Resonant expansion

Thus, the fractional improvement of the two-term resonance expansion is... 

This work discusses some rigorous analytical results for resonant states. It will be mainly concerned with full discrete resonant expansions, i.e., the second class of expansions mentioned above. We shall restrict the discussion to coherent (elastic) processes in one (ID) and three (3D) dimensions. It is worth mentioning, however, that there is work extending the formalism of resonant states to two dimensions (2D) [35,36] and also describing incoherent processes [35, 37]. 

Figure 7.1 Integration contour C = Q + cj + c used to obtain the resonant expansion of G+(r,r k ) in terms of the full set of complex poles. See text.

In this section we refer to fully one-dimensional problems to show that along the internal region of the interaction and for tunneling transmission, purely resonant expansions provide an exact description of the corresponding continuum wave solutions to the Schrbdinger equation. 

Since at x = x = 0, the expansion of the outgoing Green s function is divergent, it is not possible to obtain a purely discrete expansion of r k). As discussed in Ref. [18] for the half-line, the expansion for the reflection amplitude requires at least of two subtraction terms and will not be pursued here. Substitution of Eq. (80) into Eqs. (93) and (94) leads, respectively, to resonance expansions for the continuum wave function along the internal region and the transmission amplitude, namely. 

It turns out that one may also obtain another resonant expansions for ir k,x) and t. For scattering from the left this follows by noting that the function G+(0, x k ) exp (- ik x), with x < L, converges along all directions of the k plane ask oo, and hence, in an analogous way as discussed in Section 2.2,... 

Figure 7.4 Plot of the transmission coefficient T[E) vs E for a quadruple barrier system with parameters as discussed in the text, around the first three-resonance miniband. The exact numerical calculation (full line) is indistinguishable from the resonance expansion using the three resonant poles (dashed line). The dotted line represents the calculation without the interference resonant terms.

Figure 7.8 Plot of Ln t) as a function of time for the 5-potential with parameters X = 6 and rja =. The resonant expansion (solid line) and numerical integration calculations (dashed line) are indistinguishable. It is also shown the1/f long-time asymptotic contribution (dotted line). See text.

As another example, we consider a double-barrier resonant tunneling system in ID. These artificial quantum systems, formed of semiconductor materials, have been fabricated and studied since the 1970s of last century [61]. Sakaki and co-workers verified experimentally that electrons in sufficiently thin symmetric double-barrier resonant structures exhibit exponential decay [88]. Recent work has examined the conditions for full nonexponential decay in double-barrier resonant systems [56]. Here we want to exemplify the time evolution of the probability density in these systems along the external region using the resonant expansion given by Eq. (121) [89]. 

R. de la Madrid, G. Garcia-Galderon, J. G. Muga, Resonance expansions in quantum mechanics, Gzech. J. Phys. 55 (2005) 1146. 

The initial state populates both types of resonance states of Interference between the vibrational discrete and the vibrational continuum autoionization resonances takes place although the resonance positions of the vibrationally-discrete autoionization-resonance (disc-res) states and the vibrationally-continuum autoionization-resonance states (cont-res) are very far from one another (as compared to their width). One may think that when E en(disc — res) there is only one dominant term in the series resonance expansion of tres(E) (see Eq. 33). This is however not the case. The numerators associated with the branch-cut resonances, an(cont — res), get complex values where both the real and the imaginary parts are larger than the corresponding ones of an(disc — res) by several orders of magnitude. In the numerical calculations the ampHtudes of the continuum t3q>e resonances... 

Resonance states in the spectra, which are assignable in temis of zero-order basis will have a predominant expansion coefficient c.. Hose and Taylor [ ] have argued that for an assignable level r /,j>0.5... 

Here E(t) denotes the applied optical field, and-e andm represent, respectively, the electronic charge and mass. The (angular) frequency oIq defines the resonance of the hamionic component of the response, and y represents a phenomenological damping rate for the oscillator. The nonlinear restoring force has been written in a Taylor expansion the temis + ) correspond to tlie corrections to the hamionic... 

It turns out that the CSP approximation dominates the full wavefunction, and is therefore almost exact till t 80 fs. This timescale is already very useful The first Rs 20 fs are sufficient to determine the photoadsorption lineshape and, as turns out, the first 80 fs are sufficient to determine the Resonance Raman spectrum of the system. Simple CSP is almost exact for these properties. As Fig. 3 shows, for later times the accuracy of the CSP decays quickly for t 500 fs in this system, the contribution of the CSP approximation to the full Cl wavefunction is almost negligible. In addition, this wavefunction is dominated not by a few specific terms of the Cl expansion, but by a whole host of configurations. The decay of the CSP approximation was found to be due to hard collisions between the iodine atoms and the surrounding wall of argons. Already the first hard collision brings a major deterioration of the CSP approximation, but also the role of the second collision can be clearly identified. As was mentioned, for t < 80 fs, the CSP... 

Here, Ri f and Rf i are the rates (per moleeule) of transitions for the i ==> f and f ==> i transitions respeetively. As noted above, these rates are proportional to the intensity of the light souree (i.e., the photon intensity) at the resonant frequeney and to the square of a matrix element eonneeting the respeetive states. This matrix element square is oti fp in the former ease and otf ip in the latter. Beeause the perturbation operator whose matrix elements are ai f and af i is Hermitian (this is true through all orders of perturbation theory and for all terms in the long-wavelength expansion), these two quantities are eomplex eonjugates of one another, and, henee ai fp = af ip, from whieh it follows that Ri f = Rf i. This means that the state-to-state absorption and stimulated emission rate eoeffieients (i.e., the rate per moleeule undergoing the transition) are identieal. This result is referred to as the prineiple of microscopic reversibility. 

The section on Spectroscopy has been retained but with some revisions and expansion. The section includes ultraviolet-visible spectroscopy, fluorescence, infrared and Raman spectroscopy, and X-ray spectrometry. Detection limits are listed for the elements when using flame emission, flame atomic absorption, electrothermal atomic absorption, argon induction coupled plasma, and flame atomic fluorescence. Nuclear magnetic resonance embraces tables for the nuclear properties of the elements, proton chemical shifts and coupling constants, and similar material for carbon-13, boron-11, nitrogen-15, fluorine-19, silicon-19, and phosphoms-31. 

The first derivative is the gradient g, the second derivative is the force constant (Hessian) H, the third derivative is the anharmonicity K etc. If the Rq geometry is a stationary point (g = 0) the force constant matrix may be used for evaluating harmonic vibrational frequencies and normal coordinates, q, as discussed in Section 13.1. If higher-order terms are included in the expansion, it is possible to determine also anharmonic frequencies and phenomena such as Fermi resonance. 

Adding potassium hydroxide, KOH, to a melt containing KF and a 0.1 mol fraction of K2TaF7 leads to the appearance of an additional band at 900 cm 1, as shown in Fig. 79 [342]. This band corresponds to TaO bond vibrations in TaOF63 complex ions. Interpretation of IR spectra obtained from more concentrated melts is less clear (Fig. 80). The observed absorption in the range of 900-700 cm 1 indicates the formation of oxyfluoride polyanions with oxygen bridges. ..OTaO. The appearance of a fine band structure could be related to very low concentrations of some isolated components. These isolated conditions prevent resonance interaction between components and thus also prevent expansion of the bands by a mechanism of resonance [362]. 

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