Does uncertainty principle apply to holes/gaps in matter?

Question

Consider a hole in a semiconductor, or a missing atom in a crystal. Are their positions and momentum related by uncertainty principle? These factors would of course be determined by its surrounding matter, but it's not clear to me whether the sum of the uncertainties around it also follow HUP.

Answer 1

Holes in semiconductors move as electrons do, but act as if they were positively charged particles. The question really is, are quasiparticles, with an electron "hole" being one type, subject to the uncertainty principle?

Even though quasiparticles are not what we would class as "real" particles, they are a physical phenomena that do indeed behave like real$^1$ particles, meaning we can indeed assign to them the quantum mechanical values that we give to real particles, such as spin, charge etc., and uncertainty.

The uncertainty principle $$\Delta x\Delta p\ge \frac{\hbar}{2}$$ holds for such particles, and the same can be said for other quasiparticles. The uncertainty principle is a general relation that applies to all particles, whether they are quasiparticles or not.

$^1$ They are an example of emergent phenomena that occurs when a system of atoms or molecules behave in such a way, as though there is a particle moving through the system.

Answer 2

Yes, the uncertainty principle does apply to holes/gaps in matter.

The uncertainty principle between position and momentum is very general. It follows from the fact that the observable we call momentum is represented by the operator that generates translations in space. As a result, the momentum operator has the same kind of relationship with any observable that says where something is located in space — whether it's a localized particle, or a localized hole, or any other kind of localized structure. If the location of a structure is observable at all, then it satisfies the uncertainty principle.

Answer 3

Holes are not empty space
"Free electrons" and holes in a semiconductor are quasiparticles, that is excitations of a multi-electron system (which are mathematically identified as poles in the Green's function). These excited states can be characterized by momentum and position, which will obey the Uncertainty Principle. In other words, the Uncertainty Principle holds, because we are still talking about electrons, and "hole" does not really mean empty space.

In the simplest case, if we neglect the Coulomb interaction, the ground state of a semiconductor, $|0\rangle$ is a state with the valence band completely filled and the conduction band empty: $$ v_{\mathbf{k}}^\dagger|0\rangle=0, c_{\mathbf{k}}|0\rangle=0, $$ where $v_{\mathbf{k}}$ and $c_{\mathbf{k}}$ are the electron annihilation operators for valence and band electrons.

A state obtained by removing an electron from the valence band is $v_{\mathbf{k}}|0\rangle$, and the hole transformation is merely renaming of operators: $$h_{\mathbf{k}}=v_{-\mathbf{k}}^\dagger, h_{\mathbf{k}}^\dagger=v_{-\mathbf{k}},$$ so that the state with one electron removed is a state with one hole created: $$v_{\mathbf{k}}|0\rangle=h_{-\mathbf{k}}^\dagger|0\rangle.$$

Vacancies are atoms/ions, but atoms/ions are not always vacancies
One has to be more careful with atoms. One can conjecture a situation that is similar to the one described above, e.g., in atomic Bose-Einstein condensates, where the removal of an atom corresponds to an excited quantum state. In a semiconductor one can also think about the lattice around the removed atom, which is in a single quantum state (at low temperatures) and the state with an atom absent at a specific position can be attributed momentum and position.

However, in this cases one will talk about a vacancy rather than an atom. Calling this an atom would be misleading, since the term has meaning outside of a lattice or a coherent BEC state: we may have an atom in vacuum, or a gas with many atoms, and absence of one of them cannot be attributed neither momentum, nor position.