How are cloud chamber tracks consistent with the uncertainty principle?

Question

I have read about the uncertainty principle. As it applies to electrons, how is it that we can get exact tracks of electrons in cloud chambers? That is to say that how is it that the position is fixed?

Answer 1

In this article electrons seen in a bubble chamber are shown.

The spiral is an electron knocked off from an atom of hydrogen, a bubble chamber is filled with supercooled liquid hydrogen in this case. The accuracy of measuring the tracks is of an order of microns. The momentum of the electron can be found if one knows the magnetic field and the curvature.

The little dots on the straight tracks are electrons that have just managed to be kicked off from the hydrogen, this would give them a minimum momentum of a few keV.

The total system, picture and measurements give a space resolution of 10 to 50 microns.

$$\Delta x \sim 10^{-5}\, {\rm m}$$ $$\Delta p \sim 1\, {\rm keV}/c = 5.344286×10^{-25}\, {\rm kg\cdot m/s}$$ $$\Delta x \cdot \Delta p > \hbar/2$$ with $\hbar=1.054571726(47)\times10^{−34}\, {\rm kg\cdot m^2/s}$ is satisfied macroscopically since the value is $10^{-30}$, four orders of magnitude larger than $\hbar$.

With nanotechnology, one is getting into dimensions commensurate with the size of $\hbar$, but not with bubble chambers or cloud chambers or most particle detectors up to now.

Answer 2

The idea that an individual particle cannot have both a well-defined position and a well-defined momentum is a common misunderstanding of the Heisenberg Uncertainty Principle (HUP). If one subscribes to the notoriously opaque Copenhagen Interpretation, one must eschew all such ontological statements. If one subscribes to the much more illuminating Bohmian Mechanics, then individual particles have well-defined trajectories, as evidenced in a cloud chamber.

The HUP constrains state preparation, and refers to the properties of an ensemble of particles: one cannot create a state where the spread of position and the spread of momenta are simultaneously more constrained than the HUP limit. The HUP has nothing to say about the position or momentum of a single particle.

The track left by a particle passing through a cloud chamber is much wider than the diameter of an atom - sufficiently wide to be seen with the naked eye. The cloud chamber track is therefore not an accurate measurement. However the accuracy a cloud chamber has nothing to do with the HUP, and there is no reason in principle why a particle's position and momentum could not be determined as accurately as one likes.

Useful videos:

A clear introduction to Bohmian mechanics

Heisenberg and trajectories