Stationary states, excited

Fig. 16.6. Schematic illustration of the dissociation of Nal. The bell-shaped curve on the vertical axis represents the distribution of stationary states excited by the short pump pulse. Reproduced from Zewail (1988).

All the previous discussion in this chapter has been concerned with absorption or emission of a single photon. However, it is possible for an atom or molecule to absorb two or more photons simultaneously from a light beam to produce an excited state whose energy is the sum of the energies of the photons absorbed. This can happen even when there is no intemrediate stationary state of the system at the energy of one of the photons. The possibility was first demonstrated theoretically by Maria Goppert-Mayer in 1931 [29], but experimental observations had to await the development of the laser. Multiphoton spectroscopy is now a iisefiil technique [30, 31]. 

Final state analysis is where dynamical methods of evolving states meet the concepts of stationary states. By their definition, final states are relatively long lived. Therefore experiment often selects a single stationary state or a statistical mixture of stationary states. Since END evolution includes the possibility of electronic excitations, we analyze reaction products in terms of rovibronic states. 

The proposed model for the so-called sodium-potassium pump should be regarded as a first tentative attempt to stimulate the well-informed specialists in that field to investigate the details, i.e., the exact form of the sodium and potassium current-voltage curves at the inner and outer membrane surfaces to demonstrate the excitability (e.g. N, S or Z shaped) connected with changes in the conductance and ion fluxes with this model. To date, the latter is explained by the theory of Hodgkin and Huxley U1) which does not take into account the possibility of solid-state conduction and the fact that a fraction of Na+ in nerves is complexed as indicated by NMR-studies 124). As shown by Iljuschenko and Mirkin 106), the stationary-state approach also considers electron transfer reactions at semiconductors like those of ionselective membranes. It is hoped that this article may facilitate the translation of concepts from the domain of electrodes in corrosion research to membrane research. 

A little later (1929) the Russian physicist Andronov pointed out3 that the stationary state of self-excited oscillations discovered by van der Pol is expressible analytically in terms of the limit cycle concept of the theory of PoincarA... 

We proceed by generalizing the stationary state approach of the previous subsection to the time domain. To that end we first consider the effects of a finite laser pulsewidth and next explore the effect of a finite time delay between the pulses. The latter signal reduces in the case of identical excitation pathways to the method of two-photon interferometry that has been studied extensively in the literature of solids and surfaces [79]. 

Time-dependent response theory concerns the response of a system initially in a stationary state, generally taken to be the ground state, to a perturbation turned on slowly, beginning some time in the distant past. The assumption that the perturbation is turned on slowly, i.e. the adiabatic approximation, enables us to consider the perturbation to be of first order. In TD-DFT the density response dp, i.e. the density change which results from the perturbation dveff, enables direct determination of the excitation energies as the poles of the response function dP (the linear response of the KS density matrix in the basis of the unperturbed molecular orbitals) without formally having to calculate a(co). 

The interpretation of the correlation effects in the 3-RDM is slightly more complicated than in the 2-RDM case, although the electrons also avoid each other here by imdergoing virtual excitations. In this case, two different correlation mechanisms contribute to the overall effect. In the first mechanism, one particle is in a stationary state - the ground or an excited state - while the two other particles undergo the same kind of virtual excitations as in the 2-RDM. In the second mechanism, the cycle of transitions involves three states instead of two. Finally the role played by the hole is more complex here, since the 1-HRDM elements multiply a 2-RDM element and a cycle of two transitions. 

If a system is disturbed by periodical variation of an external parameter such as temperature (92), pressure, concentration of a reactant (41,48,65), or the absolute configuration of a probe molecule (54,59), then all the species in the system that are affected by this parameter will also change periodically at the same frequency as the stimulation, or harmonics thereof (91). Figure 24 shows schematically the relationship between stimulation and response. A phase lag <)) between stimulation and response occurs if the time constant of the process giving rise to some signal is of the order of the time constant Inim of the excitation. The shape of the response may be different from the one of the stimulation if the system response is non-linear. At the beginning of the modulation, the system relaxes to a new quasi-stationary state, about which it oscillates at frequency cu, as depicted in Fig. 24. In this quasi-stationary state, the absorbance variations A(v, t) are followed by measuring spectra... 

