L-7 - Revision history

Rosso: /* Evolution of a Gaussian wave packet */

2026-03-09T09:50:37Z

Evolution of a Gaussian wave packet

← Older revision		Revision as of 11:50, 9 March 2026
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	<math display="block">		<math display="block">
	v_s=\frac{\hbar}{2ma^2}.		v_s=\frac{\hbar}{2ma}.
	</math>		</math>

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	<math display="block">		<math display="block">
	\psi(x,t)=		\psi(x,t)=
	\frac{e^{-x^2/(4a^2(1+i v_s t))}}		\frac{e^{-x^2/(4a^2(1+i (v_s/a) t))}}
	{[2\pi a^2(1+i v_s t)]^{1/4}}.		{[2\pi a^2(1+i (v_s/a) t)]^{1/4}}.
	</math>		</math>

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	<math display="block">		<math display="block">
	\|\psi(x,t)\|^2=		\|\psi(x,t)\|^2=
	\frac{e^{-x^2/(2a^2(1+v_s^2 t^2))}}		\frac{e^{-x^2/(2a^2(1+(v_s/a)^2 t^2))}}
	{\sqrt{2\pi a^2(1+v_s^2 t^2)}}.		{\sqrt{2\pi a^2(1+(v_s/a)^2 t^2)}}.
	</math>		</math>

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	\left(\int dx\,x^2\|\psi(x,t)\|^2\right)^{1/2}		\left(\int dx\,x^2\|\psi(x,t)\|^2\right)^{1/2}
	=		=
	a\sqrt{1+v_s^2 t^2}.		a\sqrt{1+(v_s/a)^2 t^2}.
	</math>		</math>

Rosso: /* Diffusive transport */

2026-03-09T08:25:48Z

Diffusive transport

← Older revision		Revision as of 10:25, 9 March 2026
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	In most materials weak disorder does not lead to localization but to diffusive transport. A simple microscopic description is provided by the Drude model.		In most materials weak disorder does not lead to localization but to diffusive transport. A simple microscopic description is provided by the Drude model.

	Electrons in a metal do not start from rest. Even in the absence of an electric field they move with a typical velocity of order the ~~'''~~Fermi velocity~~'''~~		Electrons in a metal do not start from rest. Even in the absence of an electric field they move with a typical velocity of order the Fermi velocity

	<math display="block">		<math display="block">

Rosso: /* Diffusive transport */

2026-03-09T08:10:48Z

Diffusive transport

← Older revision		Revision as of 10:10, 9 March 2026
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	As a consequence transport far from the initial position of the particle is '''exponentially suppressed'''.		As a consequence transport far from the initial position of the particle is '''exponentially suppressed'''.
	== Diffusive transport ==		== Diffusive transport ==
			In most materials weak disorder does not lead to localization but to diffusive transport. A simple microscopic description is provided by the Drude model.

	~~In most materials weak disorder does~~ not ~~lead to localization but to diffusive transport~~. ~~A simple microscopic description is provided by~~ the ~~Drude model. In this classical picture electrons~~ move ~~freely between collisions~~ with ~~impurities or lattice defects. These scattering events occur on average after~~ a ~~characteristic time <math>\tau</math>, called~~ the ~~mean free time.~~		Electrons in a metal do not start from rest. Even in the absence of an electric field they move with a typical velocity of order the '''Fermi velocity'''

	~~Under~~ the ~~action of~~ an electric field <math>E</math>~~, an~~ electron of charge <math>-e</math> obeys		<math display="block">
			v_F=\frac{\hbar k_F}{m}.
			</math>

			Their motion is interrupted by scattering events with impurities, defects, or phonons. These collisions occur on average after a characteristic time <math>\tau</math>, called the '''mean free time'''.

			Between two collisions electrons move essentially freely. The typical distance traveled during this time is the mean free path:

			<math display="block">
			\ell = v_F \tau .
			</math>

			The motion therefore has two distinct regimes:

			* On length scales smaller than <math>\ell</math>, electrons propagate ballistically.
			* On length scales much larger than <math>\ell</math>, repeated random scattering events lead to diffusion.

