Unstable States: From Quantum Mechanics to Statistical Physics

Ivana Paidarováa ivana.paidarova@jh-inst.cas.cz; 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

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/ħ.

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

[z]=1z−H.

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=Ω(z)Pz−Heff(z);Ω(z)=P+Qz−HHP;Q=1−P.

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 Ω(z) establishes a one-to-one correspondence between the n states belonging to the inner space and the states belonging to the outer space. Ω(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)Ω(z)=P[z−Heff(z)].

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 ε(z), φ(z) an eigensolution of Heffz continued into the second Riemann sheet, and multiplying both sides of Eq. (4) (on the right) by φ(z) leads to

z−H)Ω(z)φ(z)=[z−ε(z)]φ(z).

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

ψ(z)=zψ(z);ψ(z)=Ω(z)φ(z).

ψ(z) describes a decreasing in time quasi-stationary state. Contrary to the Lippmann–Schwinger equation, which requires scattering boundary conditions, ψ(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 = PΩ(z) results in

1z−HP=Pz−Heff(z).

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...