Teaching
Prof. Dr. Martin Holthaus
Institute of Physics (» Postal address)
Lectures
Teaching
Lectures (in German) regularly given by Martin Holthaus include:
Einführung in die theoretische Physik
Introduction to Theoretical Physics
This course covers basic tools mandatory for every aspiring young physicist, namely, elements of calculus, ordinary differential equations, Newtonian mechanics, and vector calculus, culminating in electro- and magnetostatics. This is a lot of stuff indeed, but it is badly needed!
The figure depicts an archetypal example treated in this lecture series: Motion of a classical particle in a quadratic potential in both position space (above) and momentum space (below).
Teilchen und Felder I
Particles and Fields I
This lecture is divided into two parts. First we get aquainted with both the Lagrangian and the Hamiltonian formulation of classical mechanics. Then we discuss Maxwell's equations and derive the conserved quantities furnished by electromagnetic fields, before turning to the relativistic formulation of electrodynamics. Glueing these parts together, Maxwell's equations are obtained as Euler-Lagrange equations of a suitable action functional.
The figure illustrates an important topic discussed in this course, the principle of least action: From all paths between the points q1 and q2 in configuration space, a systems takes the one which renders its action functional stationary.
Teilchen und Felder II
Particles and Fields II
Particles and Fields II: Here we first take up Hamiltonian mechanics, adopting a geometrical viewpoint. After discussing canonical transformations, integrable systems, and their action-angle variables, we explore what happens when such an integrable system is perturbed, familiarizing ourselves with the celebrated KAM-theorem and chaotic dynamics.
Then we return to electrodynamics, and fortify our knowledge on fields, potentials, and gauges. Equipped with these tools, we compute the fields generated by moving point charges, and work our way through various types of radiation problems.
The figure, taken from the lecture notes of this course, roughly visualizes what happens when an integrable Hamiltonian system is perturbed: From all invariant tori in the phase space of an integrable system, indicated left by their intersection with a Poincaré plane, only the sufficiently irrational ones survive under a small perturbation, whereas the rational tori are destroyed, giving way to chains of elliptic and hyperbolic fixed points, accompanied by so-called homoclinic tangles.
Quantenmechanik
Quantum Mechanics
In this key lecture we follow the traditional approach, coping with wave functions and Schrödinger's equation (and, of course, Heisenberg's famous uncertainty relation), one-dimensional example problems (including the omnipresent harmonic oscillator), angular momentum and centrally symmetric eigenvalue problems (not avoiding the all-important Hydrogen atom), approximation procedures and applications (mostly, but not exclusively, based on time-independent and time-dependent perturbation theory), ...
Well, this is where the advanced stuff actually begins, but also where, unfortunately, the summer term usually comes to an end.
The figure illustrates one of the most important phenomena encountered in elementary quantum mechanics: The tunneling of a wave packet through a barrier in a double-well potential.
Statistische Physik
Statistical Physics
In the lecturer's humble opinion, this is the most rewarding of the basic physics courses, because one derives almost anything from almost nothing. At the end of this lecture, we (hopefully) understand semiclassical statistical mechanics as well as quantum mechanical statistical physics, formulated in the microcanonical, canonical, and grand canonical ensembles. Special emphasis is given on the study of Bose and Fermi gases, together with some of their most prominent appearances in physics.
One of the most important topics covered in this course concerns the differences between the classical Maxwell-Boltzmann (MB) and the quantum mechanical Bose-Einstein (BE) and Fermi-Dirac (FD) statistics. The figure shows average grand canonical occupation numbers pertaining to these statistics as functions of the scaled energy.
Spezielle Kapitel der theoretischen Festkörperphysik
Theoretical solid-state physics - Special topics
Well, solid-state physics is a huge field indeed. Here we deal first with some selected single-particle phenomena in periodic potentials, such as the inevitable Bloch waves, weak- and tight-binding approximations, Wannier states, Bloch oscillations, and Zener tunneling. Then we turn to the language of second quantization, which is most helpful for coming to grips with many-body physics. Based on this, we study the metal-insulator transition shown by the archetypal Bose-Hubbard model. A more recent extension of this course concerns topological insulators.