			We now introduce an external electric field <math>E</math>. An electron of charge <math>-e</math> obeys the equation of motion

	<math display="block">		<math display="block">
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	</math>		</math>

	Between two collisions the electron accelerates~~, but collisions randomize~~ its velocity. Averaging over many such events leads to a ~~stationary~~ drift velocity		Between two collisions the electron accelerates under the electric field. After each collision its velocity is randomized again. Averaging over many such scattering events leads to a small systematic drift velocity

	<math display="block">		<math display="block">
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	</math>		</math>

	If the electron density is <math>n</math>, the electric current density is		Note that this drift velocity is extremely small compared with the typical electron velocity, <math>
			v_d \ll v_F </math>. Thus electrons move very rapidly with random directions, while the electric field only produces a tiny bias in this motion. If the electron density is <math>n</math>, the electric current density is

	<math display="block">		<math display="block">
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	</math>		</math>

	This is the microscopic origin of Ohm's law		This provides the microscopic origin of Ohm's law

	<math display="block">		<math display="block">
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	</math>		</math>

	It is important to stress that the Drude model is purely '''classical'''. It ignores the wave nature of electrons and quantum interference effects. A fully quantum derivation of Ohm's law is much more subtle and typically relies on Green functions and diagrammatic methods. ~~Here~~ we will ~~instead~~ focus on the regime where quantum interference becomes ~~crucial~~ and can eventually suppress diffusion altogether.		It is important to stress that the Drude model is purely '''classical'''. It ignores the wave nature of electrons and quantum interference effects. A fully quantum derivation of Ohm's law is much more subtle and typically relies on Green functions and diagrammatic methods.

			In the following we will focus on the regime where quantum interference becomes important and can eventually suppress diffusion altogether, leading to localization.

	== Conductance ==		== Conductance ==

Rosso: /* Superposition principle */

2026-03-08T15:29:03Z

Superposition principle

← Older revision		Revision as of 17:29, 8 March 2026
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	By choosing the coefficients <math>c(k)</math> appropriately one can construct		By choosing the coefficients <math>c(k)</math> appropriately one can construct
	a localized '''wave packet''' describing a particle initially confined in space.		a localized '''wave packet''' describing a particle initially confined in space.


			In many physical situations the Hamiltonian has a mixed spectrum, containing both discrete bound states and continuum states. In this case a general wavefunction can be expressed as a superposition involving contributions from both sets of eigenstates.
			A simple example is a finite potential well: it supports a finite number of bound states with discrete energies, while particles with higher energies belong to the continuum and correspond to scattering states.

	=== Probability current ===		=== Probability current ===

Rosso: /* Eigenstates */

2026-03-08T15:28:36Z

Eigenstates

← Older revision		Revision as of 17:28, 8 March 2026
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	The probability density <math>\|\psi_n(x,t)\|^2</math> is therefore independent of time: eigenstates are stationary states.		The probability density <math>\|\psi_n(x,t)\|^2</math> is therefore independent of time: eigenstates are stationary states.
	Two qualitatively different situations may occur depending on the form of the potential.		Two qualitatively different situations may occur depending on the form of the potential.

	In many physical situations the Hamiltonian has a mixed spectrum, containing both discrete bound states and continuum states. In this case a general wavefunction can be expressed as a superposition involving contributions from both sets of eigenstates.
	A simple example is a finite potential well: it supports a finite number of bound states with discrete energies, while particles with higher energies belong to the continuum and correspond to scattering states.

	=== Discrete and continuous spectra ===		=== Discrete and continuous spectra ===

Rosso at 11:28, 8 March 2026

2026-03-08T11:28:20Z

Show changes

Rosso: /* Conductance and diffusive transport */

2026-03-08T11:19:34Z

Conductance and diffusive transport

← Older revision		Revision as of 13:19, 8 March 2026
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	As a consequence transport far from the initial position of the particle is '''exponentially suppressed'''.		As a consequence transport far from the initial position of the particle is '''exponentially suppressed'''.
	== ~~Conductance and diffusive~~ transport ==		== Diffusive transport ==

	In most materials weak disorder does not lead to localization but to ~~'''~~diffusive transport~~'''~~.		In most materials weak disorder does not lead to localization but to diffusive transport. A simple microscopic description is provided by the Drude model. In this classical picture electrons move freely between collisions with impurities or lattice defects. These scattering events occur on average after a characteristic time <math>\tau</math>, called the mean free time.

	A simple microscopic description is provided by the ~~'''~~Drude model~~'''~~. In this classical picture electrons move freely between collisions with impurities or lattice defects. These scattering events occur on average after a characteristic time <math>\tau</math>, called the ~~'''~~mean free time~~'''~~.

