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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\hbar \, \partial_t \psi(x,t) = H \psi(x,t), \quad H= -\frac{\hbar^2}{2 m} \frac{d^2}{dx^2} +V(x) }

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H } is the Hamiltonian. For a free particle the potential is absent, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(x) =0} . One first look for separable solution, 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi_k(x,t) =\frac{1}{\sqrt{2 \pi}} e^{i k x} e^{- i E_k t/\hbar},\quad \text{with} \; E_k =\frac{\hbar^2 k^2}{2 m} }

Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(x,t)=\int_{-\infty}^\infty d k c(k) \psi(x,t), \quad \text{with} \; \int_{-\infty}^\infty d k |c(k)|^2 =1}

An example is the Gaussian packet of intial spread Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_k= \left(\frac{2 a^2}{\pi}\right)^{1/4} e^{-a k^2}}

Challenge: Am I really a pro a Gaussian integrals?

• Initial state of the particle: show that at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=0 } , the wave packet and its probability density function are

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(x,t=0) = \frac{e^{-x^2/(4a^2)}}{\sqrt{2 \pi a^2}}, \quad |\psi(x,t=0)|^2=\frac{e^{-x^2/(2a^2))}}{\sqrt{(2 \pi a^2}} }

• Time evolution. Let us define Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v_s=\hbar/(2 m a^2) } show that at time Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t } , the wave packet has the form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \psi(x,t) = \frac{e^{-x^2/(4 a^2 (1+i v_s t))}}{\sqrt{2 \pi a^2(1+ iv_s t)}} }

BCE and 1D localization

BCE and 1D localization

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R } is the resistence of the sample and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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. ]