Nicolaides Unstable States in the Continuous Spectra. Analysis, Concepts, Methods and Results

1. Auflage 2010
ISBN: 978-0-12-380901-8
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, Band Volume 60, 564 Seiten

Reihe: Advances in Quantum Chemistry

Advances in Quantum Chemistry presents surveys of current developments in this rapidly developing field. With invited reviews written by leading international researchers, each presenting new results, it provides a single vehicle for following progress in this interdisciplinary area. - Publishes articles, invited reviews and proceedings of major international conferences and workshops - Written by leading international researchers in quantum and theoretical chemistry - Highlights important interdisciplinary developments

Autoren/Hrsg.

Nicolaides, Cleanthes

Inhaltsverzeichnis

1;Cover;1
2;Advances in Quantum Chemistry;2
3;Copyright;5
4;Contents;6
5;Preface;10
6;Contributors;12
7;1. Unstable States: From Quantum Mechanics to Statistical Physics;16
7.1;1. Introduction;17
7.2;2. Quantum Resonances;18
7.3;3. Collisions;41
7.4;4. Statistical Physics;48
7.5;5. Concluding Remarks;57
7.6;Acknowledgment;57
7.7;Appendices;57
7.8;References;59
8;2. Unstable States in Laser Assisted and Controlled Molecular Processes;66
8.1;1. Introduction;67
8.2;2. General Theory of Laser–Molecule Interactions;70
8.3;3. Numerical Methodologies;83
8.4;4. Processes and Mechanisms for Molecular Fragmentations in IR and UV-VisFrequency Regimes;89
8.5;5. XUV+IR Pump–Probe Spectroscopy of Molecular Dissociative Ionization;96
8.6;6. ZWRs and EPs in Molecular Photodissociation;103
8.7;7. Conclusion;114
8.8;Acknowledgments;116
8.9;References;116
9;3. Coherence Effects in Laser-Induced Continuum Structure;120
9.1;1. Introduction;121
9.2;2. A Brief Historical Survey of LICS;122
9.3;3. EIT and its Connection with LICS;123
9.4;4. QC and LICS;143
9.5;5. The Connection between LICS, CC, and AP;150
9.6;6. Summary;164
9.7;Acknowledgments;165
9.8;References;165
10;4. Theory and State-Specific Methods for the Analysisand Computation of Field-Free and Field-Induced Unstable States in Atoms and Molecules;178
10.1;1. Beyond Pure Formalism: The Importance of Solving Efficaciously the Many-Electron Problem (MEP) for Unstable (or Nonstationary, or Resonance) States in the Field-Free and Field-Induced Spectra of Many-Electron Atoms and Molecules;182
10.2;2. Principal Characteristics of the Dominant Theoretical Approaches to the Computation of Unstable States in Atoms and Molecules Up to About the End of the 1960s-Early 1970s;188
10.3;3. Field-Free Hamiltonian: The Form of Wave functions for Resonance States in the Context of Time- and of Energy-Dependent Theories and its Use for Phenomenology and Computation;201
10.4;4. Aspects of the Nature and of the Preparation of 0 and of Its Connectionto the Resonance Eigenfunction;213
10.5;5. The Form of the Resonance Eigenfunction and the Complex Eigenvalue Schrödinger Equation;223
10.6;6. Computation via the CESE SSA. Many-Body Expansion and Partial Widths with Interchannel Coupling;229
10.7;7. Two Examples of Results from the Application of the CESE Approach;232
10.8;8. The State-Specific Calculation of .0;236
10.9;9. Understanding the Electronic Structures of Resonances and Their Effectson Spectra in the Framework of the SSA;249
10.10;10. The Use of f0s and of Scattering Wavefunctions in the SSEA for the Solution of the TDSE;260
10.11;11. Field-Induced Quantities Obtained as Properties of Resonance Statesin the Framework of the CESE-SSA;261
10.12;12. Conclusion and Synopsis;269
10.13;References;273
11;5. Quantum Theory of Reactive Scattering in Phase Space;284
11.1;1. Introduction;286
11.2;2. Phase Space Structures Underlying Reaction Dynamics;289
11.3;3. Quantum Normal Form Representation of the Activated Complex;305
11.4;4. The Cumulative Reaction Probability;311
11.5;5. Gamov–Siegert Resonances;321
11.6;6. Further Challenges;323
11.7;7. Conclusions;332
11.8;Appendix;333
11.9;Acknowledgments;344
11.10;References;344
12;6. The State-Specific Expansion Approach to the Solution of the Polyelectronic Time-Dependent Schrödinger Equation for Atomsand Molecules in Unstable States;348
12.1;1. Introduction;350
12.2;2. On the Nonperturbative Solution of the TDSE for Problems Where aGround or an Excited State of an Atom (Molecule) Interacts With a Strong Electromagnetic Pulse of Short Duration;358
12.3;3. AB Initio Computation of Unusual Time-Dependent Molecular Processes Using Judiciously Chosen Expansions for the .(t);364
12.4;4. The State-Specific Expansion Approach (SSEA);370
12.5;5. The Theory and Computation of Stationary State-Specific Wave functions for Low- and High-Lying States;383
12.6;6. Applications of the SSEA;397
12.7;7. Conclusion;412
12.8;References;413
13;7. Theory of Resonant States: An Exact Analytical Approach for OpenQuantum Systems;422
13.1;1. Introduction;423
13.2;2. Properties of Resonant States in 3D (The Half-Line in 1D);426
13.3;3. Extension to 1D (the Full Line);437
13.4;4. Scattering and Tunneling in the Energy Domain;441
13.5;5. Transient Phenomena;448
13.6;6. Conclusions;461
13.7;Acknowledgments;462
13.8;Appendices;463
13.9;References;467
14;8. Quantum Electrodynamics of One-PhotonWave Packets;472
14.1;1. Introduction;473
14.2;2. Quantum Electrodynamics of a Material Two-level System—Basic Aspects;475
14.3;3. Quantum Electrodynamics with Controlled Mode Selection;482
14.4;4. Conclusions and Outlook;495
14.5;References;496
15;9. Quantum Decay at Long Times;500
15.1;1. Introduction;501
15.2;2. Examples and Simple Models;506
15.3;3. Three-Dimensional Models of a Particle Escaping from a Trap;510
15.4;4. Physical Interpretations of Long-time Decay;516
15.5;5. The Problematic Experimental Observation;526
15.6;6. Enhancing Post-Exponential Decay via Distant Detectors;535
15.7;7. Final Comments;544
15.8;Acknowledgments;544
15.9;References;544
15.10;Index;552

