# Bloch functions in Kronig-Penney model

• joel.martens

#### joel.martens

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?

Not sure what you are saying, but a band is made up of all the k-points in the first Brillouin zone. So you can't say a band is just 1 k-point.

Thats my poor explanation of the problem sorry, i get that the bands are a continuum of states from the k-points in the Brillouin zone. I was asking more about the exact meanings of the two parts of the Bloch function and how they relate to the shapes of the wavefunctions in the bands.

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.