# How big is an electron?

So I got a question about whether the fact that an electron is part of a quantum field means that the charge of an electron is “spread out” somehow. (See my article in the Boston Review for a pretty detailed discussion on what a quantum field is)

Spherical harmonic wavefunctions (from wikipedia)

Like most good questions in physics, there are a couple of ways to answer this question. The first is the quantum mechanical view. Here, we’re ignoring the “quantum field” bit, and just looking at boring old quantum effects. In this picture, yes, electrons have their charge spread out over their wavefunctions.

Electrons in quantum mechanics have a wavefunction, as do all quantum objects. This describes the probability distribution of the electron: how likely you are to find it in a particular place with a particular momentum and spin and so on. Critically, quantum mechanics means that the wavefunction isn’t just a description of our lack of knowledge: nothing in the Universe “knows” where the electron is “inside” the wavefunction before one looks for it. I’ll come back to what “look for it” means in a bit.

The most famous wavefunctions of electrons are the orbitals of atoms. In simple atoms (1 electron) these wavefunctions are spherical harmonics, and if you want to be a physicist you should get your spherical harmonic drivers license as quickly as possible.

Without getting into the spherical harmonics too deeply, the wavefunctions of electrons around an atomic nucleus sort of look like inflated balloons, which a certain symmetry as you can see looking at them. These images show where various electrons have the highest probability of being found. The actual probability distribution extends out to infinity, though you are far more likely to find an electron near the atomic nucleus than far away.

water molecule electron wavefunctions (from wikipedia)

The shape of these orbitals (and the non-analytically orbitals or more complicated atoms) tells you the shape of the electron charge distribution of an atom. For example, the water molecule is famously $\mbox{H$_2$0}$: two hydrogen and an oxygen atom. But as a result of the way that electron orbitals arrange themselves, water molecules have a “Mickey Mouse” shape: they have two “ears” of hydrogen atoms around the bigger oxygen atom. The electron orbitals are “pulled” down to the oxygen side of the molecule. That side has a net negative charge, and the hydrogen atoms have a net positive charge, because of the asymmetry in the “smearing out” of the electron wavefunctions.

This is a demonstration of reality of the wavefunction. Because water is polar, each molecule can repel each other when in solid form (when some positive sides need to be near other positive sides), causing ice to be less dense than water. It can allow strong attraction when positive and negative sides are near each other, leading to surface tension. All of these are important effects that allow life on Earth to exist. Be thankful for the reality of quantum mechanics.

So, now let’s ask about what it means to “look for an electron” inside an atom. To look for a charge, you shoot it with something that sees charge: photons. But if the photon is long-wavelength, you’ll only be able to “see” things bigger than the wavelength. You can see this effect if you watch ocean waves come in around rocks away from shore, if the rocks are small (compared to the wavelength), the water waves move past them without being seriously broken up by the existence of an obstacle. A larger rock, wavelength-sized or more, will disrupt the wave. Same with light: a photon with a wavelength longer than the size of the electron orbitals can’t “see” inside the atom. It can react to the electric fields of the atom as a whole, but it doesn’t localize the electron any smaller than the wavelength.

This is, by the way, why we can’t use photon (optical) microscopes for small objects. The wavelength is inversely proportional to the photon energy. Visible light has wavelengths between 400-700 nm (2-3 eV). Anything smaller than that can’t be resolved by visible light. To see the location of individual atoms, you can go to higher energy photons (which are difficult to produce and focus in optics), or use electrons which have much smaller wavelengths (since they have a mass, and thus much more energy than visible light photons), as in an electron tunneling microscope.

So, now let’s start throwing higher-energy photons at electrons. Eventually we’ll force the electron to respond to the photon as an individual particle, not as a part of the atom. In that case, you’ll get the electron wavefunction to collapse — to some location. You can’t predict where, but just give probabilities. Regardless, the electron is now somewhere, and the high energy photon scatters off of it.

Now the question becomes: does the electron act like a point of charge, or a charge smeared out over some region. The answer is, as far as we can see, an electron is a point particle. It scatters light as if it is a point of charge, not a smeared out object.

We know of particles that don’t act like this: protons. Protons are composite particles. They consist, on average, of two up quarks and a down quark, but also a sea of up and down and strange and charm quarks (and a few bottom and top quarks) and their antiparticles, and a ton of electrically neutral gluons. As long as you hit a proton with a photon of wavelength larger than the size of a proton (~1 fm), you see a point-like object. Higher energy photons see into the proton, and we get a measurement of the distributions of quarks that the proton contains (this is related to deep inelastic scattering which scatters electrons off of protons to measure the proton content).

Electrons, as far as we can see, don’t have internal structure. They are point particles. Which actually raises a conundrum. If you ask “how big an electron should be” quantum mechanics has an answer. You can guess that an electron should be big enough so that the energy in the electric field is equivalent to the rest mass of the electron (0.511 MeV). If the electron was some sort of charged jell, this would be the size of an pile of charged jell that had the rest energy of an electron.

What size is that? Converting a 0.511 MeV to a distance
$\frac{hc}{0.511~\mbox{MeV}} = 2426~\mbox{fm}.$
But this is hundreds of times larger than an atom. We know electrons are “smaller” than this: experimentally we have evidence for their non-compositeness down to thousandths of a femtometer. So, electrons are pretty damn point-like, if you can look with high enough energy light.
To understand how this is possible, we need to go from quantum mechanics to quantum field theory. Quantum mechanics can’t create particles. Quantum field theory can. When you start throwing photons with energy equivalent to the mass of the electron at an electron, you have (definitionally) enough energy to create electrons. Or, you have enough energy to create a positron — the antimatter counterpart of an electron, which must have the same mass as the electron.

What this means is that, when you start looking for the size of an electron inside the “classical radius” of an electron (2426 fm), you have to account for the possibility of creating pairs of electrons and positrons momentarily. When you do this, the big energy of a bunch of charge crammed into a small region cancels out, and you can have a much smaller electron without running into a contradiction about the rest energy (mass) of the particle. The electron could be point-like up to the Planck scale ($10^{19}$GeV) without any serious problem in our theory.

Interesting, we’d like to do something similar with the Higgs boson. The Higgs mass is 125 GeV, but it is hard to understand why it is not much larger — in the same way that we didn’t understand how the electron mass is so small given how tiny the electron is known to be. The electron is “allowed” to be so small because there is the positron, canceling the huge electron-charge self-energy. Unfortunately, there is no particle that can do that for the Higgs. Unless supersymmetry exists, and there’s a new particle, a Higgs partner called the higgsino, which can cancel the Higgs self-energy and resolve the contradiction of why the Higgs mass is so small.

Either that, or the Higgs is a composite particle (doesn’t seem like it after the results of the LHC), or we don’t understand something about how the Higgs self-energy should be calculated (quite possible).

(this analogy has been used by many people, but I heard it from my graduate advisor, Hitoshi Murayama).