Schulphysikalische Vertiefung

Schulphysikalische Vertiefung zur Quantenmechanik

If you still believe that waves and particles are opposites that cannot coexist in a single entity;
If quantum physics seems counterintuitive to you and far removed from classical mechanics and everyday life;
If terms like "uncertainty principle" and "Schrödinger equation" seem mysterious and hard to grasp...
...then, regardless of whether or not you have studied quantum physics before, I invite you to this course.
Here, we will discover that there is no contradiction between waves and particles and that classical mechanics is simply a limiting case of quantum mechanics; moreover, we will build a solid bridge between the two.

Course Program

1. Introduction and basics: How do we comprehend the world around us? Basic concepts for understanding the world and creating common knowledge. Fundamental problems in the study of quantum physics. Beyond everyday experience and the lack of intuition.
2. Oscillations and waves: Periodic motion of a particle between two absolutely elastic walls (Particle in a box - classical problem). The concepts: a phase plane, a phase trajectory in phase space, action, probability and probability density function. Uniform circular motion. Travelling waves. Classical wave equation. Standing waves and the relationship between discreteness and continuity. Interference and superposition of coherent and incoherent waves. Fourier series. Fourier transform. Wave packet. Complex numbers and representation of waves via complex functions. Euler’s formula.
3. Basics of quantum physics: Wave-particle duality of light and matter. The Plank-Einstein relation and the de Broglie waves. The Bohr model of the atom. Electron energy levels. Spectral lines and Balmer-Rydberg formula.
4. The Schrödinger wave equation: Derivation - plausibility argument and  time-independent Schrödinger equation. А wave function as the probability amplitude (Copenhagen interpretation).
5. The Heisenberg uncertainty principle:  A wave packet approach (a free particle). Gaussian wave packet. Time evolution of the wave packet and dispersion relation. Phase and group velocities.
6. Step potential: Analytical solution of the Schrödinger equation. The probability current and connection with classical mechanics. Transmission and reflection coefficients. Optical analogy: the boundary of two media with different refractive indices. Scattering at a negative step. Smooth step - а transition to the classical limit. Scattering of the wave packet.
7. Particle in a box:  Potential well in 1D with infinitely high walls. Stationary states, energy quantization, discreteness in dispersion dependence, quantization of action. Comparison with standing waves propagating in a string. Matching of classical and quantum solutions. The classical limit. Niels Bohr correspondence principle.
8. Quantum tunnelling: Rectangular potential barrier. Transmission and reflection. The alpha-decay and alpha-particle paradox (estimations within the framework of a simplified model). Scanning tunnelling microscopy (STM). Applications in electronics.
9. Scattering at a finite potential barrier: Rectangular potential barrier. Analytical solution. Transmission and reflection. Resonances in the transmission probability. Optical analogy. Classical limit. Scattering of the wave packet.
10. Double-barrier potential: Resonant-tunneling diode (RTD). Peculiarities in the reflection and transmission coefficients. Conditions for observing resonances. Conductivity (resistance) as an optical problem. Transmission coefficient and current-voltage characteristic.
11. Particle in a quantum finite rectangular well:  Estimation of the number of stationary levels. Qualitative consideration and analytical solution. Numerical construction of a wave packet and reproduction of a classical problem “particle in a box”. Scattering at a finite well.
12. Quantum harmonic oscillator: Аnalytical solution and semi-classical approximation. Time evolution of a wave packet composed of eigenfunctions. Comparison with classic harmonic oscillator.
13. Double-well potential: Symmetrical and asymmetrical cases. Covalent and ionic bonding. Energy level splitting. Dependence of the energy level splitting on the distance between the wells. Narrow potential barrier inside the well. Effect on symmetric and antisymmetric wave functions. Pairwise grouping of energy levels. Time evolution of the wave packet in the double-well potential.
14. From atom to crystal: Particle in a one-dimensional lattice, the Kronig–Penney model, a formation of energy bands and band gaps, the dispersion relation and effective mass. Impurity in a one-dimensional crystal. Disordered 1D lattice. Metal-insulator transition. Localization and delocalization of wave functions.

