# Does an electron flying through empty space stay on a fixed trajectory, or does it jitter around randomly?

Electrons exist as a probability cloud defined by their wave function. If an electron was flying through empty space (e.g. as beta radiation), would it (a.) follow a fixed straight path (which we could confirm every time we measured it), or would it (b.) keep jittering around in a fuzzy probability cloud, slightly varying from the straight path every time we measure it?

(b.) seems impossible because newton's first law: without any external force, how can the electron jitter around? But (a.) seems wrong too, because an electron shouldn't stay "fixed", it should occupy some fuzzy probabilistic cloud, right?

Electrons exist as a probability cloud defined by their wave function

This is true, but I suspect you have misinterpeted what it means. It is tempting to think that the electron always exists at some point in space, and the wavefunction is telling us about the probability of finding it. However this is not the case. Instead the electron simultaneously occupies many points in space. It doesn't jitter about inside the probability cloud - it exists everywhere inside the probability cloud at the same time.

More precisely the wavefunction of the electron can be written as a superposition of infinitely many position eigenstates:

$$\Psi_e = \sum_{i=0}^\infty a_i \psi_i$$

where each $$\psi_i$$ is a state with a well defined position. When we measure the position we collapse this superposition to a single position eigenstate and that is what we interpret as the position of the electron.

The collapse that happens on measurement is random but the evolution of the wavefunction with time is deterministic and is described by the Schrodinger equation. Speaking loosely we might say the "cloud" travels in a straight line at constant speed, though in fact this is only an approximation since the momentum is not precisely defined in the same way the position is not precisely defined.

To enlarge slightly on John's answer, let's take the original electron beam line from the SLAC accelerator as an example. It was 10,000 feet long from the emitter to the target. The emitter produced a collimated beam of electrons somewhat less than 1 millimeter across. That beam maintained its sub-millimeter diameter over the subsequent 10,000 feet trip with only minor focusing corrections i.e., not much jitter, if any.

BTW interesting fact- due to relativistic foreshortening, from the beam's point of view that beam line itself was only about 32 inches long.

Here you see a bubble chamber picture of $$K^-$$ particles with energy 4.2 GeV traversing the hydrogen bubble chamber,about 2m length of tracks. For our measurement accuracy, there is no dispersion,but, what are the tracks?

They are tiny bubbles of the hydrogen, created when the particle interacts with the protons "grazing" them and interacting with the hydrogen kicking off a low energy electron. When the interaction is stronger the electron is releases as in the picture , a new path.

So, within the accuracy of the possible measurement, microns in this case, the tracks follow exactly the magnetic field imposed on the chamber, and within errors no dispersion is seen.

The moral is that the wavefunction of a track in vacuum would follow a straight line within our measurement possibilities . Any "fuzziness" can only be detected if there is an interaction and the nature of the wavefunction is tested.