversely we could use a frame in which the stationary positrons are being bombarded by high energy electrons.

But regardless of our choice of frame it's the same experiment - the results can't be affected just by a choice of reference frame - so LEP did exactly the sort of measurements you describe. And it did indeed observe interactions with Z and W particles, and these do indeed mean the Coulomb potential does not apply at very short distances. LEP didn't have enough energy to observe interactions with the Higgs field (though it came close) but with a bit more energy it would have measured Higgs interactions as well.

There is a lot of confusion about what it means for a particle to be point like. Currently our best description of elementary particles is given by quantum field theory, and this describes particles as a superposition of plane waves called a coherent state. This is not a delta function, but it can be made arbitrarily close to a delta function at the cost of increasing energy. So when we say a particle is a point we don't mean it is a delta function. To realise such a state would require infinite energy so no such states exist. Instead we mean the particle can be made as close to a point as we want provided we can generate the energy required.

The real significance of point-like is that the particle is not composite. If you consider a proton it has a radius of around a femtometre due to its internal structure, and increasing the energy just probes the internal structure. Hence the proton is not point-like. As far as we know electrons are point-like since we have never measured any internal structure within them, but this does not mean the electron is physically located at a point.

Answer 2

All experiments are consistent with the point particle nature of the electron.

You are mixing up particle shape, potential function and wave function.

Answer 3

The delta functions introduced as position eigenstates in QM are never physically realised. As a rough guide, experimentally, the smallest distance scale you can probe with an experiment is on the order of the wavelength of the particle being used to do the probing. Since no particle, no matter how energetic, can have a wavelength of 0, you can't ever measure the position of an electron to that accuracy.

Also, having no internal structure, and having no extension over physical space are not mutually exclusive.

Answer 4

Are there experiments that show that the electron wave function can be point-like?

The wave function is not a measurable quantity, so there are no experiments to measure it. What is measurable is the probability distribution for a given experimental situations, $Ψ^*Ψ$, which can be compared with a large number of measurements with the same boundary conditions.

All the electron-positron collider experiments have measured the crossection of interaction for electron positron interactions at various energies, and all the experiments agree, within experimental errors, with the Quantum Field Theory predictions of the standard model which has the electron ( and the rest of elementary particles) as point particles, i.e. having no volume in space. The axiomatic point particle assumption is up to now validated.

I think if the electron was strictly Coulomb shaped

One must have clear the distinction between quantum mechanics and classical theories. There is no meaning to "coulomb shape" as in simple quantum mechanics potential problems the solutions of the wave equations with the potential included absorb its effect. That is why ground states exist in atoms and the world as we know it. The 1/r potential can be seen only in macroscopic experiments , at the quantum level it is more complicated.

In second quantization QFT models, experiments are compared with the Feyman diagram predictions for the interactions, crossections and decays, i.e. probability distributions directly and the incoming and outgoing particles from the interaction are considered point like, as the quantum fields on which creation and annihilation operators operae are plane waves , and in the energy-momentum frame where calculations are made it is not necessary to introduce the concept of wave-packet.

If one wants to model a single electron moving in vacuum one can do it with a wavepacket solution, but the theory is such that it is unnecessary to represent a beam of electrons scattering on a beam of positrons that way. The experimental errors are still much larger than the Heisenberg uncertainty widths anyway.