In Pilot wave theory how far does the wave stretch?

Question

The one thing I find difficult to understand is how far does the Pilot Wave extend. Does the Pilot Wave permeate through the entire universe?

Answer 1

The "pilot waves" are nothing more or less than the ordinary position-space wave functions from standard QM. Thus they "extend" in the exact same manner - which could be through a very large region, or confined only to a very small one.

For example, with a hydrogen atom relaxed to its ground state, the "pilot wave" for the electron is the wave described by the usual orbital

$$\psi(r, \theta, \phi) \propto e^{-a_0 r}$$

In this case the electron, perhaps counterintuitively, shows no motion, because this is a "stationary state" with $\nabla \psi = 0$. Instead it hangs at some, unknown before a measuring experiment, fixed position that is almost certainly near the nucleus.

But the situation is identical to standard QM other than that there is now an added element of a hidden well-defined particle position underneath: the wave is very localized near the atom, filling up approximately its observed volume. In exact mathematical terms it is infinite because the exponential does not actually reach zero, but it becomes completely insignificant more than a few Bohr radii $a_0$ from the nucleus.

Answer 2

It does, yes. There are two different views; either there is a single 3N-dimensional pilot wave, or there are N 3-dimensional pilot waves, where N is the number of beables in the Universe. In either case the pilot wave(s) permeate the entire Universe.