Help - Anderson localization

[weiss_tal]

Hello everyone,

I'm a graduate student and I am doing a simple work on 1D Anderson localization. I need to derive the expression for the localization length when the energies are randomly distributed in the region of $$\left[ \frac{W}{2},-\frac{W}{2} \right]$$. I know the localization length in the limit $$W<<V$$, where $$V$$ are the of diagonal elements of the hamiltonian, is $$\frac{W^2}{96V^2}$$. this expression can be derived while using the second order green function treatment. Since quantum mechanics is not my main studies, I didn't understand the derivation from the green function (I have only found it in Thouless book - Ill condensed matter). If someone know how to derive it, It will help me a lot, or at least know a good reference which explains it simply.
forgive me for my bad English.
thanks,
Tal Weiss.

 

[eljose79]

Hi Tal Weiss,

I am a scientist who specializes in quantum mechanics and I would be happy to help you with your work on 1D Anderson localization. The expression you are looking for can be derived using the second order green function treatment, as you mentioned. I will explain the derivation in a simplified manner, but if you need more details, I recommend checking out the book "Introduction to Quantum Mechanics" by David J. Griffiths, which has a clear explanation of this topic in chapter 10.

To begin, let's define the localization length as the characteristic length scale over which the wavefunction decays exponentially. In other words, if we have a wavefunction \psi(x) that describes the probability of finding a particle at position x, the localization length can be defined as the distance over which \psi(x) decreases by a factor of e. Mathematically, we can write this as:

\psi(x) = \psi(0)e^{-x/\xi}

where \xi is the localization length. Now, in the case of 1D Anderson localization, we have a random potential W(x) that varies along the x-axis. This potential can be described by a Hamiltonian, which is a matrix that represents the total energy of the system. In our case, the Hamiltonian will be a matrix of size N x N, where N is the number of lattice sites along the x-axis.

Using the Hamiltonian, we can write down the Schrodinger equation for our system:

H\psi(x) = E\psi(x)

where H is the Hamiltonian, E is the energy of the particle, and \psi(x) is the wavefunction. In the limit W<<V, as you mentioned, we can use the second order green function treatment to solve this equation. This involves writing the wavefunction as a sum of two terms: the unperturbed wavefunction \psi_0(x), which is the wavefunction in the absence of the random potential, and a correction term \psi_1(x), which takes into account the effect of the random potential. Mathematically, we can write this as:

\psi(x) = \psi_0(x) + \psi_1(x)

Substituting this into the Schrodinger equation and keeping only terms up to second order in the random potential, we get:

(H_0 - E)\psi_0(x) + H_1\psi_1(x) =