Schulphysikalische Vertiefung zur Festkörperphysik 

(Introduction to Condensed Matter Physics)

Look around you, and you will notice that everything you see - any object upon which your gaze rests - is a complex system made up of a vast number of components that interact with one another. Physicists call such systems "condensed matter." If you are curious about how physics deals with such systems, I invite you to this course.

In this course, we will explore the fundamental physical concepts underlying the description of macroscopic systems composed of large numbers of particles. Particular emphasis is placed on how particle properties transform when particles form a large collective, how system properties change with dimensionality, and how microscopic disorder influences macroscopic properties.

1. Introduction and basics: How do we comprehend the world around us? Basic concepts for understanding the world and creating common knowledge. The place of solid state physics in the system of knowledge about the Universe and the world around us. The main approaches to the description of multicomponent and complex systems: from reductionism to anti-reductionism. The role of solid state physics in everyday life.
2. Crystal structure (Periodicity in space): Short-range and long-range order. Periodic array of atoms. Translational symmetry. Crystal lattice. Unit cell and primitive lattice cell. Wigner-Seitz cell (Voronoi diagram). Fundamental types of lattices and crystallographic point groups. Bravais lattices. Close-packing of equal spheres. Examples of crystal structures. Miller indices of planes and directions.
3. Crystal binding and defects: How does a crystal form and what holds a crystal together? Crystal binding. Van der Waals-London interaction, repulsive interaction. Defects in crystals. Entropy.
4. Oscillations and waves (Periodicity in time and space): Periodic motion of a particle between two absolutely elastic walls (Particle in a box - classical problem). The concepts: a phase plane, a phase trajectory in phase space, action, probability and probability density function. Periodic travelling waves. Interference and superposition of waves. Wave packet. Standing waves and the relationship between discreteness and continuity. Complex numbers and representation of waves via complex functions. Euler’s formula.
5. Basics of quantum physics: Wave-particle duality of light and matter. The Plank-Einstein relation and the de Broglie waves. The Bohr model of the atom. The Schrödinger (matter) wave equation (Derivation) - plausibility argument and  time-independent Schrödinger equation. The Heisenberg uncertainty principle - a wave packet approach. Particle in a box - quantum consideration: stationary states, energy quantization, discreteness in dispersion dependence, quantization of action. The relationship between classical and quantum mechanics. The classical limit, Niels Bohr correspondence principle. Quantum tunnelling. From atom to crystal: Particle in a one-dimensional lattice, the Kronig–Penney model, a formation of energy bands and band gaps, the dispersion relation and effective mass.
6. Quantum statistics: Maxwell–Boltzmann statistics.Indistinguishable particles. Symmetric and antisymmetric states. Fermions and bosons. Spin. Pauli exclusion principle. Bose–Einstein statistics and Fermi–Dirac statistics and under what conditions they reproduce classical Maxwell–Boltzmann statistics. Quantum correlation and quantum degeneration. Distribution of particles over energy bands in solids. Metals, semiconductors and dielectrics from the point of view of band theory and quantum statistics.
7. Crystal vibrations: Equilibrium lattice and lattice dynamics. Harmonic oscillator. From local vibrations of atoms to collective vibrational waves propagating through the lattice. One-dimensional linear chain model. Dispersion relation. Phonon concept. Quasiparticle concept. Diatomic linear chain. Acoustic and optical phonons.
8. VIII.Free electron gas in semiconductors: Electrical conductivity and Ohm’s law. The history of the development of concepts regarding semiconductors as a distinct class of conductors. Thermal generation and generation of carriers by light in semiconductors. Comparison of the model of electron bonds with the band theory. Effective mass. Electrons and holes as quasiparticles. Doping of semiconductors. Diffusion, random walk problem. Ergodicity. Drift in an electric field, mobility. Classical Drude model. Einstein - Sutherland relation.
9. Free electron Fermi gas in three dimensions (metals): Carrier concentration, density of states, conductivity. Einstein-Kubo relation for degenerate electron gas, connection with Drude formula for conductivity. Fermi liquid theory.
10. Low-dimensional electronic systems and nanostructures: quantum wells, quantum wires, quantum dots. Subbands, carrier concentration, density of states, and conductivity of two- and one-dimensional electron Fermi gas. Heterostructures and quantum devices. Quantum point contact. Conductance quantization.