April 6, 2021 / Department of Mathematics

# Prof. Griesemer, what is Mathematical Quantum Mechanics?

Interview with members of the Department of Mathematics

Quantum mechanics is the physical theory of radiation and matter at the level of elementary particles. It is thus the basis for our entire understanding of nature, with the exception of gravity. The latter is the only one of the four fundamental forces of nature that cannot yet be described within the framework of quantum mechanics. Our understanding of the periodic table of elements, and thus all of chemistry, is based on the rules of quantum mechanics. Everything we know about the composition of the universe is based on the observation of atomic spectra whose regularities are determined by quantum mechanics. Quantum mechanical effects are also used in everyday devices (solar cells, storage media, lasers) and in medicine (magnetic resonance imaging (MRI), radiation therapy). We are currently placing great hopes in quantum computers.

*Were is the link between quantum mechanics and mathematics?*

For non-relativistic quantum mechanics as presented in physics textbooks, John von Neumann established a rigorous mathematical foundation in the 1930s. In 1951, it was recognized (Tosio Kato) that Schrödinger's description of atoms and molecules fit within von Neumann's framework. Since then, mathematicians around the world have been working out the consequences of the Schrödinger equation. Systems of more than two particles are usually not analytically solvable, and even with numerical methods one quickly reaches the limits of computer power for many particles. An important key to success is therefore effective theories that describe many-particle systems well in suitable limiting cases (large number of particles, small density, small mass ratio, etc.). Physicists are good at "guessing" such effective theories, but assessing the accuracy and validity of these effective theories qualitatively and quantitatively is a task for mathematicians. In mathematical quantum mechanics, almost the whole spectrum of known mathematical methods is applied, from analysis to geometry and stochastics to group theory.

*What are you working on, Prof. Griesemer?*

Together with colleagues and PhD students, I study, for example, energy spectra and the dynamical behavior of many-body systems. Quite often this is done via an effective model whose accuracy we first have to estimate quantitatively. The models we consider are either of current interest in physics (models for ultracold gases and quantum optics) or have been on the market for a long time and are of fundamental interest (atomic and solid state models). Sometimes there are already results in the mathematical literature for the system under consideration on which we can build, sometimes we first have to find out what the physicists or quantum chemists are doing before we can set up and analyze a mathematically clean model. We are moving in a broad field between pure analysis and theoretical physics and this is exactly what makes this research so attractive.

*What prior knowledge do students need to enter your research field in the context of a master's
or doctoral thesis?*

Good knowledge of functional analysis and some spectral theory are the basic requirements. For a PhD, prior knowledge of quantum mechanics is also required, e.g. one to two semesters of QM lecture, which also requires a basic education in classical physics (mechanics and electrodynamics). - Most importantly, however, is the motivation and willingness to fill gaps in one's own education.

*Thank you very much for the interview. *

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Prof. Marcel Griesemer
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Institute of Analysis, Dynamics and Modeling

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