Introduction
In the original Anderson tight-binding model, the evolution of the wave function ψ on the d-dimensional lattice Zd is given by the Schrödinger equation
where the Hamiltonian H is given by
with Ej random and independent, and interaction V(r) falling off as r-2 at infinity. For example, one may take Ej uniformly distributed in, and
Starting with ψ0 localised at the origin, one is interested in how fast the probability distribution |ψt|2 diffuses. Anderson's analysis shows the following:
- if d is 1 or 2 and W is arbitrary, or if d ≥ 3 and W/ħ is sufficiently large, then the probability distribution remains localized:
- uniformly in t. This phenomenon is called Anderson localization.
- if d ≥ 3 and W/ħ is small,
- where D is the diffusion constant.
Read more about this topic: Anderson Localization
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