Die beiden Säulen der modernen Physik
Vielteilchensysteme sind weit mehr
als die Summe ihrer Teile ! innere Wechselwirkungen
Korrelation: Wellenfunktionen faktorisiert nicht
Funktionenräume statt einzelner Funktionen:
dramatischer Anstieg der Komplexität bei
der theoretischen Beschreibung.
Relativitätstheorie (Einstein 1905 und 1916):
große Geschwindigkeiten und Massen
Quantenmechanik (Schrödinger und Heisenberg, 1924):
mikroskopische Welt ist eine Vielteilchenwelt
Quarks, Kerne, Atome, Moleküle, Cluster, Festkörper, ...
Computer-algebraische Ansätze und Werkzeuge
zur Beschreibung quantenmechanischer VielteilchensystemeS. Fritzsche, Universität Kassel
am 17. Mai 2003
Reduce (seit Ende der 60er)Matlab, Mathematica, Axiom, DeriveMaple
Polynome, Differentiation und IntegrationLineare AlgebraGewöhnliche (und partielle) Dgls.Funktionen der mathematischen PhysikKombinatorik, Graphentheorie(formale) Gruppentheorie
Formel- und Programmiersprache enthält einige Tausend Einzelfunktionen
Vorteile für die theoretische Physik
Kenntnis des mathematischen Regelwerkes
Erleichtert aufwendige Herleitungen.
Oftmals dort verwendet, wo der algebraische
Weg prinzipiell bekannt ist.
Zuverlässigkeit, einfache Reproduzierbarkeit
Computer-algebraische Ansätze und Werkzeuge
zur Beschreibung quantenmechanischer VielteilchensystemeS. Fritzsche, Universität Kassel
am 17. Mai 2003
Reduce (seit Ende der 60er)Matlab, Mathematica, Axiom, DeriveMaple
Polynome, Differentiation und IntegrationLineare AlgebraGewöhnliche (und partielle) Dgls.Funktionen der mathematischen PhysikKombinatorik, Graphentheorie(formale) Gruppentheorie
Formel- und Programmiersprache enthält einige Tausend Einzelfunktionen
Vorteile für die theoretische Physik
Kenntnis des mathematischen Regelwerkes
Erleichtert aufwendige Herleitungen.
Oftmals dort verwendet, wo der algebraische
Weg prinzipiell bekannt ist.
Zuverlässigkeit, einfache Reproduzierbarkeit
Wie können wir diese Vorteile für die Beschreibung
von Vielteilchensystemen nutzen ? Von der formalen Theorie zum phys. Verständnis
Racah's Algebra: Werkzeuge für die Atom-, Molekül-
und Kernstruktur
Numerical studies -- An accepted route in theoretical physics ?
About 40 years ago, (pure) numerical studies became an accepted instrument in theoretical physics; they -- in fact -- often provide the only route to obtain sufficient information about many systems.
Matrix diagonalization:
10 4...7 (today's dimensions)
Numerical libraries: LU decomposition, Davidson algorithm, ...
1 0 00 0 10 1 0
+1 0 00 +1 00 0 B1
Numerical studies -- An accepted route in theoretical physics ?
About 40 years ago, (pure) numerical studies became an accepted instrument in theoretical physics; they -- in fact -- often provide the only route to obtain sufficient information about many systems.
Matrix diagonalization:
10 4...7 (today's dimensions)
Numerical libraries: LU decomposition, Davidson algorithm, ...
Symbolic manipulations: automatic search for symmetries and appropriate coordinates
simplification of expressions, operators and/or matrix elements
classification of (many-particle) quantum states
coordinate transformations
...maple, mathematica, matlab, derive, ... (included in present-day curricula)
1 0 00 0 10 1 0
+1 0 00 +1 00 0 B1
Applications in many-particle physics and dynamics
Advanced calculus for hydrogenic systems
Angular momentum in physics (Racah's algebra)
Perturbation expansions in many-particle physics
Use of point-group symmetries in physics, chemistry and biology
Classification and topology of Feynman graphs
Use of hyperspherical coordinates
Experience: Implementation and computations often require the dominant effort in studying (quantum-) many-particle systems.
