61

Although it's commonly said that fundamental particles are point particles you need to be clear what this means. To measure the size of the particle to within some experimental error $d$ requires the use of a probe with a wavelength of $\lambda=d$ or less i.e. with an energy of greater than around $hc/\lambda$. When we say particles are pointlike we mean ...


50

Under special relativity nothing can be incompressible: consider any object of nonzero size and finite mass in its rest frame; when you apply a force to it on one side it will start moving. If it were completely incompressible, the other end would start moving simultaneously. Since the ends are spatially separated, there is a frame in which the other end ...


48

As far as we know the electron is a point particle - this is addressed in the question Qmechanic suggested: What is the mass density distribution of an electron? However an electron is surrounded by a cloud of virtual particles, and the experiments in the links you provided have been studying the distribution of those virtual particles. In particular they ...


37

If something is 'Electrically neutral' this means that the algebraic sum of its electric charges, however distributed, is zero. This does not imply that there is no electric field in its vicinity. Plenty of neutral bodies – even, it is believed, the neutron – have electric fields, for just the reason you have pointed out.


35

Can the center of charge and center of mass of an electron differ in quantum mechanics? They can. Particle physics does allow for electrons (and other point particles) to have their centers of mass and charge in different locations, which would give them an intrinsic electric dipole moment. For the electron, this is unsurprisingly known as the electron ...


34

It is said that atoms with the same number of electrons as protons are electrically neutral, so they have no net charge or net electric field. This is a great over-simplification, which I am sure you have already determined (based on why you are asking this question). You can have objects that are polarized where, overall, they have no "net charge", yet the ...


34

In quantum field theory, an elementary particle doesn't have one precise location and size in space. The quantum of an electron field in free space has different extent compared to the electron around a hydrogen atom, for example (i.e. it's harder to bounce an electron off a free electron than off a hydrogen atom). While in one very real way, an electron's ...


26

One who is familiar with the history of particle physics, and physics in general, knows that physics is about observations fitted with mathematical models. This review examines the limits on size we presently accept for the fundamental particles which presently are at the foundation of the present standard model of particle physics. This analysis of what &...


26

It's true that a point particle with finite charge is problematic in electromagnetism because of the infinite field and associated energy near such a particle. However, we don't need that concept in order to make a defining statement about the electric field. Rather, we can use $$ {\bf E} = \lim_{r \rightarrow 0} \frac{\bf f}{q} $$ where $\bf f$ is the force ...


24

The shape of a distribution of charges is described in terms of multipole expansion, which you can think of as similar to Fourier expansion but in two dimensions. The total charge gives you the "monopole term", whose interaction is spherically symmetric. If there's an offset between the center of the mass distribution and the center of the charge ...


23

Scattering experiments can be used to determine the size of a particle. The results for an extended object are different than that of a point particle. But all of these scattering experiments depend on getting the probe particle "close" to the scattering object. In the case of electrons, that means launching the probe with enough energy to overcome the ...


19

We study it using deep inelastic scattering. It was this type of experiment that first revealed the proton had an internal structure, and if the electron has an internal structure it will be this type of experiment that reveals it. No experiment has yet discovered evidence for an internal structure in the electron, however that doesn't prove the electron ...


18

If the particles are not point-like, they will take up some volume. As the gas is compressed, the collision frequency will rise more quickly, which will make the pressure-volume curve change. The corrections in the Van der Waals model of a real gas account for the volume of the particles. Also if they have internal structure, that structure can have ...


18

There is no "going" going on in field-line diagrams. The direction of the field lines indicates, by convention, the direction of the electrostatic force experienced by a positive test charge at that location. Field lines do not indicate the 'flow' of any physical quantity, and there is nothing being 'generated'; instead, all you have is a force field, and ...


16

To represent the states of a single particle in the $3$-space you should exploit the so called Newton-Wigner position representation. The point is that the so obtained wavefunction, for a KG field, is a complex function defined in the rest space of the reference frame you are adopting and it has a strongly non-local behavior under the action of Lorentz ...