There is another way of looking at this coupled ion system, namely, in terms of stationary states. From this point of view, one considers that the excitation belongs to both ions simultaneously. To determine the wave functions of the two-ion system, one resorts to degenerate perturbation theory. The coupling H can be shown to remove the degeneracy, and two new states that are mixtures of X20 and X11 are formed. For each the excita-... 

Another form of behaviour exhibited by a number of chemical reactions, including the Belousov-Zhabotinskii system, is that of excitability. This concerns a mixture which is prepared under conditions outside the oscillatory range. The system sits at the stationary state, which is stable. Infinitesimal perturbations decay back to the stationary state, perhaps in- a damped oscillatory manner. The effect of finite, but possibly still quite small, perturbations can, however, be markedly different. The system ultimately returns to the same state, but only after a large excursion, resembling a single oscillatory pulse. Excitable B-Z systems are well known for this propensity for supporting spiral waves (see chapter 1). 

The driving force for this excitable behaviour can be revealed from the model studied above, again assuming e is small and looking at the nullclines in the phase plane. A suitable orientation of the curves /(a, 0) = 0 and g(tx, 0) = 0 is shown in Fig. 5.11. The stationary-state intersection lies outside the range of instability , i.e. just before the maximum in the g(a, 0) = 0 nullcline, reflecting stability. If a small perturbation momentarily decreases a, or induces either a decrease or a very small increase in 0, the system merely jumps back on to that nullcline and then moves along it to the intersection... 

Fig. 5.11. Excitability in a chemical system, (a) The nullclines /(a, 0) = 0 and g(a,0) = 0 intersect just to the left of the maximum. A suitable perturbation must make a full circuit, as shown by a typical trajectory, before returning to the stable stationary state, (b), (c) The corresponding evolution of the concentration of intermediate A and the temperature excess in time showing the large-amplitude excursion.

The semiclassical treatment just given has the defect of not predicting spontaneous emission. According to (3.13), if there is no outside perturbation, that is, if // (0 = 0, then dcm/dt = 0 for all m if the atom is in the nth stationary state at / = 0, it will persist in that state forever. However, experimentally we find that unperturbed atoms in excited states spontaneously radiate energy and drop to lower states. Quantum field theory does predict spontaneous emission. Since quantum field theory is beyond us, we shall use an argument given by Einstein in 1917 to find the spontaneous-emission probability. 

In the experiments described thus far, the pump laser simply populates the intermediate excited state. The consequence is that the experiment becomes a means to study that excited state. Often we are more concerned with learning about the ground state than about excited states. For this purpose, it is useful to prepare a vibrational wavepacket of that ground state. One useful means to do this is to excite the species of interest to an allowed excited state and then to down-pump from that excited state back to the ground state, with a pulse that generates a packet rather than a stationary state. The simplest way to do this currently seems to be to raise the power level of the pulsed pump laser [26]. This process is shown schematically in Fig. 7. 

It is very important, in the theory of quantum relaxation processes, to understand how an atomic or molecular excited state is prepared, and to know under what circumstances it is meaningful to consider the time development of such a compound state. It is obvious, but nevertheless important to say, that an atomic or molecular system in a stationary state cannot be induced to make transitions to other states by small terms in the molecular Hamiltonian. A stationary state will undergo transition to other stationary states only by coupling with the radiation field, so that all time-dependent transitions between stationary states are radiative in nature. However, if the system is prepared in a nonstationary state of the total Hamiltonian, nonradiative transitions will occur. Thus, for example, in the theory of molecular predissociation4 it is not justified to prepare the physical system in a pure Born-Oppenheimer bound state and to force transitions to the manifold of continuum dissociative states. If, on the other hand, the excitation process produces the system in a mixed state consisting of a superposition of eigenstates of the total Hamiltonian, a relaxation process will take place. Provided that the absorption line shape is Lorentzian, the relaxation process will follow an exponential decay. 

ATOMIC ENERGY LEVELS. 1. The values of the energy corresponding to the stationary states of an isolated atom. 2. The set of stationary states in which an atom of a particular species may be found, including the ground state, or normal state, and the excited states,... 

Fig. 4.1. Schematic illustration of the evolution of a one-dimensional time-dependent wavepacket in the upper electronic state. The wavepacket is complex for t > 0 only its real part is shown here. Note that the upper horizontal axis does not correspond to a particular energyl The wavepacket is a superposition of stationary states corresponding to a broad range of energies, which are all simultaneously excited by the infinitely short light pulse indicated by the vertical arrow.

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