L-7

Goal : This is the first lecture about the localization. Localization is a wave phenomenon induced by disorder.

The Gaussian packet of free particles: the ballistic behaviour

The Schrodinger equation govern the evolution of the quantum state of a particle in 1D:

$i\hbar \,\partial _{t}\psi (x,t)=H\psi (x,t),\quad H=-{\frac {\hbar ^{2}}{2m}}{\frac {d^{2}}{dx^{2}}}+V(x)$

Here $H$ is the Hamiltonian. For a free particle the potential is absent, $V(x)=0$. One first look for separable solutions, or eigenstates or stationary solutions. These solutions are stationary because if the particle is in an eigenstate, its physical properties are time independent. For a free particle, the separable solutions has the form

$\psi _{k}(x,t)={\frac {1}{\sqrt {2\pi }}}e^{ikx}e^{-iE_{k}t/\hbar },\quad {\text{with}}\;E_{k}={\frac {\hbar ^{2}k^{2}}{2m}}$

Here $k$ is a real number. These solutions are delocalized on the entire real axis and they cannot be normalized.

The physical states of the particle can be decomposed on the eigenstate of the Hamiltonian. Indeed, For the superposition principle, any linear combination of separable solution is also a solution of the Schrodinger equation. However this superposed solution are not separable and the particle's properties will evolve in time. Hence, we can costruct a localized wave packet, with the correct normalization:

$\psi (x,t)=\int _{-\infty }^{\infty }dkc(k)\psi (x,t),\quad {\text{with}}\;\int _{-\infty }^{\infty }dk|c(k)|^{2}=1$

An example is the Gaussian packet of intial spread $a$:

$c_{k}=\left({\frac {2a^{2}}{\pi }}\right)^{1/4}e^{-ak^{2}}$

Challenge: Am I really a pro with Gaussian integrals?

• Initial state . Show that at time $t=0$, the wave packet and its probability density function are

$\psi (x,t=0)={\frac {e^{-x^{2}/(4a^{2})}}{(2\pi a^{2})^{1/4}}},\quad |\psi (x,t=0)|^{2}={\frac {e^{-x^{2}/(2a^{2}))}}{\sqrt {2\pi a^{2}}}}$

• Time evolution . We define the spreading speed $v_{s}=\hbar /(2ma^{2})$. Show that the time evolution of the Gaussian wave packet has the form:

$\psi (x,t)={\frac {e^{-x^{2}/(4a^{2}(1+iv_{s}t))}}{[2\pi a^{2}(1+iv_{s}t)]^{1/4}}}$

• Ballistic Spreading. Show that the Probability density function of the position is

$|\psi (x,t)|^{2}={\frac {e^{-x^{2}/(4a^{2}(1+v_{s}^{2}t^{2}))}}{[2\pi a^{2}(1+v_{s}^{2}t^{2})]^{1/4}}}$

Hence, the particle spreads on a distance

${\sqrt {\langle x^{2}\rangle }}\equiv \left(\int _{-\infty }^{\infty }dxx^{2}|\psi (x,t)|^{2}\right)^{1/2}=a{\sqrt {1+v_{s}^{2}t^{2}}}$

Localization experiment

BCE coondensate confined with an harmonic trap in a 1D disordered geometry. Billy et al, Nature 2008

In 1958 P. Anderson proposed that disorder potential can quench the dynamics of non ineracting particles. This phenomenon, called Anderson localization, has been observed in numerical simulations, and recenlty in experiments. A Gaussian packet of a BCE can be constructed with an harmonic trap. When the trap is removed particles start to spread but remain localized in space.

To explain how this is possible we need to solve the eigenstate problem of the hamiltonian with a disorder potential $V(x)$, e.g. a Gaussian white noise. We will study this problem in different dimensions and see that eigenstates can be localized in space.

Semilog plot of the particle concentration. Billy et al, Nature 2008

In this case an eigenstate $\psi _{k}(x,t)$ is concentrated around some position ${\overline {x}}$:

The exponential decay of the tails is the fingerprint of a finite localization lenght

The conductance and the diffusive behaviour

Ohm's laws characterize electric transport of (good or bad) conductors:

• First law:

${\frac {V}{I}}=R,\quad {\text{or}}\quad {\frac {I}{V}}=G$

Here $R$ is the resistence of the sample and $G$ is its conductance.\

• Second law

$R=\rho {\frac {L}{S}}\sim \rho L^{2-d},\quad {\text{or}}\quad G=\sigma {\frac {S}{L}}\sim \sigma L^{d-2}$

Here $\rho ,\sigma$ are the resistivity and the conductivity. These are material properties, independent of the geometry of the sample These phenomenological laws are a macroscpic manifestation of the diffusive motion of the electrons. From the Drude model we know that disorder is the crucial ingredient to justify the presence of diffusion. ]