Atoms in different environments have led to quitedifferent communities, although they apply very similar theoretical concepts.
From formal theory to physical understanding
-- that also means quantitative predictions
Atomic Physics Case
Atomic structure and properties
Ion-atom collisions and dynamics of many-particle systems
Atoms and ions in external(electric and magnetic) fields
Interaction of atoms with particles and light
Atoms in strong laser fields
Experimental requirementsLevel and transition
propertiesdifferential and total
cross sectionsrate coefficients
angular distributionsion yields
characteristic timescales
wave packet dynamics
Schrödinger theory
Dirac Equation
Racah's algebra andtheory of spherical tensors (SO3 group)
Atomic shell model
Statistical models
fundamental concepts
From formal theory to physical understanding
-- that also means quantitative predictions
Atomic Physics Case
Atomic structure and properties
Ion-atom collisions and dynamics of many-particle systems
Atoms and ions in external(electric and magnetic) fields
Interaction of atoms with particles and light
Atoms in strong laser fields
Experimental requirementsLevel and transition
propertiesdifferential and total
cross sectionsrate coefficients
angular distributionsion yields
characteristic timescales
wave packet dynamics
Schrödinger theory
Dirac Equation
Racah's algebra andtheory of spherical tensors (SO3 group)
Atomic shell model
Statistical models
fundamental concepts
Werkzeuge der Computeralgebra
Atomic Physics Case
Atomic structure and properties
Ion-atom collisions and dynamics of many-particle systems
Atoms and ions in external(electric and magnetic) fields
Interaction of atoms with particles and light
Atoms in strong laser fields
Experimental requirementsLevel and transition
propertiesdifferential and total
cross sectionsrate coefficients
angular distributionsion yields
characteristic timescales
wave packet dynamics
Schrödinger theory
Dirac Equation
Racah's algebra andtheory of spherical tensors (SO3 group)
Atomic shell model
Statistical models
fundamental concepts
Werkzeuge der Computeralgebra
Reduce (seit Ende der 60er)Matlab, Mathematica, Axiom, DeriveMaple
Polynome, Differentiation und IntegrationLineare AlgebraGewöhnliche (und partielle) Dgls.Funktionen der mathematischen PhysikKombinatorik, Graphentheorie(formale) Gruppentheorie
Formel- und Programmiersprache enthält einige Tausend Einzelfunktionen
Anwendungen in der PhysikTensoralgebra und Analysis (ART)Quantenmechanik des H-AtomsTheorie des DrehimpulsesPunktsymmetrien der MoleküleSymmetrieorbitale
Vielteilchenstörungstheorie Operatorprodukte (Wick-Theorem) Erwartungwerte Klassifizierung und Topologie von
Diagrammen, etc.
Properties and behaviour of hydrogen-like ions
-- in different environments
H-like Uranium
EK = -132 · 103 eV, Z·α ∼ 1<E>= 1.8 · 1016 V/cm
Z = 92
Z = 1 Hydrogen
EK = -13.6 eV, Z·α « 1
<E>= 1 · 1010 V/cm
Increase in the field strengthby six orders of magnitude
1 10 20 30 40 50 60 70 80 90109
1010
1011
1012
1013
1014
1015
1016
1s
<E
>
[V/c
m]
Nuclear Charge, Z
Hydrogen-like ions (i.e. an effective one-partilce model) have a very wide of applications, including atomic and molecular collisions, plasma physics, quantum optics, ...
