From a co-moving frame, what does a free electron look like, is it a wave or a point?

Question

I am not asking about an electron around a nucleus, whether it is a standing wave or not. I am specifically asking about a free electron, viewed from a co-moving frame. I am not asking whether an electron is a wave or a point, that has been asked a lot of times on this site. None of those questions answer my question about the co-moving frame's view.

I have read this question:

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. 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.

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

And this one:

Quantum objects are not waves. Quantum obejcts are not classical point-like particles. They are quantum objects, which may show wave-like and particle-like properties. You may represent a quantum state by its "probability wave" or wavefunction, whose square gives the probability density to find the object "as a particle" at certain locations. It is not a wave in the classical sense that anything physical would be oscillating here, and the Schrödinger equation does not always look like a wave equation.

Electron as a standing wave and its stability

Now as far as I understand, electrons do have rest mass and do have a rest frame. So it does have a meaning when we say, "in the co-moving frame of the electron". This is specifically what I am asking about, what does an electron look like in a co-moving frame. In this frame as I understand, the electron has to be completely stationary.

And this is where the confusion comes from. The electron does not cease to be a quantum particle. It is just that we view it from a co-moving frame.

So in this frame, the electron can be:

1. completely stationary, just a point, but then it cannot be subject to the HUP

2. not completely stationary, and still subject to the HUP, that is, an oscillating wave (maybe a standing wave)

As far as I understand, a co-moving frame means that we view the electron from its own frame (or one that always co-moves with it), and in this frame, we do know the electron's position, so it cannot be subject to the HUP. So basically what I am asking is, if we view the electron from its own frame, does it look like a point? Or would it still show wave characteristics?

Question:

1. From a co-moving frame, what does a free electron look like, is it a wave or a point?

Answer 1

The only way to meaningfully speak of a comoving frame of an electron is to take it to mean the frame in which the momentum of the electron vanishes. This, cannot always be arranged because an electron need not always be in an eigenstate of its momentum operator. However, if you imagine an electron in one of its momentum eigenstates then it can always be arranged to boost to a frame where it is in a momentum eigenstate with momentum zero. But, since it is in a momentum eigenstate, it simply doesn't have a definite position. So, even in such a comoving frame (in fact, especially when such a comoving frame can be arranged), there is no definite position of the electron. And, as always, HUP is not violated.

Finally, it depends on what you mean by "look like". As such, the position wavefunction of the electron from a comoving frame of the kind described above would be a (complex) wave of unit magnitude everywhere. But if looking is taken to mean measuring its position then, of course, it would collapse the wavefunction of the electron to a Dirac delta at a point and the electron would look like a point. However, at this moment, you'd no longer be in its comoving frame because the electron would have stopped being in a momentum eigenstate and there would not exist any comoving frame of that electron.

Answer 2

Insofar as being a superposition of eigenstates makes something a probability wave, an electron is always a wave.

Insofar as having no eigenvalue(s) corresponding to "size" makes something a point, an electron is always a point.

Insofar as definitely being an object that is sitting still in definitely one particular place, an electron is never a point.

You can't know you're in a comoving frame and know the electron's position at the same time. If you can be sure you're in a comoving frame, then you know the electron's momentum exactly, so by $\sigma_x \sigma_p \ge \hbar / 2 $ the probability that the electron is in any one specific place is zero.

If you're exactly sure where the electron is, then you know its position exactly, so the probability that the electron is in any one specific frame is zero.