Two bound state

Figure 7.24 (a) The repulsive ground state and a bound excited state of Hc2. (b) Two bound states... 

This expression is valid for any state, as long as ( n. The factor v%2 accounts for the fact that v2t) is normalized per unit energy. The overlap integral (vb v2tB) between two bound state wavefunctions would not have the factor v22 and would be equal to one for v2 = vh and zero for v2 different from vb by an integer, as expected. We can write the cross section as... 

The unique properties of dilute metal-ammonia solutions depend not upon the nature of the metal species, but upon the solvated electron common to all these solutions. Thus, the electron-in-a-cavity model (17, 19, 21) seems best suited to describe the species present in these solutions since the model is independent of the type of cation present. Jortner and his associates (15, 16) have extended this model by assuming that the cavity arises from polarization of the medium by the electron. The energy levels of the bound electrons are obtained by using a potential function containing the static and optical dielectric constants of the bulk medium as parameters. Using one-parameter hydrogen-like wave functions for the first two bound states of the electron, the total energy of the ith state is expressed as... 

A popular case studied is V(r) = 7.5r2 exp(—r), which does not contain any bound states (only resonances, see more below) and modifies the Coulomb spectrum accordingly. As we will see later these formulas are easily generalized to the complex plane by contour integration. In Figure 2.4, we show the integration contour for the so-called Cauchy representation of m, in the simple case of two bound states, and the cut along the positive real axis. 

The intensities factorize into an ordinary (multi-dimensional) FC overlap factor involving two bound states and a dynamical, energy-dependent factor which reflects the quenching of each state. 

Here 3/, = Ex — av = tuox — (Ek + ))/2. The second term in Eq. (5.47), in brackets, is a function of TxSEfi. If two bound state levels dominate the pump excitation, then this term contributes a scaling characteristic to control plots. That is, if we plot contours of constant dissociation probability as a function of A, and 5E.k then, barring the first term, plots with different Tx will appear similar, with a new range scaled by 5E, = (Tx/T )2SEjt. >... 

To appreciate the essence of this control scenario, recall the results of applying a laser pulse a(t) to induce a transition between two bound states Ef) and , ). We denote the dipole transition matrix element between these two states by d,- - and define k,j = 2dpopulation transfer between these levels can be accomplished by using a n pulse, that is, a pulse of duration t satisfying... 

Since at t = 0 the first element of U and of U(3) is zero, then the two nontrapped adiabatic states are orthogonal to the initial state [fij) at the beginning of the process. > Hence, the only adiabatic state populated initially is the trapped state ) A, (r)). If there f is no coupling between the three adiabatic states, the system will continue to evolve t fas the trapped state [Ai (/)), executing an adiabatic passage to the E2) state as / t - oo. Thus we can achieve the control objective of complete population transfer k between two bound states. 

The one-dimensional potential depicted in Fig. 7(a) provides an illustration of this effect. The Schrodinger equation can be solved with the method used for the square-well case above. Each well gives rise to a nearly independent progression of states. For fi = 2p = 1 and other potential parameters indicated in Fig. 7 one finds that the system has two bound states and a resonance state at 9.46 — i 0.11 localized above the deep outer well. There is also another resonance in the system, E = 9.8 - i 0.002. Its width is very small because this state belongs to the shallow inner well, which is separated from the continuum by a potential barrier. Suppose that we force — by varying a parameter in the Hamiltonian — the narrow state (denoted n) in the shallow well to move across the broader resonance (b) belonging to the deep minimum. The relative positions of the two states can be, for example, controlled by shifting the infinite wall at the... 

The pair of levels 21s - (16,3) is exactly analogous to the extreme blue and red Na Stark states of n and + 1. The fact that only one has a permanent dipole moment is of no consequence it is only the difference in the permanent moments which is significant. Based on the single cycle Landau-Zener description of microwave ionization we expect that if atoms in the 18s state are exposed to a microwave field of amplitude equal to the crossing field, Eq = 753 V/cm, they would make transitions to the (16,3) state. On the other hand, if a static field is present as well as the microwave field it should be possible to see resonant microwave multiphoton transitions between these two bound states, and seeing the connection between these processes is part of our objective. 