Correlation effects are often negligible
Explicitly time-dependent processes
Properties and behaviour of hydrogen-like ions
-- in different environments
H-like Uranium
EK = -132 · 103 eV, Z·α ∼ 1<E>= 1.8 · 1016 V/cm
Z = 92
Z = 1 Hydrogen
EK = -13.6 eV, Z·α « 1
<E>= 1 · 1010 V/cm
Increase in the field strengthby six orders of magnitude
1 10 20 30 40 50 60 70 80 90109
1010
1011
1012
1013
1014
1015
1016
1s
<E
>
[V/c
m]
Nuclear Charge, Z
Hydrogen-like ions (i.e. an effective one-partilce model) have a very wide of applications, including atomic and molecular collisions, plasma physics, quantum optics, ...
Correlation effects are often negligible
Explicitly time-dependent processes However: sophisticated calculus wave functions and matrix
elements in different representations; combination of analytic and
numerical methods
Zwei- und Mehrphotonenionisation im Innerschalenbereich
Zirkular
Linear
Zweiphotonenquerschnitte zum H-Atom
Wirkungsquerschnitte
dWdt
= ∑σn⋅F n
Z 6σ 2 Z, Eph
= σ 2 1, Eph⁄Z 2
»1013 W/cm2
σ2 =cm4 sec; σ2I
=cm4 WB1
Skalierung (Zernik, Phys. Rev. 1964)
Aktuelle Fragen:
Innerschalenionisation bei höheren Z ?
Resonante vs. nichtresonante Prozesse bei Mehrelektronenatomen ?
Experimentatoren: Welche Systeme und Energiebereiche sind von Interesse, um nichtlineare
Prozesse auch im Röntgenbereich tatsächlich nachweisen zu können ?
λ⁄ A
σ2
Theorie der Multiphotonenionisation
-- am Beispiel von Zweiphotonenprozessen
Skalierungen mit effektiven Ladungen:
Direkte Summation:
Greensche Funktionen:
Für das H-Atom bekannt
Abschätzung der Größenordnung der Querschnitte im nichtresonanten Bereich sowie von relativistischen Effekten.
... sehr schwer handhabbar !
Summation über Vielteilchenspektrum
Floquet-Theorie:
Schrödinger-Entwicklung in einer harmonisch (-zeitabhängigen) Basis und Berechnung geeigneter Mittelwerte (höhere Harmonische, ATI, Pulsabhängigkeiten, ...)
-- allerdings kaum praktikabel für Vielteilchensysteme !
σ2 = C
Eph2
Mν
2
Mν = ∑ψ
feikr u
λ⋅p ψν ψν eikr u
λ⋅p ψ
i
Ei+E
phBEν
G r, r’; E = ∑lg r, r’; E ∑m
Ylm
Ω Ylm
+ Ω
Theorie der Multiphotonenionisation
-- am Beispiel von Zweiphotonenprozessen
Skalierungen mit effektiven Ladungen:
Direkte Summation:
Greensche Funktionen:
Für das H-Atom bekannt
Abschätzung der Größenordnung der Querschnitte im nichtresonanten Bereich sowie von relativistischen Effekten.
... sehr schwer handhabbar !
Summation über Vielteilchenspektrum
σ2 = C
Eph2
Mν
2
Mν = ∑ψ
feikr u
λ⋅p ψν ψν eikr u
λ⋅p ψ
i
Ei+E
phBEν
G r, r’; E = ∑lg r, r’; E ∑m
Ylm
Ω Ylm
+ Ω
Relativistische Greensche Funktionen
Näherungsverfahren fürGreensche Vielteilchenfunktionen
Theory of angular momentum (Racah's algebra)
Physicist's view on Racah's algebra: Very large number of algebraic relations among the Wigner n-j symbols, spherical harmonics of various kinds, rotation and spin matrices, ... and their appropriate application (Racah algebra techniques). (Varshalovich et al. 1988) suitable for symbolic manipulations
J p , Jq = iεp q r Jr J ... angular momentum operator and generator of the rotation group.
Well known group theoretical background for a long time !
