die beiden säulen der modernen physik - uni-kassel.decompmath/tagung2003/vortrag-fritzs… · die...

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




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












Schrödinger theory

Dirac Equation


Atomic shell model

Statistical models


Werkzeuge der Computeralgebra

Atomic Physics Case












Schrödinger theory

Dirac Equation


Atomic shell model

Statistical models


Werkzeuge der Computeralgebra




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.

Explizite Darstellung der Wigner 3-j Symbole

Summation über Produkte von Fakultäten:

Special values, orthogonality and sum rules

Special values:

Orthogonality rules:

Sum rules:

....

Sum rules and graphical techniques

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.


Provides data structures which are flexible enough to cover most applications.

Helpful for both, occasional use as well as advanced research work.


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


Provides data structures which are flexible enough to cover most applications

Helpful for both, occasional use as well as advanced research work


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) .....

Interaction matrix for coupled two-particle states

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




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.

die beiden säulen der modernen physik - uni-kassel.decompmath/tagung2003/vortrag-fritzs… · die...

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