Applications of Group Theory to Quantum Physics : Algebraic Apects
THEORIE DES GROUPES
THEORIE QUANTIQUE
CONGRES ET CONFERENCES
ATOMES
MOLECULES
PHYSIQUE NUCLEAIRE
SYMETRIE
HADRONS
Lectures given at the 1969 Battelle Summer Rencontres in Mathematics and Physics. Seattle - Washington - USA
MICHEL
P/69/X032
IHES
09/1969
A4
97 f.
EN
TEXTE
PREPUBLICATION
P_69_X032.pdf
1969
Structure and classification of band representations
CRISTALLOGRAPHIE
SOLIDES
ASTRONOMIE
ORBITE
ETOILES
MODELISATION
ENERGIE
ATOMES
Abstract : Band representaitons in solids are investigated in the general framework of induced representations by using the concepts of orbits (stars) and strata (Wyckoff positions) in their construction and classification. The connection between band representations and irreducible representations of space groups is established by reducing the former in the basis of quasi-Bloch functions wich are eigenfunctions of translations but ar not, in general, eigenfunctions of the Hamiltonian. While irreducible representations of space group ar fnite-dimensional and are induced from infinite-order little groups Gk for vectors K in the Brillouin zone, band representations are infinite-dimensional adn are induced from finite-order little groups Gr for vectors r in the Wigner-Seitz cell. This connection between irreductible representations and band representations of space groups shedd new light on the duality properties of the Brillouin zone and the Wigner-Seitz cell. As an introduction to band representations the induced representations of point groups wich id applied to the investigation of th equivalency of band representations. Based on this connection and on the properties of the crystallographic point groups a necessary condition id established for the inequivalency of band representations induced from maximal isotropy groups. For using this condition there is need fro the induced representaitons of oint groups and a full list of them is given in the paper. One is especially interested in irreducible-band representations which form the elementary building bricks for band representations. From the point of view of the physics, irreducible-band representations correspond to energy bands with minimal numbers of branches. A method id developped for finding all the inequivalent irreducible-band representations of space groups by using the induction from maximal isotropy groups. As a rule the latter leads to inequivalent irreducible-band representations. There are, however, few exceptions to this rule. A full list of such exceptions is tabulated in the paper. With this list at hand one can construct all the different irreducible-band representations of 2-dimensional space groups. For them we list the continuity chords of all their irreducible-band representations.
MICHEL
BACRY
ZAK
P/86/35
IHES
06/1986
A4
46 f.
EN
TEXTE
PREPUBLICATION
P_86_35.pdf
1986
Rydberg states of atoms and molecules. Basic group theoretical and topological analysis
ETATS DE RYDBERG
ATOMES
HYDROGENE
CHAMPS ELECTROMAGNETIQUES
Abstract : Rydberg states of atoms and molecules are studied within the qualitative approach-based primarily on topological and group theoretical analysis. The correspondence between classical and quantum mechanics is explored to apply the results of qualitative (topological) approach to classical mechanics developed by PoincarĂ©, Lyapounov and Smale to quantum problems. The study of the action of the symmetry group of the problems considered on the classical phase space enables us to predict qualitative features of the energy level patterns for quantum Rydberg operators.
MICHEL
ZHILINSKII
P/97/54
IHES
07/1997
A4
41 f.
EN
TEXTE
PREPUBLICATION
P_97_54.pdf
1997
Collapse of the Zeeman structure of the hydrogen atom in external electric field
CHAMP ELECTRIQUE CRISTALLIN
ATOMES
HYDROGENE
STRUCTURE MAGNETIQUE
Abstract : From Zeeman to Stark structure of a weakly split Rydberg n multiplet of the H atom in parallel magnetic and electric fields is analyzed. Classical mechanics together with topologiacl and group theoretical arguments enable us to describe in details the modifications of dynamics under the variation of electric field near the point where the collapse of magntic Zeeman structure is observed. Sequence of classical bifurcations responsible for the transition between different dynamic regimes is given. Comparison with quantum picture is done.
MICHEL
SADOVSKII
SHILINSKII
P/95/86
IHES
09/1995
A4
4 f.
EN
TEXTE
PREPUBLICATION
P_95_86.pdf
1995