Leseproben

Chapter 1

Unstable States: From Quantum Mechanics to Statistical Physics

Ivana Paidarováa [e-mail]; Philippe Durandb    a J. Heyrovský Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, v.v.i. 182 23 Praha 8, Czech Republic
b Laboratoire de Chimie et Physique Quantiques, IRSAMC, Université de Toulouse et CNRS, 31062 Toulouse cedex 4, France

Abstract

Influenced by the ideas of Jaynes and Prigogine from the mid-1950s, we present a unified formulation of dynamics and thermodynamics of irreversible processes. Our approach originates in the quantum theory of resonances described by effective Hamiltonians. The concept of effective Hamiltonian is extended to the concept of effective Liouvillian that deals with macroscopic observables and brings insight into the dissipative nonequilibrium thermodynamics. The time-energy/frequency Fourier–Laplace transformation and the use of projectors focus on the variables of interest. The line profiles and dynamics in quantum mechanics are treated on the same footing. The long macroscopic times in statistical physics are derived from short microscopic times by means of hierarchies of effective Liouvillians and perturbation theory in the complex plane. The theory is illustrated on solvable models of quasi–continua and continua related to fluctuations and dissipation and on a model of kinetics of a chemical reaction implying a short-lived (resonance) transition state.

1 INTRODUCTION

Unstable states and resonances are ubiquitous in nature. Their timescales vary in a huge range from a few femtoseconds for molecular excited states, over seconds in our world, to millions of years for the solar system stability. The present study is devoted to resonances in molecules. Within this domain the characteristic times vary typically from femtoseconds to seconds. We focus on the dynamical and spectroscopic properties of typical quantum irreversible processes. Another aspect of this study is understanding the pathway from the microscopic to the macroscopic world, the traditional subject of statistical physics. From a mathematical viewpoint the irrelevant degrees of freedom are described in terms of discrete and continuous spectra. With the aim to provide generic results, we use a unique theoretical scheme based on the Fourier–Laplace transformation and projectors. We advocate the advantages to work with the variable energy extended in the complex plane instead of solving directly the Schrödinger equation with the variable time. The theory benefits in this way from the “rigidity” properties of analytic functions, in particular, from analytical continuation. In addition, the Fourier–Laplace transform in quantum mechanics establishes a direct link between the dynamics and the spectroscopies. The systematic use of projectors focuses on the observables of interest and on the derivation of their effective interactions. There is some freedom in the choice of these observables which may be, however, crucial and depends on the relevant timescales. For example, the chemical kinetics can be investigated by either considering or excluding the transition states: the choice depends on the characteristic times of the experiment.

