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
Famous quotes containing the word introduction:
“My objection to Liberalism is this—that it is the introduction into the practical business of life of the highest kind—namely, politics—of philosophical ideas instead of political principles.”
—Benjamin Disraeli (1804–1881)
“For better or worse, stepparenting is self-conscious parenting. You’re damned if you do, and damned if you don’t.”
—Anonymous Parent. Making It as a Stepparent, by Claire Berman, introduction (1980, repr. 1986)
“We used chamber-pots a good deal.... My mother ... loved to repeat: “When did the queen reign over China?” This whimsical and harmless scatological pun was my first introduction to the wonderful world of verbal transformations, and also a first perception that a joke need not be funny to give pleasure.”
—Angela Carter (1940–1992)