16

Because of the Pauli exclusion principle, it's extremely difficult to compress atomic matter beyond a certain density. It's not impossible, because there are always higher-energy electron states available, but there's a very strong force opposing it (called electron degeneracy pressure). This is what it means for space to be full. If you define "empty space"...


16

Building upon other answers, we must first differentiate between net charge and electric field - an atom with an equal number of equally charged positive and negative particles will have no net charge, but may still have an electric field, depending on the arrangement of the charge, as in a dipole. Now, your intuition is correct, it doesn't seem valid that ...


15

You're forgetting one thing: a particle cannot feel its own electric field, so a point charge that generates a $1/r^2$ field doesn't do anything unless acted upon by an external field. You also can't place a particle at $r=0$ of another particle's $1/r^2$ electric field, because, well, there's already a particle there. (Also, how are you going to get it ...


13

Electrons, and such small things :-) are handled by quantum mechanics. Quantum mechanics differs very, very much from the classical, Newtonian mechanics and from our intuition based on our experience. In QM, although the electron is handled as if it were a point-like body, it doesn't have an exact location. Instead of it, its location is described by a ...


12

Composite particles in QFT have size in the sense that cross sections are not scale-independant (because they have a radius that breaks that invariance). The radius of the proton was first measured by Robert Hofstadter. He studied the scattering of electrons and atomic nuclei. The Fourier transform of the cross-section is just (proportional to) the charge ...


12

Before addressing the pointlike nature of the electron, let's consider the proton. It was found that when the energy with which particles (such as electrons) scatter off a proton exceeds a certain level (about 1 GeV), it starts to resolve the proton. What we mean by that is that, below this energy the scattering cross-section seems to follow a scale ...


12

When you start compressing ordinary matter, you first start by decreasing the space between atoms (after you have, almost mechanically, broken the bonds between molecules). This gets increasingly harder because the atoms are bouncing around and they are repelling each other, because when two atoms get close enough to each other, their electron clouds see ...


11

This is one of the key results of quantum field theory: particles are not single points, they are disturbances in quantum fields that are spread out over space. Typically the disturbance is not spread out very much, otherwise it looks more like what we know as a wave than a particle. The technical term for what you're calling a "smear" is a wavepacket.


11

Elementary particles are understood today as the quanta of quantum fields. The fields are ontologically primary and exist even when there are no particles, but a quantum field is not “a wave-only model” as is, say, a classical electromagnetic field. Instead, a quantum field is a continuous field, existing everywhere in spacetime, of operators that create ...


10

Yes, elementary particles such as electrons and quarks (inside protons) are point-like or at least, their internal structure is incomparably smaller than the size of the atom. So the atom is mostly empty space. However, that doesn't mean that atoms may penetrate each other. Matter is impenetrable because of a combination of the uncertainty principle that ...


10

It depends on how small the size of the cylinder is compared to the rope/string. You want the rope as long and light as possible, and the bob as small and heavy as possible. You also want the bob to be much heavier than the string. Mathematically, you want the moment of inertia of the bob about the rotation point to be as close to $mL^2$ as possible, where $...


9

The missing piece of information is that the point-particle stress-energy tensor is actually $$T^{\mu\nu} (x^\mu) = \int \frac{m}{\sqrt{-g(x^\mu)}} \frac{d \gamma^\mu}{d \tau}(\tau) \frac{d \gamma^\nu}{d \tau}(\tau)\delta^{4}[x^\mu - \gamma^\mu(\tau)] d\tau $$ where $\gamma^\mu(\tau)$ is the world-line of the particle and the integration bounds goes through ...


8

There's never any direct experimental proof that anything does not exist, including a nonzero electron radius. But we have a very good (you might even say "Standard") Model that describes the electron as a point particle and accurately explains all known experimental data (at least, data describing processes involving electrons). With no experimental ...


7

In fact any object with non-zero physical extent is required to be deformable by relativity (but see below); otherwise pushing on it can transmit energy and information faster than light. And quantum mechanics rules at those scales so everything does have fuzzy edges. At the scale of the very small there are no sharply defined objects. A consequence of this ...


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