4 - Probability of quantum diffusion

Summary

This chapter contains a description of essential concepts and tools which will be used throughout the book. We take ħ = 1.

The average Green function describes the evolution of a plane wave in a disordered medium, but it does not contain information about the evolution of a wave packet. For optically thick media, or for metals, most physical properties are determined not by the average Green function, but rather by the probability P(r, r′, t) that a particle will move from some initial point r to a point r′, or possibly return to its initial point.

In this chapter, starting from the Schrödinger equation (or the Helmholtz equation), we establish a general expression describing the quantum probability for the propagation of a particle, i.e., a wave packet, from one point to another. When this probability is averaged over a random potential we can identify, in the weak disorder limit kl_e ≫ 1, three principal contributions:

• the probability of going from one point to another without any collision;

• the probability of going from one point to another by a classical process of multiple scattering;

• the probability of going from one point to another by a coherent process of multiple scattering.

We will show that the last two processes, called respectively Diffuson and Cooperon, satisfy a diffusion equation in certain limits.

Definition

Starting from the Schrödinger equation, we want to determine the probability of finding a particle of energy ∈₀ at point r₂ at time t, if it was initially at r₁ at t = 0.