Quantum Electronics

Main reference:

  • A. F. J. Levi, “Applied Quantum Mechanics”, Second Edition, 2012, 9780521183994

Abstract

Quantum mechanics is the basis for understanding physical phenomena on the atomic and nano-meter scale.  There are numerous applications of quantum mechanics in biology, chemistry and engineering.  Those with significant economic impact include semiconductor transistors, lasers, quantum optics and photonics.  As technology advances, an increasing number of new electronic and opto-electronic devices will operate in ways that can only be understood using quantum mechanics.  Over the next twenty years fundamentally quantum devices such as single-electron memory cells and photonic signal processing systems will become common-place.  The purpose of this course is to cover a few selected applications and to provide a solid foundation in the tools and methods of quantum mechanics.  The intent is that this understanding will enable insight and contributions to future, as yet unknown, applications.

Prerequisites

  • Mathematics:
    • A basic working knowledge of differential calculus, linear algebra, statistics, and geometry.
  •  Computer skills:
    • An ability to program numerical algorithms in C, MATLAB, FORTRAN or similar language and display results in graphical form.
  • Physics background:
    • Should include a basic understanding of Newtonian mechanics, waves, and Maxwell’s equations.
  • Introduction:  Lectures 1 – 5
  • Lecture 1-2
  • REVIEW OF CLASSICAL CONCEPTS
    • The linear and nonlinear oscillator
    • The one-dimensional simple harmonic oscillator
    • Harmonic oscillation of a diatomic molecule
    • The monatomic linear chain
    • The diatomic linear chain
    • Classical electromagnetism
  • Lecture 3        
  • TOWARDS QUANTUM MECHANICS – PARTICLES AND WAVES
    • Diffraction and interference of light
    • Black-body radiation and evidence for quantization of light
    • Photoelectric effect and the photon particle
    • The link between quantization of photons and quantization of other particles
    • Diffraction and interference of electrons
    • When is a particle a wave?
  • Lecture 4-5        
  • WAVE-PARTICLE DUALITY
  • THE SCHRÖDINGER WAVE EQUATION
    • The wave function description of an electron of mass m0 in free-space
    • The electron wave packet and dispersion
    • The Bohr model of the hydrogen atom
    • Calculation of the average radius of an electron orbit in hydrogen
    •  Calculation of energy difference between electron orbits in hydrogen
    • Periodic table of elements
    • Crystal structure
    • Three types of solid classified according to atomic arrangement
    •  Two-dimensional square lattice, cubic lattices in three-dimensions
    •  Electronic properties of semiconductor crystals
    • The semiconductor heterostructure
  •  =========================================================================
  • Using the Schrödinger wave equation: Lectures 6 – 8 
  • Lecture 6-8
    • The effect of discontinuities in the wave function and its derivative
    • Wave function normalization and completeness
    • Inversion symmetry in the potential
    • Particle in a one-dimensional square potential well with infinite barrier energy
    • NUMERICAL SOLUTION OF THE SCHRÖDINGER EQUATION
      • Matrix solution to the discretized Schrödinger equation
      • Nontransmitting boundary conditions. Periodic boundary conditions
    • CURRENT FLOW
      • Current flow in a one-dimensional infinite square potential well
      • Current flow due to a traveling wave
  • DEGENERACY IS A CONSEQUENCE OF SYMMETRY
    •  Bound states in three-dimensions and degeneracy of eigenvalues
    • BOUND STATES OF A SYMMETRIC SQUARE POTENTIAL WELL
    • Symmetric square potential well with finite barrier energy
    • TRANSMISSION AND REFLECTION OF UNBOUND STATES
      • Scattering from a potential step when effective electron mass changes
      • Probability current density for scattering at a step
      •  Impedance matching for unity transmission
    • PARTICLE TUNNELING
      • Electron tunneling limit to reduction in size of CMOS transistors
      • THE NONEQUILIBRIUM ELECTRON TRANSISTOR
  •  =========================================================================
  • Scattering in one-dimension:  The propagation method:  Lectures 9 – 11
  • Lecture 9-11
  • THE PROPAGATION MATRIX METHOD
    • Writing a computer program for the propagation method
    • Time reversal symmetry
    • Current conservation and the propagation matrix
    • The rectangular potential barrier
      • Tunneling
  • RESONANT TUNNELING
    • Localization threshold
      • Multiple potential barriers
  • THE POTENIAL BARRIER IN THE d-FUNCTION LIMIT
  • ENERGY BANDS IN PERIODIC POTENTIALS:  THE KRONIG-PENNY POTENTIAL
    • Bloch’s theorem
    • Propagation matrix in a periodic potential
    • Real and imaginary band structure
  • THE TIGHT BINDING MODEL FOR ELECTRONIC BAND STRUCTURE
    • Nearest neighbor and long-range interactions