Since the field investigated in this review is broad, it is difficult to attribute a precise definition to the resonances, quasi-bound states, unstable states, etc. We have in mind that the terms unstable states and resonances belong to the general scientific and technical vocabulary. Narrow and broad resonances are found in electricity, mechanics, as well as in the spectroscopies where they are associated with the narrow and broad profiles (bumps). On the other side, the terms bound states and quasi-bound states will be used in their usual meaning in quantum mechanics. We will not discuss the mathematical properties of the wavefunctions of the quasi-bound states and of the continua. Regarding the differences between resonances, quasi-bound states and the mathematical aspects of the theory, the reader is advised to read the proceedings of the Uppsala Resonance Workshop in 1987 [1, 2]. These two references, which represent the state of the art in the 1980s, focus on the spectral theories of resonances, whereas our review is mainly centered on Green functions.

To follow the scale of complexity, the review is divided into three parts. The first two parts deal with the key concept of effective Hamiltonians which describe the dynamical and spectroscopic properties of interfering resonances (Section 2) and resonant scattering (Section 3). The third part, Section 4, is devoted to the resolution of the Liouville equation and to the introduction of the concept of effective Liouvillian which generalizes the concept of effective Hamiltonian. The link between the theory of quantum resonances and statistical physics and thermodynamics is thus established. Throughout this work we have tried to keep a balance between the theory and the examples based on simple solvable models.

2 QUANTUM RESONANCES

2.1 Theory

The dynamics of a system described by the Hamiltonian H is characterized by the evolution operator

(t)=e-iHt/h.

  (1)

Instead of considering U(t) it is useful to investigate its Fourier–Laplace transform, the resolvent or Green operator

[z]=1z-H.

  (2)

z is the energy extended in the complex plane. The operator U(t) is recovered by the inverse Laplace transformation (see Appendix A). If we are interested in the dynamics of a small number of n quasi-bound states (resonances) there is too much information in G[z] and we project the Green function into the inner space of these states. The partition technique (see appendix B and Chap. 4 of Ref. [3]) provides

z-HP=O(z)Pz-Heff(z);O(z)=P+Qz-HHP;Q=1-P.

  (3)

P projects into the inner space of n unstable quasi-bound states which are either true observable resonances or wave packets in the continuum (decay channels) taking a significant part in the dynamics. These latter may be either strongly or weakly coupled to the resonances. We shall treat in the same way the resonances and the quasi-bound states of interest and hence we call equally “resonance” or “quasi-bound state” any state belonging to the inner space. Q = 1 – P projects onto the complementary outer space. The energy-dependent wave operator O(z) establishes a one-to-one correspondence between the n states belonging to the inner space and the states belonging to the outer space. O(z) extends the concept of wave operator previously defined for bound states [4,5]. Let us transform the expression on the left of Eq. (3) into the basic equation

z-H)O(z)=P[z-Heff(z)].

  (4)

This equation generalizes an inhomogeneous Schrödinger equation which was given, almost one half a century ago, by Löwdin in the framework of the partitioning technique (see Eq. (46) in Ref. [6] and Appendix B). Denoting by e(z), f(z) an eigensolution of Heffz continued into the second Riemann sheet, and multiplying both sides of Eq. (4) (on the right) by f(z) leads to

z-H)O(z)f(z)=[z-e(z)]f(z).

  (5)

At the poles of the resolvent, when z = e(z), Eq. (5) reduces to the Gamow–Siegert equation

?(z)=z?(z);?(z)=O(z)f(z).

  (6)

?(z) describes a decreasing in time quasi-stationary state. Contrary to the Lippmann–Schwinger equation, which requires scattering boundary conditions, ?(z) does require outgoing boundary conditions commensurate with the Gammow–Siegert method. It is inherent in the complex technique and defined in a nonambiguous manner as a continued wavefunction in the second Riemann sheet.

To investigate the resonances, the useful part of the resolvent is projected into the inner space. Multiplying Eq. (3) on the left by P and using the intermediate normalization P = PO(z) results in

1z-HP=Pz-Heff(z).

  (7)

Equation (7) shows the significance of the effective Hamiltonian which is directly related to the spectroscopic and dynamical observables, as lineshapes (see the end of this section) and transition probabilities. The effective Hamiltonian can be written as the sum of the projection of the exact Hamiltonian into the inner space and of the energy-shift operator...