• Support PF! Buy your school textbooks, materials and every day products Here!

Bloch functions in Kronig-Penney model

  • #1
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?
 

Answers and Replies

  • #2
674
2
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.
 
  • #3
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.
 
  • #4
674
2
Usually you look at the probability density, which is just [tex]\left|\Psi_{nk}(x)\right|^2[/tex]. So the phase factor out front disappears and you are just left with the periodic charge density [tex]\left|u_{nk}(x)\right|^2[/tex]. And the shape of that depends on the potential.
 

Related Threads for: Bloch functions in Kronig-Penney model

  • Last Post
Replies
1
Views
4K
  • Last Post
Replies
5
Views
10K
  • Last Post
Replies
1
Views
6K
Replies
1
Views
2K
Replies
2
Views
2K
Replies
1
Views
579
Replies
2
Views
15K
Replies
0
Views
2K
Top