# Bloch functions in Kronig-Penney model

I'm writing a report for a computer lab where we ran simulations of the wavefunction of an electron in an array of square wells as per the Kronig-Penney model and i'm just looking for some verification of my interpretation of Bloch's Theorem as it applies to the solutions of the schrodinger equation in this case.

## Homework Equations

ψ_k (x)=u_k (x)e^ikx , solution to the SE for the periodic potential.

## The Attempt at a Solution

My understanding of it is that the e^ikx is the 'envelope' for the solution and takes the shape of the solution of the SE for an equivalent single well and the u_k(x) is the periodic function that modulates the wavefunction with the same periodicity of the lattice.
So for the lower energy band, is the envelope function the familiar 1/2 wave for all states in the lower band and the 1 wavelength wavfunction the envelope for all the states in the higher band?

Usually you look at the probability density, which is just $$\left|\Psi_{nk}(x)\right|^2$$. So the phase factor out front disappears and you are just left with the periodic charge density $$\left|u_{nk}(x)\right|^2$$. And the shape of that depends on the potential.