Coupling of two angular momenta (i.e. two particles)
Wigner 3-j symbol
Clebsch-Gordan coefficient
Applications of Racah's algebra in many-particle physics
Theory of great elegance and power analytic integration over angular coordinates
3N spatial coordinates --> N radial coordinates
conservation of angular momentum
utilization of given symmetries (scattering, collision)
--> coordinate rotations
Applications in Physics: Evaluation of many-electron matrix element
Correlation functions: Products of spherical harmonics which are coupled to a total angular momentum J = 0
Polarisation reactions and transfer
--> cross sections
Classical field theory: Earth quakes
Re-coupling of angular momenta
Coupling of three angular momenta: J = j1 + j2 + j3, Jz = j1z + j2z + j3z
Applications in atomic structure theory:1 Transformation from LS- to Jk-coupling <L(s1s2) S,J | (s1L) K, s2 J > coupling of the angular momenta L, s1, s2
1 Transformation from LS- to jj-coupling <(l1l2)L (s1s2) S, J | (l1s1) j1 (l2s2) j2, J> coupling of the angular momenta l1, l2, s1, s2
General Racah expressions
Traditional pathes for simplifying Racah expressions Replacement of special values by much simpler expressions Orthogonality relations Sum rule evaluation Application of graphical rules
Aim in the manipulation of Racah expressionsReduce the number of summation indices and/or Wigner symbols, spherical harmonics, ...
In general, a Racah expression might include any number of Wigner n-j symbols of differentkinds as well as Kronecker and triangular deltas δ(a,b,c) symbols.
Symmetries of the Wigner n-j symbols
Classical vs. Regge (1958) symmetries
Complex Racah expressions can therefore be writtenin a very large (huge) number of equivalent forms.
Each 3-j symbol can be written in 72equivalent forms which all representthe same value !
Similar symmetries are known for theWigner 6-j and 9-j symbols.
The RACAH package -- based on Maple
Large set of functions and symbols.
Modular structure: with(Racah);
Provides data structures which are flexible enough to cover most applications.
Helpful for both, occasional use as well as advanced research work.
More than 280 Maple procedures within a hierarchical structure; however, only about 10 commands need to be known by the user.
Part of the CPC-Library (Computer Physics Communications).
The RACAH package
Large set of functions and symbols.
Modular structure: with(Racah);
Provides data structures which are flexible enough to cover most applications.
Helpful for both, occasional use as well as advanced research work.
More than 280 Maple procedures within a hierarchical structure; however, only about 10 commands need to be known by the user.
Part of the CPC-Library (Computer Physics Communications)Main commands of the RACAH packageRacah_compute() Computes the numerical value of some Racah expression.Racah_evaluate() Attempts to simplify a general Racah expression by using a list of special values, orthogonality relations, and a variety of sum rules which are known for the Wigner n-j symbols, spherical harmonics, ...Racah_help() Displays a list of all commands which are implemented in the current version.Racah_print() Prints a Racah expression in a neat format.Racah_recursion() Applies recursion relations to some Wigner n-j symbol.Racah_set() Enters some Racah expressionRacah_tensorYlm() tensor spherical harmonicRacah_w3j(), Racah_w6j(), Racah_w9j(), Racah_w12j(), ...
The RACAH package
Large set of functions and symbols
Modular structure: with(Racah);
Provides data structures which are flexible enough to cover most applications
Helpful for both, occasional use as well as advanced research work
More than 280 Maple procedures within a hierarchical structure; however, only about 10 commands need to be known by the user.
Part of the CPC-Library (Computer Physics Communications)
Maple procedures for the coupling of angular momenta
I. Data structures and numerical computations (1997) II. Evaluation of sum rules (1998) III. Standard quantities for many-electron matrix elements (2000) IV. Spherical harmonics (2001) V. Recoupling coefficients (2001) VI. LS-jj transformation matrices (2002) VII. Accelerated and extended computations (2003) .....
