Bloch functions in Kronig-Penney model

[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 Schrödinger 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?

 

[nickjer]

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.

 

[joel.martens]

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.

 

[nickjer]

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.