German Research School for Simulation Sciences and RWTH Aachen University (RWTHonline)

Lectures Thu 09:30-12:00 and Exercises 13:15-15:45, Lecture Hall, GRS Jülich

Lectures Thu 09:30-12:00 and Exercises 13:15-15:45, Lecture Hall, GRS Jülich

**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.:**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**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*k*in Å_{n}^{-1}

reading: Schwabl: Sec. 3.2, (3.3), 3.4, Griffiths 2.5

slides and exercises**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**harmonic oscillator**

analytic solution, Hermite polynomials

algebraic solution, ladder operators

reading: Griffiths Sec. 2.3 or Schwabl Sec. 3.1

slides and exercises**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**common eigenfunctions of commuting operators**

simultaneous diagonalization of operators

uncertainty relations

reading: Schwabl Sec.4.1/3, Griffiths Sec. 3.5

slides and exercises**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**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**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**Xmas Lecture: Quantum Computing**

IBM Quantum Experience: Web access to IBM toy quantum computer

Lecture series on Quantum Information

slides**perturbation theory**

first and second order, non-degenerate and degenerate

reading: Griffiths Sec. 6.1/2 or Schwabl Sec. 11.1

slides and exercises**time-dependent perturbation theory**

first order; harmonic perturbation, Fermi's golden rule

reading: Griffiths Ch. 9 or Schwabl Sec. 16.3

slides and exercises**basis sets and tight-binding**

chemical bonds: covalent, polar, ionic

Born-Oppenheimer approximation

Hellmann-Feynman theorem

slides

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