The figure illustrates the celebrated Bloch-Zener scenario: The wave packet of a quantum particle in a periodic lattice potential subjected to a static external force is depicted as a contour plot vs. position (horizontal) and time (vertical). The wave packet performs Bloch oscillations, here with a fairly small amplitude because the force is quite strong. Once per period, it undergoes a Zener transition to a higher band, then escaping to infinity on a parabolic trajectory corresponding to a free particle accelerated by the force.
Theorie der Supraleitung
Theory of Superconductivity
This course is targeted towards the explanation of superconductivity given by Bardeen, Cooper, and Shrieffer. Their theory, arguably, constitutes one of the most beautiful achievements of theoretical physics, capturing the delicate interplay between several basic mechanisms and effects, reducing them to the bare essentials, and making rock-hard predictions. After recapitulating some key experimental findings, we study electron-phonon interaction, and verify the existice of Cooper pairs. We then turn to the mean-field approximation, utilize the Bogoliubov transformation, and formulate the BCS self-consistency equation. Having constructed approximate solutions of the gap equation we explore the thermodynamics of BCS superconductors, and (if time permits) investigate the current-carrying state.
The figure depicts an elementary process underlying superconductivity: An first electron (symbolized by the straight lines right) emitts a phonon (wavy line), which then is absorbed by a second electron. One of the key topics covered in this course is the finding that this phonon exchange gives rise to an attractive force between the negatively charged electrons, leading to the formation of the famous Cooper pairs.
Quantenmechanik mit dem Computer: Elemente der Vielteilchentheorie
Quantum mechanics by computer: Elements of many-body theory
Well, admittedly this lecture still is in an experimental stage. The idea is to employ one single model which is conveniently tractable on a modest laptop, the so-called Bose-Hubbard dimer, and to explore features of many-body physics with the help of numerical calculations. Along the way, we concern ourselves with the self-trapping effect, the accuracy of the mean-field approximation, and "atomic" coherent states. When augmented with a time-periodic driving force, the model leads directly to the field called "quantum chaos".
The figure shows the color-coded Husimi distribution of an exact quantum mechanical Floquet state of a periodically driven Bose-Hubbard dimer, superimposed on a Poincaré section pertaining to the mean-field dynamics. Figures such as this one, which actually require far more explanation, can be computed routinely by the participants of this course after mastering the theoretical basics.
Topics covered in this course may well serve as starting point for further Bachelor's and Master's theses.
Periodisch angetriebene Quantensysteme
Periodically driven quantum systems
Quantum mechanical systems under the influence of a periodically time-dependent force, such as atoms or molecules exposed to strong classical laser fields, or ultracold atoms in periodically "shaken" optical lattices, can be theoretically investigated by means of the Floquet theory, which is the temporal analog of the Bloch theory known from quantum particles in spatially periodic potentials. By suitably adjusting, e.g, the strength and/or the frequency of the drive, one can systematically equip the driven system with properties which its undriven antecessor does not possess. This line of research, named "Floquet engineering", has become a "hot topic" recently, with many problems still to be solved. We cover both basics and key applications of the Floquet approach, starting from the driven two-level system and gathering knowledge on ac Stark shifts and multiphoton resonances, then proceeding to periodically driven anharmonic oscillators and discussing, among others, periodic thermodynamics and time crystals. This blossoming field, which combines elements of quantum optics, nonlinear dynamics, and non-equilibrium statistical physics, provides ample opportunities for many more Master's and PhD-theses!
The figure depicts subharmonic response of a quantum mechanical many-body system to an external time-periodic driving force: It takes three cylces of the drive before the system returns to its initial state. This phenomenon, which is closely related to the so-called time crystals, is among the topics treated in this course.