Point-group symmetries in physics and chemistry
Determination of normal coordinates
Classification of molecular states
Selection rules and spectroscopic activities
Construction of symmetry and hybrid orbitals
Ligand field theory: splitting of atomic level energies in external crystal fields
...
Since many molecules exhibit some type of symmetry, the theory of the point groups can be usedto predict the properties and behaviour of molecules and to simplify molecular computations.
We presently develope the BETHE toolbox to support applications of the molecular point groups in physics, chemistry, and biology.
Applications in many-particle physics and dynamics
Experience: Implementation and computations often require the dominant effort in studying (quantum-) many-particle systems.
Center of scientific computing -- need or luxury ?
Concur of algebraic and numerical methods !
Advanced calculus for hydrogenic systems
Angular momentum in physics (Racah's algebra)
Perturbation expansions in many-particle physics
Use of point-group symmetries in physics, chemistry and biology
Classification and topology of Feynman graphs
Use of hyperspherical coordinates
Zusammenfassung und Ausblick
Anforderungen der Vielteilchenphysik an CA-Werkzeuge:
Eine der Gemeinschaft angepaßte Sprache.
Einfachheit und Nutzerfreundlichkeit
(vorbereitete) Datenstrukturen
skalierbare Algorithmen
Fehlerausbreitung
Testszenarien
CA-Werkzeuge fürquantenmechanischeVielteilchensysteme
Modell des H-Atoms(DIRAC) ....
Störungsreihen(GOLDSTONE)
Spinsysteme
Punktgruppen(Bethe)
Physik der Drehgruppe (RACAH)
Dank für Zusammenarbeit
-- mit experimentellen und theoretischen Arbeitsgruppen
Drittmittel:DFG, GSI, BMBF, Hessen
Vielteilchenphysik in Kassel
Andrey Surzhykov (GSI)
Peter Koval (DFG)
Katja Rykhlinskaya (DFG)
Alexander Uvarov (DFG)
Thorsten Inghoff (Otto-Braun)
Chenzhong Dong (bis 2001)
Maple procedures for the coupling of angular momenta
I. Data structures and numerical computations (1997) II. Evaluation of sum rules (1998) III. Standard quantities for many-electron matrix elements (2000) IV. Spherical harmonics (2001) V. Recoupling coefficients (2001) VI. LS-jj transformation matrices (2002) VII. Accelerated and extended computations (2003) .....
Main commands of the RACAH package
Racah_compute() Computes the numerical value of some Racah expression.Racah_evalute() Attempts to simplify a general Racah expression by using a list of special values, orthogonality relations, and a variety of sum rules which are known for the Wigner n-j symbols, spherical harmonics, ...Racah_help() Displays a list of all commands which are implemented in the current version.Racah_print() Prints a Racah expression in a neat format.Racah_recursion() Applies recursion relations to some Wigner n-j symbol.Racah_set() Enters some Racah expressionRacah_tensorYlm() tensor spherical harmonicRacah_w3j(), Racah_w6j(), Racah_w9j(), Racah_w12j(), ...
Computeralgebra-Werkzeuge zur Herleitung und
Manipulation der Formeln
Reduce (seit Ende der 60er)Matlab, Mathematica, Axiom, DeriveMaple
Polynome, Differentiation und IntegrationLineare AlgebraGewöhnliche (und partielle) Dgls.Funktionen der mathematischen PhysikKombinatorik, Graphentheorie(formale) Gruppentheorie
Formel- und Programmiersprache enthält einige Tausend Einzelfunktionen
Anwendungen in der PhysikTensoralgebra und Analysis (ART)Quantenmechanik des H-AtomsTheorie des DrehimpulsesPunktsymmetrien der MoleküleSymmetrieorbitale
Vielteilchenstörungstheorie Operatorprodukte (Wick-Theorem) Erwartungwerte Klassifizierung und Topologie von
Diagrammen, etc.