Physicists think they’ve resolved the proton size puzzle.

This story. Oof. They should have someone who understands quantum mechanics write about it. So many problems.

But quantum mechanics gives us a much more precise (albeit weirder) description. The electrons aren’t really orbiting the nucleus; they are technically waves that take on particle-like properties when we do an experiment to determine their position. While orbiting an atom, they exist in a superposition of states, both particle and wave, with a wave function encompassing all the probabilities of its position at once. A measurement will collapse the wave function, giving us the electron’s position. Make a series of such measurements and plot the various positions that result, and it will yield something akin to a fuzzy orbit-like pattern.

“They are technically waves….”

Nope. Electrons are not ordinary classical waves like ripples in water. They are quantum objects described by a wavefunction. That is a totally different thing. There is a quantum state in Hilbert space, represented in position space by a wavefunction, whose squared magnitude gives the probability density for finding the electron at different locations.

“Take on particle-like properties when we do an experiment….”

This is way too crude to be accurate. That suggests the electron was truly a wave and then becomes a particle only because we took a look at it. Quantum mechanics does not work that way. What it predicts is that measurements yield discrete, localized outcomes. That is different from saying the electron was previously just a classical wave. (It wasn’t.)

“While orbiting an atom….”

In modern quantum mechanics the whole point is that atomic electrons are not moving on definite classical orbits. They occupy orbitals, which are stationary quantum states with definite energy, angular momentum properties, and spatial probability distributions. Quantum state, not orbit.

“They exist in a superposition of states, both particle and wave….”

Not conceptually sound. “Particle” and “wave” are not usually the two states in a superposition. Superposition refers to combinations of quantum states such as different energy eigenstates, angular momentum states, spin states, or position states. Wave-particle duality is really not well described as “being in a superposition of wave and particle.” That doesn’t make a lot of sense.

“With a wave function encompassing all the probabilities of its position at once….”

That is imprecise. The wavefunction does not directly list probabilities. Its squared magnitude gives the probability density for position. More generally, the wavefunction encodes the probabilities for many possible measurement results, not just position.

“Make a series of such measurements and plot the various positions that result, and it will yield something akin to a fuzzy orbit-like pattern….”

This is misleading in two ways. First, repeated position measurements on the same electron do not reveal some hidden orbit. That’s just not how reality is, unfortunately. The measurements disturb the state. Second, what you recover from many measurements on many identically prepared atoms is the orbital probability distribution, not an orbit-like path. It is not revealing of a blurred trajectory around the nucleus. It is a cloud-like spatial distribution characteristic of the quantum state.

My corrected, accurate version of that portion would read:

“Quantum mechanics replaces the antiquated picture of electrons orbiting the nucleus like planets orbiting the sun. In the quantum way of doing things, an electron in an atom is described by a wavefunction, which encodes the possible outcomes of measurements and their probabilities. Bound electrons occupy orbitals, which are standing-wave-like quantum states with discrete energies. These orbitals are not paths through space. They are stationary state descriptions whose squared magnitude gives the probability density for finding the electron at different locations. When a position measurement is performed, the electron is detected at a particular place as a localized event. Repeating the experiment across many identically prepared systems does not reveal a smeared-out orbit, but rather the characteristic spatial probability pattern of the orbital. The electron therefore does not fit neatly into the classical categories of either a tiny orbiting particle or a literal extended wave. Instead, it is a quantum object with behavior that shows aspects of both, depending on how it is probed.

If you measure position across many identically-prepared atoms and plot the results, you recover the orbital’s probability distribution, which looks like a cloud or density pattern, not a fuzzy track traced out by an electron in orbit.”