How can a single electron in a crystal become excited, given Bloch's theorem?

Question

Suppose I have an infinite crystal. A physically realistic, though non-stationary, state for the crystal is to have one electron excited, and the rest of the electrons not excited.

However, Bloch's theorem states that there is an eigenbasis of periodic states. Any linear combination of periodic states will also be periodic, so a state with just one excited electron cannot be in their span.

It seems like I should be able to describe a single excitation as a state of an infinite crystal, and Schrodinger's equation should tell me how that excitation will spread through the crystal. But Bloch's theorem seems to contradict that. Which of my assumptions is wrong here?

Answer 1

A single electron can easily have a periodic state, regardless of whether it's ground state or excited. But from what you're asking, I assume you wonder how a localized wave packet can arise from Bloch states.

Bloch's theorem states that there is an eigenbasis of periodic states

Not all Bloch states are exactly periodic. In fact, most are not. Each Bloch wavefunction has a periodic component, but it's multiplied by a complex exponential, which for most of the states has a spatial period incommensurable with the lattice constant, thus making the product non-periodic.

Any linear combination of periodic states will also be periodic

This is also not true. Take e.g. $x\mapsto\sin(x)$ and $x\mapsto\sin(x\sqrt2).$ Both of these functions are periodic. But any nontrivial linear combination of them is not.