	Under the action of an electric field <math>E</math>, an electron of charge <math>-e</math> obeys		Under the action of an electric field <math>E</math>, an electron of charge <math>-e</math> obeys
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	</math>		</math>

	~~To relate this~~ to the ~~conductance~~ of ~~a sample, consider a wire~~ of ~~length <math>L</math>~~ and ~~cross section~~ <math>S</math>. ~~The total current~~ is		It is important to stress that the Drude model is purely '''classical'''. It ignores the wave nature of electrons and quantum interference effects. A fully quantum derivation of Ohm's law is much more subtle and typically relies on Green functions and diagrammatic methods. Here we will instead focus on the regime where quantum interference becomes crucial and can eventually suppress diffusion altogether.

			==Conductance==



			The conductance <math> G</math> is the inverse of the resistance. We will determine is scaling, it will be the starting point for the scaling theory of localization.

			* Diffusive Transport: Consider a wire of length <math>L</math> and cross section <math>S</math>. The total current is
	<math display="block">		<math display="block">
	I=jS,		I=jS,
	</math>		</math>

	while the voltage drop is		while the voltage drop is

	<math display="block">		<math display="block">
	V=EL.		V=EL.
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	I=\sigma \frac{S}{L} V.		I=\sigma \frac{S}{L} V.
	</math>		</math>

	Using <math>G=I/V</math>, one obtains		Using <math>G=I/V</math>, one obtains

	<math display="block">		<math display="block">
	G=\sigma \frac{S}{L}.		G=\sigma \frac{S}{L}.
	</math>		</math>

	~~Equivalently,~~

	~~<math display="block">~~
	~~R=\frac{1}{G}=\rho\frac{L}{S},~~
	~~\qquad~~
	~~\rho=\frac{1}{\sigma}.~~
	~~</math>~~

	For a sample of linear size <math>L</math> in spatial dimension <math>d</math>, the cross section scales as <math>S\sim L^{d-1}</math>. Hence		For a sample of linear size <math>L</math> in spatial dimension <math>d</math>, the cross section scales as <math>S\sim L^{d-1}</math>. Hence
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	</math>		</math>

	This scaling is the characteristic signature of diffusive transport~~, and it will be the starting point for the scaling theory of localization.~~		This scaling is the characteristic signature of diffusive transport.

	It is important to stress that the Drude model is purely '''classical'''. It ignores the wave nature of electrons and quantum interference effects. A fully quantum derivation of Ohm's law is much more subtle and typically relies on Green functions and diagrammatic methods. Here we will instead focus on the regime where quantum interference becomes crucial and can eventually suppress diffusion altogether.

	== Conductance in the localized regime ==		== Conductance in the localized regime ==

Rosso: /* Conductance and diffusive transport */

2026-03-08T11:11:44Z

Conductance and diffusive transport

← Older revision		Revision as of 13:11, 8 March 2026
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	In most materials weak disorder does not lead to localization but to '''diffusive transport'''.		In most materials weak disorder does not lead to localization but to '''diffusive transport'''.

	In the Drude picture electrons ~~scatter randomly on~~ impurities. ~~After many~~ scattering events ~~their motion becomes~~ a ~~random walk. Beyond~~ the mean free ~~path the motion is therefore diffusive~~.		A simple microscopic description is provided by the '''Drude model'''. In this classical picture electrons move freely between collisions with impurities or lattice defects. These scattering events occur on average after a characteristic time <math>\tau</math>, called the '''mean free time'''.

	~~In this regime Ohm's laws hold.~~		Under the action of an electric field <math>E</math>, an electron of charge <math>-e</math> obeys

	* First law		<math display="block">
			m\frac{dv}{dt}=-eE.
			</math>

			Between two collisions the electron accelerates, but collisions randomize its velocity. Averaging over many such events leads to a stationary drift velocity

	<math display="block">		<math display="block">
	\frac{V}{I}=R,		v_d=-\frac{eE\tau}{m}.
			</math>

			If the electron density is <math>n</math>, the electric current density is

			<math display="block">
			j=-ne\,v_d.
			</math>

			Substituting the drift velocity gives

			<math display="block">
			j=\frac{ne^2\tau}{m}E.
			</math>

			This is the microscopic origin of Ohm's law

			<math display="block">
			j=\sigma E,
	\qquad		\qquad
	\frac{I}{V}=G.		\sigma=\frac{ne^2\tau}{m}.
	</math>		</math>

	* Second law		To relate this to the conductance of a sample, consider a wire of length <math>L</math> and cross section <math>S</math>. The total current is

