260172 VO Many-body quantum Physics - ESI Lecture Course (2014S)
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Vorbesprechung: 05.03.14 um 15:00 Uhr, Erwin Schrödinger Institut f. Math. Physik (ESI), Boltzmanngasse 9, 1090 Wien, Schrödinger HS/Lecture Room.
Vorlesung: Dienstag 16:15 - 18:15, Maerz - Mai 2014
Uebung: Donnerstag 13:00 - 14:00, Maerz - Mai 2014.
Termin-Aenderungen auf http://www.math.ku.dk/~solovej/MANYBODY/
Vorlesung: Dienstag 16:15 - 18:15, Maerz - Mai 2014
Uebung: Donnerstag 13:00 - 14:00, Maerz - Mai 2014.
Termin-Aenderungen auf http://www.math.ku.dk/~solovej/MANYBODY/
Details
Sprache: Englisch
Prüfungstermine
Lehrende
Termine
Zur Zeit sind keine Termine bekannt.
Information
Ziele, Inhalte und Methode der Lehrveranstaltung
Art der Leistungskontrolle und erlaubte Hilfsmittel
Mindestanforderungen und Beurteilungsmaßstab
The aim of the lectures is to introduce the students to the concepts and methods of quantum many-body systems. The main methods the students will learn are second quantization, reduced density matrices, and Bogolubov transformations. The students will learn concepts such as thermodynamic stability, Bose condensation and superfluidity and superconductivitiy.
Prüfungsstoff
Literatur
The main source of literature is:
Many-Body Quantum Mechanics, Lecures notes by J.P. Solovej.Further reading on Bose Gases:
Elliott H. Lieb, Robert Seiringer, J.P. Solovej, and J. Yngvason, The Mathematics of the Bose Gas and its Condensation. Series: Oberwolfach Seminars, Vol. 34.Further reading on Spectral Theory and Schrödinger Operators:
Gerald Teschl, Mathematical Methods in Quantum Mechanics With Applications to Schrödinger Operators. Graduate Studies in Mathematics, Volume 99, Amer. Math. Soc., Providence, (2009).Original papers:
N. N. Bogoliubov, On the theory of superfluidity. J. Phys. (USSR), 11, (1947), p. 23;
J. Bardeen, L. Cooper, J. Schrieffer, Theory of Superconductivity. Phys. Rev. 108, 1175–1204, (1957).Newer papers of interest:
M. Lewin, P. T. Nam, S. Serfaty, J. P. Solovej, Bogoliubov spectrum of interacting Bose gases. To appear in Comm. Pure and Applied Math. arXiv:1211.2778;
R. Seiringer, The excitation spectrum for weakly interacting bosons. Commun. Math. Phys., 306, (2011), pp. 565-578.
Many-Body Quantum Mechanics, Lecures notes by J.P. Solovej.Further reading on Bose Gases:
Elliott H. Lieb, Robert Seiringer, J.P. Solovej, and J. Yngvason, The Mathematics of the Bose Gas and its Condensation. Series: Oberwolfach Seminars, Vol. 34.Further reading on Spectral Theory and Schrödinger Operators:
Gerald Teschl, Mathematical Methods in Quantum Mechanics With Applications to Schrödinger Operators. Graduate Studies in Mathematics, Volume 99, Amer. Math. Soc., Providence, (2009).Original papers:
N. N. Bogoliubov, On the theory of superfluidity. J. Phys. (USSR), 11, (1947), p. 23;
J. Bardeen, L. Cooper, J. Schrieffer, Theory of Superconductivity. Phys. Rev. 108, 1175–1204, (1957).Newer papers of interest:
M. Lewin, P. T. Nam, S. Serfaty, J. P. Solovej, Bogoliubov spectrum of interacting Bose gases. To appear in Comm. Pure and Applied Math. arXiv:1211.2778;
R. Seiringer, The excitation spectrum for weakly interacting bosons. Commun. Math. Phys., 306, (2011), pp. 565-578.
Zuordnung im Vorlesungsverzeichnis
MF 5, MaG 15
Letzte Änderung: Mi 19.08.2020 08:06
Tensor products and the formulation of many-body quantum mechanics,
Identical particles and statistics,
Fock spaces and second quantization,
Bogolubov transformations,
Quasi-free states,
Quadratic Hamiltonians,
The Bogolubov variational principle,
The Bogolubov approximation for bosonic systems,
Fermionic systems and the BCS, and
Hartree-Fock-Bogolubov models.The lectures introduce the formalism and concepts of many-body quantum mechanics. The basic notions of Hilbert spaces and operators as well as the principles of quantum mechanics will be reviewed. In particular, the concept of many-body systems and the notion of indistinguishable particles will be discussed and bosons and fermions will be introduced. This requires introducing tensor products of Hilbert spaces and simple representations of the permutation group. Some time will be spent on semibounded operators and the min-max principle for describing their eigenvalues below the continuum. The general theory will be applied to Schrödinger operators. The lectures will, however, not give a full account of spectral theory. The formalism of second quantization will be introduced and basic properties of many-body quantum states will be analyzed. In particular we will discuss their one- and two-particle reduced density matrices and their eigenvalues. Different classes of states will be discussed and, in particular, the notion of quasi-free states and Bogolubov transformations will be introduced and analyzed. The role of quasi-free states as equilibrium states of quadratic Hamiltonians will be emphasized and the method of diagonalizing these using Bogolubov transformations will be described. This leads to introducing the Bardeen-Cooper-Schrieffer theory or more generally the Bogolubov-Hartree-Fock theory for fermions and the Bogolubov theory for bosons. Hopefully time will permit to consider the theory applied to simple physical systems to illustrate such phenomena as super-conductivity and super-fluidity.For Course Website including Lecture Notes visit the ESI website http://www.esi.ac.at/activities/events/2014/many-body-quantum-physics-1