Resonance state quantum mechanical, time-dependent

A QUANTUM MECHANICAL RESONANCE STATE FROM A TIME-DEPENDENT PERSPECTIVE... 

Different theoretical methods have been used to calculate the complex energies, Eq. (8.1), for compound-state resonances. They can be divided into time-independent and time-dependent methods. A standard quantum mechanical time-independent method is a close-coupling calculation (Stechel et al., 1978) which considers resonant state formation as a result of a collision such as A + BC —> ABC AB + C. Determined... 

Time-dependent quantum mechanical calcnlations have also been perfomied to study the HCO resonance states [90,91]. The resonance energies, linewidths and quantum number assigmnents detemiined from these calcnlations are in excellent agreement with the experimental results. 

Quantum mechanical effects—tunneling and interference, resonances, and electronic nonadiabaticity— play important roles in many chemical reactions. Rigorous quantum dynamics studies, that is, numerically accurate solutions of either the time-independent or time-dependent Schrodinger equations, provide the most correct and detailed description of a chemical reaction. While hmited to relatively small numbers of atoms by the standards of ordinary chemistry, numerically accurate quantum dynamics provides not only detailed insight into the nature of specific reactions, but benchmark results on which to base more approximate approaches, such as transition state theory and quasiclassical trajectories, which can be applied to larger systems. 

A relaxation process will occur when a compound state of the system with large amplitude of a sparse subsystem component evolves so that the continuum component grows with time. We then say that the dynamic component of this state s wave function decays with time. Familiar examples of such relaxation processes are the a decay of nuclei, the radiative decay of atoms, atomic and molecular autoionization processes, and molecular predissociation. In all these cases a compound state of the physical system decays into a true continuum or into a quasicontinuum, the choice of the description of the dissipative subsystem depending solely on what boundary conditions are applied at large distances from the atom or molecule. The general theory of quantum mechanics leads to the conclusion that there is a set of features common to all compound states of a wide class of systems. For example, the shapes of many resonances are nearly the same, and the rates of decay of many different kinds of metastable states are of the same functional form. 

The first volume contained nine state-of-the-art chapters on fundamental aspects, on formalism, and on a variety of applications. The various discussions employ both stationary and time-dependent frameworks, with Hermitian and non-Hermitian Hamiltonian constructions. A variety of formal and computational results address themes from quantum and statistical mechanics to the detailed analysis of time evolution of material or photon wave packets, from the difficult problem of combining advanced many-electron methods with properties of field-free and field-induced resonances to the dynamics of molecular processes and coherence effects in strong electromagnetic fields and strong laser pulses, from portrayals of novel phase space approaches of quantum reactive scattering to aspects of recent developments related to quantum information processing. 

In this chapter, we discussed the principle quantum mechanical effects inherent to the dynamics of unimolecular dissociation. The starting point of our analysis is the concept of discrete metastable states (resonances) in the dissociation continuum, introduced in Sect. 2 and then amply illustrated in Sects. 5 and 6. Resonances allow one to treat the spectroscopic and kinetic aspects of unimolecular dissociation on equal grounds — they are spectroscopically measurable states and, at the same time, the states in which a molecule can be temporally trapped so that it can be stabilized in collisions with bath particles. The main property of quantum state-resolved unimolecular dissociation is that the lifetimes and hence the dissociation rates strongly fluctuate from state to state — they are intimately related to the shape of the resonance wave functions in the potential well. These fluctuations are universal in that they are observed in mode-specific, statistical state-specific and mixed systems. Thus, the classical notion of an energy dependent reaction rate is not strictly valid in quantum mechanics Molecules activated with equal amounts of energy but in different resonance states can decay with drastically different rates. 

The main obstacle for calculating temperature and pressure dependent rate constants according to Eq. (76) or variants of it is the need for state-resolved dissociation rates for high rotational states. To perform exact quantum mechanical calculations for J = 40, for example, is not possible at present time even for a triatomic molecule, especially when it consists of three heavy atoms like O3. Until now, except for very few studies — HCO and HOCl, for example, discussed in Sect. 5 — most studies of resonance widths have been performed for J = 0. However, even at temperatures well below room temperature the molecules with J — 0 form only a small fraction of the ensemble. The common way of evaluating the resonance... 

The two previous secfions were devoted to modeling quantum resonances by means of effective Hamiltonians. From the mathematical point of view we have used two principal tools projection operators that permit to focus on a few states of interest and analytic continuation that allows to uncover the complex energies. Because the time-dependent Schrodinger equation is formally equivalent to the Liouville equation, it is attractive to try to solve the Liouville equation using the same tools and thus establishing a link between the dynamics and the nonequilibrium thermodynamics. For that purpose we will briefly recall the definition of the correlation functions which are similar to the survival and transition amplitudes of quantum mechanics. Then two models of regression of a fluctuation and of a chemical kinetic equation including a transition state will be presented. 

The relationship of the Rigged-Hilbert space formulation of quantum mechanics with the formalism of resonant states discussed here requires to be clarified. For example, it is not clear that the complex poles seated on the third quadrant of the complex k plane and their corresponding resonant states U-n play a role in that formalism for times f > 0. As we have shown, these poles are essential to obtain the long time 1/P behavior of decay. In facf the time evolution of decay in terms of fhe resonant state formalism coincides exactly with the numerical solution to the time-dependent Schrbdinger equation of the problem. At a more fundamental level, the above issue is related to the understanding of irreversibility at the quantum level, where the formation and decay of transient states play a fundamental role [93,99]. Further work is required on this issue. 

Experimental and theoretical interest in USCSs has existed since the early days of quantum mechanics. For example, a textbook picture of such an unstable state is that of the one-dimensional potential with a local minimum and a finite barrier that is used to explain, in terms of quantum mechanical tunneling, the instability of a nucleus, the concomitant emission of an alpha particle, and ifs energy. Another textbook example of basic importance is the formal construction of a wave packet from a superposition of a complete set of stationary states and the determination, at least for simple one-dimensional motion, of its time evolution. Finally, another example often presented in books is the appearance of structures ("peaks") in the energy-dependent transition rates (cross sections) over the smoothly varying continuum characterizing a physico-chemical process, which are normally called resonances and which are associated with the transient formation of USCSs. 

From time-dependent perturbation theory of quantum-mechanics, it can be stated that a transition between two states ir) and ) is allowed provided that (Vf 77p ) 0. This takes place if v vq (ie, the resonance condition) and the alternative magnetic field Bi(t) is polarized perpendicularly to the static magnetic field Bo. Concerning a spin 7 = 1 (Fig. lb), similar calculations show that only the single-quantum transitions 0) 1> and -1) 0> (and those in the opposite directions) are allowed in the first approximation and occur at the same frequency, given by equation 3. 

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