One-dimensional periodic case

From Academic Kids

In quantum mechanics, when talking about solid materials, the discussion is mainly about crystals - periodic lattices. Here we will discuss a 1-dimensional lattice of positive ions. The one-dimensional particle lattice is a simplified version of the 3D infinite potential barrier problem (particle in a box). While the "particle in a box" assumes the potential inside the box is 0, that is not the case when looking inside a solid material.

Assuming the spacing between two ions is a, the potential in the lattice will look something like this:

Missing image
Potential-actual.PNG
Image:Potential-actual.PNG

The mathematical representation of the potential is a periodic function with a period a.

According to Bloch's theorem, the wavefunction solution of Schrödinger equation when the potential is periodic, can be written as:

<math> \psi (x) = e^{ikx} u(x) <math>

where u(x) is a periodic function which satisfies

u(x+a)=u(x) and u'(x+a)=u'(x)

When nearing the edges of the lattice, there are problems with the boundary condition. Therefore, we can represent the ion lattice as a ring. If L, the length of the lattice, is such that L>>a, then the number of ions in the lattice is so large, that when considering one ion, its environment is almost constant, and the wavefunction of the electron is unchanged. So now, instead of two boundary conditions we get one circular boundary condition

<math> \psi (0)=\psi (L) <math>
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