# Do particles have spatial extent or are they point-like?

• jamesb99

#### jamesb99

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?

Proton is not an elementary particle, so it has a certain size.

Elementary particles, like the electron, are considered point-like.
But it doesn't mean they're infinitely small. The smallest meaningful length is the Plank length.

Quoted from:
http://en.wikipedia.org/wiki/Point_particle

"In quantum mechanics, the concept of a point particle is complicated by the Heisenberg uncertainty principle: Even an elementary particle, with no internal structure, occupies a nonzero volume."

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 $\psi(x_a,y_a,z_a)$ and the impossibility of localizing both function $\psi$ 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 $\psi(x_a, y_a, z_a)$. But these $x_a, y_a, z_a$ are not arbitrary spatial coordinates x like that in electric field function $\mathbf E(\mathbf x)$, but the coordinates $\textit{of the electron}$. For two electrons a,b we have $$\psi(x_a,y_a,z_a,x_b,y_b,z_b).$$

This wavefunction gives a complex number for every possible combination of coordinates of both electrons, similarly to action in classical mechanics; $S(x_a,y_a,z_a)$ can look like a field, but $S(x_a,y_a,z_a,x_b,y_b,z_b)$ 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.

The standard model of particle physics is based on local quantum field theory describing point particles. No experiment has ever detected a finite size for one of the elementary particles. This has been verified experimentally up to the current energy (7 TeV) for the Large Hadron Collider, corresponding to a distance scale of about 10-17 cm.

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?

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). Momentum and energy, and therefore by implication mass by E=mc^2, are two sides of the same coin according to relativity, so saying that a particle's momentum has infinite uncertainty is effectively like saying that it could potentially be infinitely massive.

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 $\psi(x_a,y_a,z_a)$ and the impossibility of localizing both function $\psi$ 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 $\psi(x_a, y_a, z_a)$. But these $x_a, y_a, z_a$ are not arbitrary spatial coordinates x like that in electric field function $\mathbf E(\mathbf x)$, but the coordinates $\textit{of the electron}$. For two electrons a,b we have $$\psi(x_a,y_a,z_a,x_b,y_b,z_b).$$

This wavefunction gives a complex number for every possible combination of coordinates of both electrons, similarly to action in classical mechanics; $S(x_a,y_a,z_a)$ can look like a field, but $S(x_a,y_a,z_a,x_b,y_b,z_b)$ 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 really a philosophical question. The wavefunction measures the probability of finding a particle in a certain state (technically the square of the norm does this, but whatever). The wavefunction measures the probability of finding a point particle at a position. So there's some ambiguity as to whether a particle is indeed a point even though it always behaves like [\i] a point. In quantum field theory we learn that position isn't the true dynamical variable, but a field of which what we call a wavefunction in qm is a component in fock space, and that given the proper Fourier transform we can use whatever the hell coordinates we like - space-time and energy-momentum are most common - as long as certain properties of the field remain Lorentz invariant. So in a way, whether it's pointlike or not isn't meaningful because all of the physics is in the field, not the particle, which isn't pointlike at all. Just giving my two cents from the theoretician's side.

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?

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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)
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.

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?
As I said, the LHC at its current energy can resolve any particle structure down to 10-17 cm.

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.

But point-like particles don't behave like they're only at one position at once, which is the point, I assume, of the question that started this discussion. The Heisenberg uncertainty principle isn't just some arbitrary rule - it comes from the wave-character of the particles. I know what point you're trying to make but really it obfuscates the issue far more to say that these particles have no size, because the way we observe them guarunties they have size. I guess what I'm trying to say is - just because something is pointlike doesn't mean it exists at a point.

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.

My understanding is that despite an extended wavefunction, particles interact as if they are pointlike. Radius can't be determined to infinite precision, so the best one can say is that no nonzero radius has been found yet.

The wavefunction is an important part of scattering calculations as I understand it -- the shorter the de Broglie wavelength, the finer the interaction scale (and radius measurement) can be. So it is accounted for, but I don't think it contributes a cross-section to the particle. (Edit: My guess is that it does contribute an effective cross-section at each energy, but the cross-section diminishes with wavelength, towards a limit of 0. I'd appreciate someone with particle physics experience describing this better.)

Which is small, but not infinitely small - which was my point exactly.

The way you respond to questions is eerily similar to the way someone in my lab group does.

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.

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?

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.

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).

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.

"the number of electrons produced approaches infinity"
It doesn't approach infinity at all.

Yeah, it does. That's how I set up the experiment...

"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.

If he won't detect any electrons within Δt = (100 m)/c, then the domain that the electron wavefunction amplitude is integrated over is not infinity, which was exactly my point.