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 schrodinger equation in this case.

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


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?

 

[BigJDubs]

Yes, that is correct. Bloch's Theorem states that the solutions of the Schrödinger equation for a periodic potential have the form ψ_k (x)=u_k (x)e^ikx, where u_k(x) is the periodic function with the same periodicity as the lattice and e^ikx is the 'envelope' wavefunction which takes the shape of the solution of the Schrödinger equation for an equivalent single well. For the lower energy band, the envelope wavefunction is the familiar 1/2 wave for all states in the lower band, and the 1 wavelength wavefunction is the envelope for all states in the higher band.

 

[charng]

I can confirm that your understanding of Bloch's Theorem in the context of the Kronig-Penney model is correct. Bloch's Theorem states that the solutions to the Schrodinger equation for a periodic potential can be expressed as a product of a periodic function and a plane wave with a wave vector k. In this case, the periodic function is represented by u_k(x) and the plane wave is represented by e^ikx.

The envelope function, e^ikx, takes the shape of the solution for a single well potential and is responsible for the overall behavior of the wavefunction. The periodic function, u_k(x), modulates the wavefunction with the same periodicity as the lattice potential.

For the lower energy band, the envelope function will indeed take the shape of a 1/2 wave for all states in that band. And for the higher energy band, the envelope function will take the shape of a 1 wavelength wave for all states in that band.

I hope this helps to verify your interpretation of Bloch's Theorem in the Kronig-Penney model. Keep up the good work in your computer lab simulations!