Applied Quantum Mechanics

German Research School for Simulation Sciences and RWTH Aachen University
Lectures Thu 09:30-12:00 and Exercises 13:15-15:45, Lecture Hall, GRS Jülich (first lecture: 17 Oct 2024)

Lectures

  1. Why Quantum Mechanics?
    Particle-waves and Schrödinger equation:
       time-dependent (initial value problem), time-independent (eigenvalue problem)
    Particle in a box (example: ground state energy for L=1nm:
       google (hbar*pi/1 nm)^2/(2*electron mass) in eV)
    reading: Griffiths Sec. 1.1, 1.2, 2.1, 2.2
    slides and exercises
    links to movies etc.:
  2. time-dependent Schrödinger equation
    probability interpretation and continuity equation
    separation of variables and time-independent Schrödinger equation
    initial value problem (Crank-Nicolson vs. eigenstate expansion, Numerical Recipes Sec. 19.2)
    Gaussian wave packets (animation)
    reading: Schwabl Sec. 2.7, 2.3, 2.10.2 or Griffiths Sec. 2.4
    slides and exercises
  3. piece-wise constant potentials
    matching of wave functions
    potential step, tunneling
    finite potential well (example: 20 Å wide, 4 eV deep:
       google x*tan(x*20/2), -x/tan(x*20/2), sqrt((0.262468435*4)-x^2) and read off kn in Å-1
    reading: Schwabl: Sec. 3.2, (3.3), 3.4, Griffiths 2.5
    slides and exercises
  4. linear potentials and numerical solution of the Schrödinger equation
    linear potential and dimensionless units
    Airy functions (NIST Digital Library of Mathematical Functions)
    asymptotics of wave functions
    numerical solution: finite differences and Numerov trick
    stability of integration
    reading: Griffiths Sec 8.3 and Schwabl Sec. 3.6
    slides and exercises
  5. harmonic oscillator
    analytic solution, Hermite polynomials
    algebraic solution, ladder operators
    reading: Griffiths Sec. 2.3 or Schwabl Sec. 3.1
    slides and exercises
  6. formalism of quantum mechanics
    measurement and expectation value
    Dirac notation, inner product, Hilbert space
    linear operators, inverse, unitary, Hermitian
    reading: Griffiths Sec. 3.1-4+6, Schwabl Sec. 8.1-3
    slides and exercises
  7. common eigenfunctions of commuting operators
    simultaneous diagonalization of operators
    uncertainty relations
    reading: Schwabl Sec.4.1/3, Griffiths Sec. 3.5
    slides and exercises
  8. spherical symmetry and angular momentum
    spherical coordinates
    radial Schödinger equation
    angular momentum algebra
    reading: Griffiths Sec. 4.1 or Schwabl Sec. 5.3, 6.1
    slides and exercises
  9. spherical harmonics
    calculating spherical harmonics
    interactive visualization and how it is done: tutorial (zip)
    reading: Schwabl Sec. 5.2/3 or Griffiths Sec. 4.1 and 4.4
    slides and exercises
  10. hydrogen atom
    radial equation
    analytic solution, Laguerre polynomials
    atomic orbitals, periodic table
    self-consistent calculations for many-electron systems
    reading: Griffiths Sec. 4.2 or Schwabl Sec. 6.3/4
    slides and exercises
    interactive page for calculating many-electron atoms
  11. Xmas Lecture: Quantum Computing
    IBM Quantum Experience: Web access to IBM toy quantum computer
    Lecture series on Quantum Information
    slides
  12. perturbation theory
    first and second order, non-degenerate and degenerate
    reading: Griffiths Sec. 6.1/2 or Schwabl Sec. 11.1
    slides and exercises
  13. time-dependent perturbation theory
    first order; harmonic perturbation, Fermi's golden rule
    reading: Griffiths Ch. 9 or Schwabl Sec. 16.3
    slides and exercises
  14. basis sets and tight-binding
    chemical bonds: covalent, polar, ionic
    Born-Oppenheimer approximation
    Hellmann-Feynman theorem
    slides