Although the spectroscopy of combination modes in molecular crystals has been studied in some detail, information on the decay times, and hence the dynamics, of these states is available only in a few cases and specifically in the Vi-2v2 region of CSj, NjO, and COj, three almost isomorphous crystals. Nevertheless the dynamics of the bound states, as exhibited by the temperature variation of their relaxation times, varies dramatically from one crystal to another. Thus while the decay time of the 2v2 mode of N2O is practically independent of temperature, the relaxation times of the two bound states 2 andfi" associated with the v,2v2 Fermi resonance in CO2 show a very strong temjjerature dependence (see Fig. 7). 3,144... 

This behavior may be understood intuitively by supposing that the natural relaxation channel for the bound states will be into the parent two-phonon quasi-continuum. In the case of N2O, where it appears that the bound state (or resonance) may be degenerate with the tail of the two-phonon quasi-continuum, this mechanism is im.mediate leading to the experimentally observed temjjerature-insensitive lifetime. For CO2, where strong Fermi resonance produces two bounds states and 2, and pushes them outside the continuum, an additional phonon is required to access the continuum. As Q lies below, and fi above the two-phonon continuum, this process requires the absorption or emission, respectively, of a low-frequency phonon leading to the following simplified expression for the inverse life-times F and F+... 

We focus here into the radiative processes that modify the ion charge state, i.e., those that involve the capture of an electron from the target valence band. We do not include radiative relaxation processes in which one electron decays between two bound states of the ion, although they can be of the same importance in the description of a realistic neutralization and relaxation... 

In this way, we may picture orbital contraction as a purely quantum-mechanical effect, which arises from the existence of a short range well within the atom. The binding strength of this well increases with atomic number and, as a result, the critical condition for the appearance of the first bound state is satisfied around Z = 56. The condition for two bound states to occur inside the inner well is satisfied in a similar way at the onset of the 5/ period, giving rise to the actinide sequence. 

Another way of considering the problem which is perhaps physically more meaningful is that in fact the two bound states 0 > and 1 > are coupled to each other by the two lasers, in one case via a bound virtual state (the Raman path) and in the other via the continuum (the autoionising path) as marked in fig. 8.4. It turns out that, if the Raman channel dominates, the resulting lineshapes tend to become symmetric, while if the autoionisation channel dominates, the characteristic interference asymmetries of Beutler-Fano resonances emerge. ... 

In terms of the simple model of adsorbed water (8), environmental state B (bulk-like water) is first removed from the pore during dehydration leaving behind the two bound states Ai and 2 (or the combined state A). The combined state A is assumed to occupy two layers of water and is independent of porous glass pore diameter. See Table II in Reference (. ... 

To date only three reaction-dissociation calculations have been reported by Manz and Romelt /Kaye and Kuppermann and Leforestier. The former two used the hyperspherical coordinates method to study model X-X2 systems,bearing respectively one and two bound states asymptotically. The latter one,whose results are presented on figure 3,will be discussed below it corresponds to a model H-HD system,HD and H2 bearing respectively 7 and 6 bound states. 

As discussed in Section II, the Ij and CS2 molecules show intramolecular magnetic quenching while the radiationless process associated with the magnetic quenching of I2 involves the transition between the bound state and the dissociating state, the radiationless process associated with the magnetic quenching of CS2 involves the transition between the two bound states. 

Figure 1 Cuts through the potential energy surfaces for the ground and excited states of CHj I. Also shown are the locations of two bound states at Ej and Eg and the particular excitation energy E=37593.9 cm associated with Figures 2-4.

The spectral distribution of the radiant flux from a source is called its emission spectrum. The thermal radiation discussed in Sect. 2.2 has a continuous spectral distribution described by its spectral energy density (2.13). Discrete emission spectra, where the radiant flux has distinct maxima at certain frequencies Vik, are generated by transitions of atoms or molecules between two bound states, a higher energy state Ek and a lower state Ei, with the relation... 

Dirac Equation Generalized for Two Bound-State Electrons... 

Finally, a brief comment on the case of multiple bound states is made. Suppose there are two bound states whose eigenvalues are k and A2 (< Ai). Quite analogously to (6.3.26), we have the following approximate expression for 0 ... 

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