Macroscopic systems and quantum theory

Research projects

Spin Systems

The synthesis of molecular magnets has undergone rapid progress in recent years. Each of the identical molecular units can contain as few as two and up to several dozens of paramagnetic ions (``spins"). Although these materials appear as macroscopic samples, i.e. crystals or powders, the intermolecular magnetic interactions are utterly negligible as compared to the intramolecular interactions. Therefore, measurements of their magnetic properties reflect mainly ensemble properties of single molecules.

We are interested in static and dynamical properties of magnetic molecules, our research focuses on:

  • properties of spin rings like the popular ferric wheels;
  • rigorous and numerical results on properties of the spectrum for Heisenberg spin arrays, like bounding parabolas, rotational bands, quantum numbers of low-lying states;
  • high N limits for spin chains;
  • comparison of classical and quantum Heisenberg model;
  • spin-spin correlation functions and related quantities like spin-lattice relaxation times or neutron scattering cross sections;
  • approximate methods for spin systems like DMRG, high temperature expansion etc. (German Molecular Magnetism)
Mohammed Allalen - PhD Student @ UOS
Klaus Bärwinkel - Professor, Senior Fellow @ UOS
Mirko Brüger - Diploma Student @ UOS
Matthias Exler - Graduate Student @ UOS
Peter Hage - Diploma Student @ UOS
Frank Hesmer - Diploma Student @ UOS
Paul Kögerler - PostDoc, Fellow @ Ames Lab, Iowa
Marshall Luban - Professor, Senior Fellow @ Ames Lab, Iowa
Detlef Mentrup - PhD Student @ UOS, now Philips Research
Robert Modler - Assistant Professor @ Ames Lab, Iowa
Heinz-Jürgen Schmidt - Professor, Senior Fellow @ UOS
Jürgen Schnack - Professor @ Bielefeld University
Christian Schröder - Professor, Senior Fellow @ FH Bielefeld
Stefan Torbrügge - Master Student @ UOS/UoG Athens

Thermostated Dynamics

Statistical properties of finite interacting systems are of great interest. The aim is to describe the behaviour of systems like atomic clusters, atomic vapours or atomic nuclei at finite temperatures and to investigate properties like the specific heat or phase transitions. For realistic systems like atomic clusters or nuclei where the Hamilton function or operator contains a (two-body) interaction it is hard or impossible to evaluate the partition function especially for the quantum description.

Equations of motion for the investigated system are often much easier; either they are exactly known and can be integrated at least numerically as it is the case with the classical Hamilton's equation or they can be approximated with standard methods like Time-dependent Hartree-Fock (TDHF) or quantum molecular dynamics methods as it is the case on the quantum side. The idea then is to extract the desired thermodynamic quantities from the time evolution of the system. If the system is ergodic, ensemble averages can be replaced by time averages.

Our current research focuses on:

  • thermodynamic equlibrium properties of small quantum systems by time averaging together with Fermionic Molecular Dynamics (FMD);
  • determination of the caloric curve of finite nuclei and investigation of the nuclear liquid-gas phase transition;
    View movies. View more movies.
  • thermalization of a quantum system with the help of additional degrees of freedom and complex time steps;
  • Nose-Hoover-like thermostat for ideal quantum gases in harmonic oscillator potentials;
  • Nose-like thermostat for general quantum systems.
Detlef Mentrup - PhD Student @ UOS, now Philips Research
Hans Feldmeier - Professor, Senior Fellow @ GSI/TUD
Jürgen Schnack - Professor @ Bielefeld University

Harmonic Oscillator - Fermions and Bosons in Traps

Ideal quantum gases are usually treated in the thermodynamic limit, i.e. occupying an infinite volume but maintained at a given density, since all applications which were important in the past, like the electron gas, phonons or photons, deal with huge particle numbers. Only the experimental attempts of the last years to investigate finite Fermi and Bose systems and to describe them in terms of thermodynamics called for new theoretical effort. Interesting finite Fermi systems are for instance nuclei, which behave like a liquid drop and therefore can undergo a first order liquid-gas-like phase transition. On the low excitation site of the caloric curve the nuclear systems can be very often well described as an ideal Fermi gas in a common harmonic oscillator potential (shell model). Small Bose systems became available through the development of traps. Here the focus is on the Bose-Einstein condensation which for instance could be found investigating dilute atomic vapours (alkali atoms) in magnetic traps. Again the system can be well described as an ideal quantum gas contained in an external harmonic oscillator potential.

With the help of recursion formulae analytical and approximative results are obtained for small non-interacting Fermi and Bose systems.

A closer inspection of the canonical partition function uncovers a surprising symmetry property which connects fermions and bosons contained in harmonic oscillator potentials of odd space dimensions. Simply speaking, it turns out that the properties of N fermions at temperature T are related to the properties of N bosons at the respective negative temperature -T.

Heinz-Jürgen Schmidt - Professor, Senior Fellow @ UOS
Jürgen Schnack - Research Associate @ UOS

Transport Theory

A quasi-particle theory for monatomic gases in equilibrium is formulated and evaluated to yield the exact virial contributions to the thermodynamic state functions in lowest order of the density. Van der Waals blocking has necessarily to be accounted for in occupation number statistics. The quasi-particle distribution function differs from the Wigner function by a bilinear functional thereof. The progress made so far is promising with respect to a corresponding version of kinetic theory.
Klaus Bärwinkel - Professor, Senior Fellow @ UOS
Jürgen Schnack - Professor @ Bielefeld University

Fermionic Molecular Dynamics (FMD)

A new type of molecular dynamics is proposed to solve approximately the many-body problem of interacting identical fermions with spin 1/2 using variational principles. The interacting system is represented by an antisymmetrized many-body wave function consisting of single-particle states which are localized in phase space. The equations of motion for the parameters characterizing the many-body state (e.g. position, momentum, width and spin of the particles) are derived from a quantum variational principle. The model is designed to describe ground state properties of nuclii as well as heavy ion reactions. Therfore the ansatz is extended towards correlated many-body states, in order to include short-range correlations. Due to its non-linear equations of motion the model shows large fluctuations in the final stage as it is seen in fragmentation reactions. Not only heavy-ion reactions may be addressed, but also properties of excited nuclii like the nuclear liquid-gas phase transition.
FMD home page, local copy
detailed description (PDF with hyperlinks)
Hans Feldmeier - Professor, Senior Fellow @ GSI/TUD
Robert Roth - Junior Professor @ TUD
Jürgen Schnack - Professor @ Bielefeld University
Thomas Neff - PostDoc @ GSI/TUD

Unitary Correlation Operator Method (UCOM)

The short range repulsion between nucleons is treated by a unitary correlation operator which shifts the nucleons away from each other whenever their uncorrelated positions are within the replusive core. By formulating the correlation as a transformation of the relative distance between particle pairs, general analytic expressions for the correlated wave functions and correlated operators are given. The decomposition of correlated operators into irreducible n-body operators is discussed. The one- and two-body-irreducible parts are worked out explicitly and the contribution of three-body correlations is estimated to check convergence. Ground state energies of nuclei up to mass number A=48 are calculated with a spin-isospin-dependent potential and single Slater determinants as uncorrelated states. They show that the deduced energy- and mass-number-independent correlated two-body Hamiltonian reproduces all "exact" many-body calculations surprisingly well.
UCOM home page
Hans Feldmeier - Professor, Senior Fellow @ GSI/TUD
Robert Roth - Junior Professor @ TUD
Jürgen Schnack - Professor @ Bielefeld University
Thomas Neff - PostDoc @ GSI/TUD

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Last change 15. 01. 2009 JS, HJS