	<math display="block">		<math display="block">
	R=\rho\frac{L}{S},		I=jS,
			</math>

			while the voltage drop is

			<math display="block">
			V=EL.
			</math>

			Therefore

			<math display="block">
			I=\sigma \frac{S}{L} V.
			</math>

			Using <math>G=I/V</math>, one obtains

			<math display="block">
			G=\sigma \frac{S}{L}.
			</math>

			Equivalently,

			<math display="block">
			R=\frac{1}{G}=\rho\frac{L}{S},
	\qquad		\qquad
	G=~~\sigma~~\frac{S}{L}.		\rho=\frac{1}{\sigma}.
	</math>		</math>

	~~Here <math>\rho</math> and <math>\sigma</math> are the resistivity and conductivity of the material.~~		For a sample of linear size <math>L</math> in spatial dimension <math>d</math>, the cross section scales as <math>S\sim L^{d-1}</math>. Hence

	For a sample of linear size <math>L</math> in spatial dimension <math>d</math>, the cross section scales as <math>S\sim L^{d-1}</math>. ~~Therefore~~

	<math display="block">		<math display="block">
	G \sim \sigma L^{d-2}.		G(L)\sim \sigma L^{d-2}.
	</math>		</math>

	This scaling ~~behavior~~ is the characteristic signature of diffusive transport.		This scaling is the characteristic signature of diffusive transport, and it will be the starting point for the scaling theory of localization.

			It is important to stress that the Drude model is purely '''classical'''. It ignores the wave nature of electrons and quantum interference effects. A fully quantum derivation of Ohm's law is much more subtle and typically relies on Green functions and diagrammatic methods. Here we will instead focus on the regime where quantum interference becomes crucial and can eventually suppress diffusion altogether.

	== Conductance in the localized regime ==		== Conductance in the localized regime ==

Rosso: /* Short recap: wavefunctions and eigenstates */

2026-03-08T10:25:30Z

Short recap: wavefunctions and eigenstates

← Older revision		Revision as of 12:25, 8 March 2026
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	</math>		</math>

	where <math>H</math> is the Hamiltonian of the system.		where <math>H</math> is the Hamiltonian of the system. For a particle moving in one dimension in a potential <math>V(x)</math>, the Hamiltonian reads

			<math display="block">
			H = -\frac{\hbar^2}{2m}\partial_x^2 + V(x).
			</math>

	=== Eigenstates ===		=== Eigenstates ===
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	</math>		</math>

	The probability density <math>\|\psi_n(x,t)\|^2</math> is therefore independent of time: eigenstates are ~~'''~~stationary states~~'''~~.		The probability density <math>\|\psi_n(x,t)\|^2</math> is therefore independent of time: eigenstates are stationary states.
	Two qualitatively different situations may occur depending on the form of the potential.		Two qualitatively different situations may occur depending on the form of the potential.

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	and the particle remains localized in space. These states are called		and the particle remains localized in space. These states are called
	~~'''~~bound states~~'''~~.		bound states.

	* '''Continuous spectrum (continuum states)'''		* '''Continuous spectrum (continuum states)'''
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	E=\frac{\hbar^2 k^2}{2m}.		E=\frac{\hbar^2 k^2}{2m}.
	</math>		</math>
			These states are not normalizable in the usual sense. They are instead
			normalized using Dirac delta functions,

	~~These states are not normalizable in the usual sense~~. They ~~are instead~~		<math display="block">
	~~normalized using Dirac delta functions and serve as~~ a basis ~~for~~		\int dx\, \psi_k^*(x)\psi_{k'}(x)=\delta(k-k').
	~~constructing physical~~ wave packets.		</math>

			They form a continuous basis that can be used to construct physical
			wave packets.

	=== Superposition principle ===		=== Superposition principle ===
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	The Schrödinger equation is linear. As a consequence, any linear combination		The Schrödinger equation is linear. As a consequence, any linear combination
	of solutions is again a solution. This property is known as the		of solutions is again a solution. This property is known as the
	~~'''~~superposition principle~~'''~~.		superposition principle.

	If the spectrum is discrete, an arbitrary wavefunction can be expanded		If the spectrum is discrete, an arbitrary wavefunction can be expanded
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	are the reflection and transmission probabilities.		are the reflection and transmission probabilities.


	== Free particles and ballistic behaviour ==		== Free particles and ballistic behaviour ==

Rosso at 10:17, 8 March 2026

2026-03-08T10:17:01Z

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