Stationary resonance state

In a time-dependent picture, resonances can be viewed as localized wavepackets composed of a superposition of continuum wavefimctions, which qualitatively resemble bound states for a period of time. The unimolecular reactant in a resonance state moves within the potential energy well for a considerable period of time, leaving it only when a fairly long time interval r has elapsed r may be called the lifetime of the almost stationary resonance state. 

This is the manner at which the wave packet in Section 2 evolves just outside the interaction region as portrayed in Figure 1.7 and Eq. (23), where rj(t) = e rt/2 A . There we got that the exponential increase of the amplitude goes exactly as the imaginary part of the complex momentum of the stationary state = Im[kres] = 0.0048[a.u.]. This shows that even outside the barriers, there is a region where the evolution is similar to that of the stationary resonance state. 

Atomic and Molecular Energy Levels. Absorption and emission of electromagnetic radiation can occur by any of several mechanisms. Those important in spectroscopy are resonant interactions in which the photon energy matches the energy difference between discrete stationary energy states (eigenstates) of an atomic or molecular system = hv. This is known as the Bohr frequency condition. Transitions between... 

At this point, we have a comprehensive picture of quantum resonance states. The discussion above shows that the non-Hermitian stationary resonance solutions of Section 3 are real flesh and blood beings even in the Hermitian world dictated by the TDSE. As a wave packet evolves with time,... 

Despite being called a state, a resonance does not show up as an eigenstate of an Hermitian Hamiltonian. However, as it represents a particle state that is localized in space for some time and that delocalizes with a small but finite rate, a resonance is reminiscent of a stationary state, but with a decaying norm. Indeed, it can be shown that we can represent resonance states as eigenstates of a non-Hermitian Hamiltonian, whose complex eigenvalues lie in the lower half of the complex plane. 

The decay of Nal can be described in an alternative way [K.B. Mpller, N.E. Henriksen, and A.H. Zewail, J. Chem. Phys. 113, 10477 (2000)]. In the bound region of the excited-state potential energy surface, one can define a discrete set of quasi-stationary states that are (weakly) coupled to the continuum states in the dissociation channel Na + I. These quasi-stationary states are also called resonance states and they have a finite lifetime due to the coupling to the continuum. Each quasi-stationary state has a time-dependent amplitude with a time evolution that can be expressed in terms of an effective (complex, non-Hermitian) Hamiltonian. 

When the reaction probability of the activated molecule A E) is sufficiently small, see Example 7.1 for an illustration, one can introduce the concept of quasi-stationary states. These states, n), associated with the slowly decaying bound states of A (E) are also called resonance states. The dynamics of the states is, essentially, given by [3] U(t) n) = e l(E lTnl 1 )tlh n). Note that, if n) was a true stationary state, i.e., an eigenstate of the full Hamiltonian Hi, then T = 0. The probability of observing the molecule in a quasi-stationary state is given by U(t) n) 2 = c that is, a... 

The general depiction of a one-color, three-photon ionization contains four experimental outcomes. These occur when the energy of the multiple of excitation photons (a) does not match a stationary excited state of the system or (b) matches some stationary resonance RS state of the system or when the ion is produced (c) in an excited, ion state (/ ), or (d) in excited, continuum states which, thereafter, ionize (i.e. ATI). The kinetic energy of the ejected electron in (b) is KE = 3hv - /. In cases (c) and (d), electrons of energy KE2 = 3hv -I and KE3 = 4hv - /, respectively, are observed (Figure 4). 

At the end of this section, one should note, that the quantum mechanical system in a potential cavity of large size may also be described as a system of discrete spectrum states with energies being almost constant and a system of decreasing stationary states for an enlarged "potential box". It is well known that the system of typical "ladder" structures in the spectrum gives rise to resonant states of molecular systems (see e.g. [92,93] and references therein), but the resonant states (the poles of continuation in a complex plane for the resolvent matrix elements) are a special field of Quantum Science and we shall not consider them here (see [5], Sect. XII.6 for discussions, or [55], Sect. 3). 

Now we are ready to discuss the energy splitting of molecular Fermi resonance states. From the classical point of view, the stationary states correspond to a purely periodic dependence of the amplitudes A and B on time ... 

Figure 4 A diagram of the four-level model of a dye molecule in solution is shown. The equilibrated ground state, 1, is surrounded by a solvent cage. Upon transition to the Franck-Condon excited state, 2, the nuclear configuration of the dye and the solvent cage remains stationary. Resonance fluorescence from the Franck-Condon state is shown. Equilibration in the excited state of the dye molecule and the surround solvent is obtained in level 3. The major portion of fluorescence takes place between levels 3 and 4, although emission also occurs continuously from the intermediate levels (Reproduced by permission from Chem. Phys. Letters, 1975, 32, 476)...

We illustrate this approach with a model that has been developed (Qian et al., 1990 Ogai et al., 1992 Solter et al., 1992 Reisler et al., 1994) to interpret rotational state distributions of the diatomic fragment BC following the dissociation of resonance states in the triatomic molecule ABC. The resonances (see chapter 8) are identified by the quantum numbers v, Vj) and Vj, which refer to BC stretch, AB stretch, and bending vibrations, respectively. In the model, very weak coupling is assumed between the A— BC bending mode and the BC vibration, the decay is viewed as a direct dissociation process which starts at the transition state, with the stationary wave function for the... 

BEYOND PURE FORMALISM THE IMPORTANCE OF SOLVING EFFICACIOUSLY THE MANY-ELECTRON PROBLEM (MEP) FOR UNSTABLE (OR NON STATIONARY, OR RESONANCE) STATES IN THE FIELD-FREE AND FIELD-INDUCED SPECTRA OF MANY-ELECTRON ATOMS AND MOLECULES... 

In Fano s [29] formal theory of resonance states, the energy-dependent wavefunctions are stationary, the energies are real, and the formalism is Hermitian. The observable quantities, such as the photoabsorption cross-section in the presence of a resonance, are energy-dependent and the theory provides them in terms of computable matrix elements involving prediagonalized bound and scattering N-electron basis sets. The serious MEP of how to compute and utilize in a practical way these sets for arbitrary N-electron systems is left open. 

In the first case, the energy corresponds to the position of the doubly excited resonance state. He "2s2p" P°. The energy-dependent, stationary state description of the excitation is... 

Quantum transients are temporary features that appear in the time evolution of matter waves before they reach a stationary regime. They usually arise as a result of a sudden switch interaction that modifies the confinement of particles in a spatial region or after the preparation of a decaying sfafe [48]. The archetypical quantum transient phenomena is diffraction in time which consists of the sudden opening of a shutter to release a semi-infinite beam producing temporal and spatial oscillations of the time evolving wave [49]. A common feature in the mathematical description of quantum transient phenomena is the Moshinsky function, which as we have seen is closely related to the Faddeyeva function. Since in a recent review [48] there appears a discussion on transient phenomena for the dynamics of tunneling based on the present resonant state formalism [54, 66-76], here we restrict the discussion to the time evolution of quantum decay. 

Thus, resonance states may also hide in that part of the eontinuum that has energy higher than the barriers (with a short lifetime because sueh resonances are wide cf.. Fig. 4.9). They are also a reminder of the stationary states of the particle in a box longer than the separation of the barriers and infinite well depth. 

Above the dissociation limit, one is dealing with a continuum of states of the dissociation products with kinetic energy. In such a continuum, one may have also the resonance states, which may have wave functions that resemble those of stationary states but differ from them by having finite lifetimes. 

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