Quantum well states periodic potential

As the particle traverses from one barrier to the next it changes its energy. The conditional probability kernel P(E E ) that the particle changes its energy from E to E is determined by the energy loss parameter 8 = pA and a quantum parameter a = The quantum kernel is as in Eq. 38. The main difference between the double and single well cases and the periodic potential arises in the steady state equation for the fluxes ... 

Accounting for the potential profile of the nueleobase sequences, a DNA molecule can be treated as a periodic system with some potential barriers and wells, which are empty at a non-exited state. The carrier transport in an A-period structure can be described by kinetic equations accounting for the carrier concentration in the j-th quantum well [5]. 

Fig. 6.6 (a) Resonance fluorescence of atoms trapped in periodic potential wells. The spectrum is shown near the central component of the Mollow triplet, (b) The quantum transitions of an atom between the vibrational states responsible for the side components. (Reprinted from Jessen et al. 1992 with courtesy and permission of the American Physical Society.)... 

The potential well of an individual atom with boimd states A, B, C can be represented in Figure 19.10. When assembled into a crystal, the potentials of the adjacent atoms overlap to form a periodic potential as illustrated in Figure 19.11. In metals, the outer s-electrons represented as C become delocalized and form the Fermi band. The electronic states represented as B are only partially localized. Recall from quantum mechanics that an electron wavefunction does not go to zero at a finite potential barrier and a portion of its wavefunction exists outside the barrier the lower and thirmer the barrier, the more wavefunction is outside of the potential barrier. For this reason, the B electrons are only partially localized. 

The classical counterpart of resonances is periodic orbits [91, 95, 96, 97 and 98]. For example, a purely classical study of the H+H2 collinear potential surface reveals that near the transition state for the H+H2 H2+H reaction there are several trajectories (in R and r) that are periodic. These trajectories are not stable but they nevertheless affect strongly tire quantum dynamics. A study of tlie resonances in H+H2 scattering as well as many other triatomic systems (see, e.g., [99]) reveals that the scattering peaks are closely related to tlie frequencies of the periodic orbits and the resonance wavefiinctions are large in the regions of space where the periodic orbits reside. 

Figure 13 Comparison of the experimental and a quantum mechanically computed (by exact wave packet propagation using an ab initio computed potential energy) spectrum of a nonrotating Na, molecule pumped to its B electronic state. (Courtesy of Experiment by S. Rutz, E. Schreiber, and L. Woste Computations by B. Reischl, all of the Free University of Berlin) (a) The short time dynamics Shown is the population of the excited state vs. time as determined by a pump-probe experiment and by the computation (points connected by a straight-line segments). The periodicity (about 320 fs) is due to the symmetric stretch motion, (b) A frequency spectrum. The long time dynamics (as reflected in the well-resolved spectrum) show the contribution of a different set of vibrational modes. The dominant peaks can be identified as the radial pseudorotation motion of Na,(B) while the splittings are due to the angular pseudorotational motion. (Adapted from B. Reischl, Chem. Phys. Lett., 239 173 (1995) and V. Bonacic-Koutecky, J. Gaus, J. Manz, B. Reischl, and R. de Vivie-Riedle, to be published.)...

Similarly to the result obtained for QRs, these unitary DSs are just the spatial transformations dictated by the symmetry of the nanotube potential and compensated by the appropriate translation in time. It is interesting to examine the quantum numbers associated with the DSs Rjv and Poo- Note that the Floquet states 0 (r, t) are eigenstates of P and as well. Recall, that for QRs we have = I and, therefore, the eigenvalues of Pjv are the Mh order roots of - 1. The situation is more intricate for nanotubes in circularly polarized fields, where we find P P = I. Owing to the foim of the interaction term, equation (28), and the periodicity embedded in P, it is natural to transform from z and t to another set of orthogonal coordinates o)t — Icqz and cjt + koz)/2. Afterwards, it is possible to rewrite a Floquet state as... 

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