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In the standard formulation of QM, do particles like the electron, proton, etc. have size (e.g. radius), or are they thought of as infinitely small points in space?
In the standard formulation of QM, do particles like the electron, proton, etc. have size (e.g. radius), or are they thought of as infinitely small points in space?
Often people tell that in QT electrons are not point-like, because the wavefunction is not point-like.
But the wavefunction is not the electron. Using Heisenberg's principle for arguing for non-zero size of electron is really talking only about the size of the wavefunction [itex]\psi(x_a,y_a,z_a)[/itex] and the impossibility of localizing both function [itex]\psi[/itex] and its spatial Fourier transform.
In non-relativistic quantum theory the wavefunction is not a spatial field, none the more the potential energy of interacting many-particle system is a spatial field. Of course, for one electron a there is a function [itex]\psi(x_a, y_a, z_a)[/itex]. But these [itex]x_a, y_a, z_a[/itex] are not arbitrary spatial coordinates x like that in electric field function [itex]\mathbf E(\mathbf x)[/itex], but the coordinates [itex]\textit{of the electron}[/itex]. For two electrons a,b we have [tex]\psi(x_a,y_a,z_a,x_b,y_b,z_b).[/tex]
This wavefunction gives a complex number for every possible combination of coordinates of both electrons, similarly to action in classical mechanics; [itex]S(x_a,y_a,z_a)[/itex] can look like a field, but [itex]S(x_a,y_a,z_a,x_b,y_b,z_b)[/itex] clearly is only a function of the space-time coordinates of the particles. What else in this theory than point-like could these particles be?
Besides, experimentally electrons are well point-like - I'm thinking of the point spots on the detector screens, photographs of trajectories, indivisible elementary charge, etc.
This is a good illustration why I don't teach the uncertainty principle - people invoke it in circumstances where it does not apply. Whether a particle is pointlike and whether it can be localized are two entirely separate issues.No particle is truly pointlike because the Heisenberg Uncertainty principle tells us that in order for there to be absolute spatial localization (ie 0 Δx), we would need there to be zero specification of the momentum (ie 0 Δp)
As I said, the LHC at its current energy can resolve any particle structure down to 10^{-17} cm.BTW, experimentally, you always have finite resolution (instrument uncertainty) on your detectors, so how are we able to detect whether something is point-like or not?
This is a good illustration why I don't teach the uncertainty principle - people invoke it in circumstances where it does not apply. Whether a particle is pointlike and whether it can be localized are two entirely separate issues.
As I said, the LHC at its current energy can resolve any particle structure down to 10^{-17} cm.
Which is small, but not infinitely small - which was my point exactly.
So I could say EVERY particle has a spherical probability wave that permeates the whole universe, and EVERY particle has an above zero chance to become physical anywhere in the universe. To say we are all made up of the stuff of stars is quite an understatement.
Yep, but what bothers me is that nothing can propagate faster than c, so how can the domain of an electron be infinity?
I think I understand what you are bothered by but that speed of propagation is irrelevant. If a particle could at a given instant in time be either HERE or HERE plus 10 cm in some direction, how is that different qualitatively from saying it can be either HERE or HERE plus 10 light years?
The probability distribution has notthing to do with travel. If the particle IS HERE plus 10 light years, that's just where it is. It didn't travel to get there.
It seems to me that if this were true, then FTL communication would be possible.
As a thought-experiment, consider an experimenter who has a container PACKED full of photons. He can release photons at will so that they collide with a nuclei, and via pair production, this produces an electron and a positron.
Let's say we want to transmit the message "101" faster than the speed of light to an observer 100 meters away, who has an electron detector.
At t=0, the experimenter releases a huge number of these photons all at once. By pair production, an equally huge number of electrons is produced. At the instant these electrons are produced, the observer 100 meters away would INSTANTLY register a statistical anomaly at his electron detector, simply because as the number of electrons produced approaches infinity, the probability that the observer detects one of these at his detector approaches 1.
At t=1, the experimenter does nothing. The observer notices no statistically significant change in his data at this time.
At t=2, the experimenter again releases a huge number of photons. The observer now would see another statistical anomaly.
So, how do we explain this then? The only thing I can think of is that the probability domain of measuring an electron is not the entire universe and that the theory is either too simplified to make it easy to understand or that there is something fundamentally wrong with it (I'm guessing the former).
"the number of electrons produced approaches infinity"
It doesn't approach infinity at all.
"the probability that the observer detects one of these at his detector approaches 1"
Only if they had enough time to travel those 100 meters, with a speed less than c, else he won't detect any of the electrons.
A particle can be anywhere, as long as it is possible for it to get there, given the observations that were previously made on it.