Technical prelude for Penning-trap enthusiasts

I could try to weasel out and say "A Penning-trap is not a real trap! The magnetron mode is unstable and the particle will escape at some point!". But in theory and in practice you you can make the trap arbitrarily stable with the help of suitable RF-shielding.

A typical Penning trap (6 Tesla field, 4 Kelvin temperature, 20 MHz axial frequency) confines an otherwise free electron to something like 20 nanometers in the radial plane and 200 micrometers in the axial direction, assuming we are using a classical picture of the electron's motion, and assuming that the cyclotron mode and axial mode are thermalized to 4 Kelvin (and that the magnetron mode is centered as much as possible through sideband cooling with the cyclotron mode). Describing the electron in a quantum picture leads to similar numbers, but we have to concede that "thermal amplitude" is not a good way to think about the amplitude, since the quantum numbers of the cyclotron mode and the magnetron mode are both only ~2.

These values are, of course, arbitrary. The thermal amplitudes depend on the trap-frequencies, which in turn can be chosen somewhat freely by using suitably strong B- and E-fields. In principle, the confinement can be made aribtrarily small by scaling up the B- and E-fields to ridiculous values. At some point, this leads to QED-effects that make everything quite complicated, but I won't weasel out using some obscure QED-argument either, since the Heisenberg uncertainty principle should hold irregardless.

The argument

Let's imagine a trap with ridicilously strong trapping-fields and an extremely good confinement. Let's assume the electron is in the motional ground-states in all three trap-modes (axial, cyclotron, magnetron). The ground-state energy, however, depends on the frequency of the individual modes, and (assuming the ridicilously strong trapping fields), the ground-state energy is ridicilously large. So you can imagine the electron bouncing up and down, left and right, front and back inside the trap, with high velocity, but never going very far before it bounces back towards the trap center.

If the trap is switched of instantly at some point in time, we can be fairly certain that, at that moment, the electron's position is very, very close to the trap center. But at the same time, the electron will likely start to fly off at a high speed into a random direction. In other words, at the time we switch off the trap, we know the electron's position very well, but its momentum is uncertain -- this is the gist of the Heisenberg uncertainty principle.

And if we never switch off the trap? Then we know the position of the electron very well, and we can easily predict that it will be in exactly the same place next year. Does this mean that we know the momentum of the electron really well? No, it still bounces around the trap, and we know very little about the instaneous momentum of the electron. But if the electron is still in the same place next year, we can infer something about the momentum of the electron + Penning-trap system. But since superconducting magnets and so on tend to be large and heavy and not well isolated from the observer, quamtum effects are heavily suppressed.