    • Crystal momentum and effective electron mass
  •  =========================================================================
  • Related mathematics:  Lectures 12 – 17
  • Lecture 12-13
  • ONE PARTICLE WAVE FUNCTION SPACE
  • PROPERTIES OF LINEAR OPERATORS
    • Hermitian operators
    • Commutator algebra
  • DIRAC NOTATION
  • MEASUREMENT OF REAL NUMBERS
    • Time dependence of expectation values. Indeterminacy in expectation value
    • The generalized indeterminacy relation
  • DENSITY OF STATES
    • Density of states of particle mass m in 3D, 2D, 1D and 0D
    • Quantum conductance
    • Numerically evaluating density of states from a dispersion relation
    • Density of photon states
  •  =========================================================================
  • The harmonic oscillator: 
  • Lecture 14-17
  • THE HARMONIC OSCILLATOR POTENTIAL
  • CREATION AND ANNIHILATION OPERATORS
    • The ground state. Excited states
  • HARMONIC OSCILLATOR WAVE FUNCTIONS
    • Classical turning point
  • TIME DEPENDENCE
    • The superposition operator. Measurement of a superposition state
    • Time dependence in the Heisenberg representation
    • Charged particle in harmonic potential subject to constant electric field
  • ELECTROMAGNETIC FIELDS
    • Laser light     
    • Quantization of an electrical resonator
    • Quantization of lattice vibrations
    • Quantization of mechanical vibrations
  •  =========================================================================
  • Time dependent perturbation theory and the laser diode (parts of this lectures will be taught in Optoelectronics): Lectures 18 – 20
  • Lecture 18-20
  • FIRST-ORDER TIME-DEPENDENT PERTURBATION THEORY
    • Abrupt change in potential
    • Time dependent change in potential
  • CHARGED PARTICLE IN A HARMONIC POTENTIAL
  • FIRST-ORDER TIME-DEPENDENT PERTURBATION
  • FERMI’S GOLDEN RULE
  • IONIZED IMPURITY ELASTIC SCATTERING RATE IN GaAs
    • The coulomb potential. Linear screening of the coulomb potential
    • Correlation effects in position of dopant atoms
    • Calculating the electron mean free path
  • EMISSION OF PHOTONS DUE TO TRANSITIONS BETWEEN ELECTRONIC STATES
    • Density of optical modes in three dimensions
    • Light intensity
    • Background photon energy density at thermal equilibrium
    • Fermi’s golden rule for stimulated optical transitions
    • The Einstein A and B coefficients
    • Occupation factor for photons in thermal equilibrium in a two-level system
    • Derivation of the relationship between spontaneous emission rate and gain
  • THE SEMICONDUCTOR LASER DIODE
    • Spontaneous and stimulated emission. Optical gain in a semiconductor. Optical gain in the presence of electron scattering
  • DESIGNING A LASER CAVITY
    • Resonant optical cavity.  Mirror loss and photon lifetime
    • The Fabry-Perot laser diode. Rate equation models
  • NUMERICAL METHOD OF SOLVING RATE EQUATIONS
    • The Runge-Kutta method. Large-signal transient response. Cavity formation
  • NOISE IN LASER DIODE LIGHT EMISSION
    • Effect of photon and electron number quantization
    • Langevin and semiclassical master equations
  • QUANTUM THEORY OF LASER OPERATION
    • Density matrix
    • Single and multiple quantum dot, saturable absorber
  •  =========================================================================
  • Time independent perturbation theory:  Lecture 21
  • Lecture 21
  • NON-DEGENERATE CASE
    • Hamiltonian subject to perturbation W
    • First-order correction.  Second order correction
    • Harmonic oscillator subject to perturbing potential in x, x2 and x3
  • DEGENERATE CASE
    • Secular equation
    • Two states
    • Perturbation of two-dimensional harmonic oscillator
    • Perturbation of two-dimensional potential with infinite barrier
  •  =========================================================================
  • Angular momentum, the hydrogenic atom, and bonds:  Lectures 22
  • Lecture 21
  • ANGULAR MOMENTUM
    • Classical angular momentum
    • The angular momentum operator
    • Eigenvalues of the angular momentum operators Lz and L2
    • Geometric representation
  • SPHERICAL HARMONICS AND THE HYDROGEN ATOM
    • Spherical coordinates and spherical harmonics
    • The rigid rotator
    • Quantization of the hydrogenic atom
    • Radial and angular probability density
Lecture Notes
 L06.pdf
 L07.pdf
 QM_L01_Introduction.pdf
 QM_L02_Classical mechanicswith notes.pdf
 QM_L02.pdf
 QM_L03_Tward Quantum Mecanicswithnotes.pdf
 QM_L04_Tward Quantum Mecanics.pdf
 QM_L06_Using the Schrödinger wave equationwith notes.pdf
 QM_L08_Schrodinger Wave Equation.pdf
 QM_L09_Electron Propagation1.pdf
 QM_L10_Electron Propagation2.pdf
 QM_L11_Electron Propagation3.pdf
 QM_L12_Eigenstates and Operators1.pdf
 QM_L13_Eigenstates and Operators2.pdf
 QM_L14_The Harmonic Oscillator01.pdf
 QM_L15_The Harmonic Oscillator02.pdf
 QM_L16_The Harmonic Oscillator03.pdf
 QM_L20_Time-Independent Perturbation.pdf
 QM_L23_Time Dependent Pertubation.pdf