Forbidden region

The accuracy of the CSP approximation is, as test calculations for model. systems show, typically very similar to that of the TDSCF. The reason for this is that for atomic scale masses, the classical mean potentials are very similar to the quantum mechanical ones. CSP may deviate significantly from TDSCF in cases where, e.g., the dynamics is strongly influenced by classically forbidden regions of phase space. However, for simple tunneling cases it seems not hard to fix CSP, by running the classical trajectories slightly above the barrier. In any case, for typical systems the classical estimate for the mean potential functions works extremely well. 

Equation (2.2) defines the statistically averaged flux of particles with energy E = P /2m -f V Q) and P > 0 across the dividing surface with Q =0. The step function 6 E — Vq) is introduced because the classical passage is possible only at > Vq. In classically forbidden regions, E < Vq, the barrier transparency is exponentially small and given by the well known WKB expression (see, e.g., Landau and Lifshitz [1981])... 

In the classically forbidden region, on the other hand, E" has the same sign as W and it follows from Equation 33.12 that p" is positive. 

The electronic properties of solids can be described by various theories which complement each other. For example band theory is suited for the analysis of the effect of a crystal lattice on the energy of the electrons. When the isolated atoms, which are characterized by filled or vacant orbitals, are assembled into a lattice containing ca. 5 x 1022 atoms cm 3, new molecular orbitals form (Bard, 1980). These orbitals are so closely spaced that they form essentially continuous bands the filled bonding orbitals form the valence band (vb) and the vacant antibonding orbitals form the conduction band (cb) (Fig. 10.5). These bands are separated by a forbidden region or band gap of energy Eg (eV). 

Fig. 7. The tunneling paths in a double minimum potential like that of Fig. 5 may be classified as having one or more (odd) numbers of instantons, or tunneling segments. Three such traverses of the classically forbidden region are shown above. The classification of all paths according to the number N of instantons is the basis for evaluating the path integral as the sum in Eq. 29 note however that the therein is constructed to include non-harmonic (beyond semiclassical) fluctuations around the minimum action instanton paths, which are evaluable by Metropolis Monte Carlo...

These are plotted in Fig. 2 for = 1. Localized states occur in the six regions indicated. (P means that there are two (P states, CP31 that there is one (P state and one 31 state, and so forth. We see that the system may have two or one or no localized states. The area of the region where there are no localized states decreases as rj increases, and such a forbidden region exists only if < 2. 

The Heisenberg space defines the available uncertainty space where, in quantum mechanics, it is possible to perform, direct or indirect, measurements. Outside this space, in the forbidden region, according to the orthodox quantum paradigm, it is impossible to make any measurement prediction. We shall insist that this impossibility does not result from the fact that measuring devices are inherently imperfect and therefore modify, due to the interaction, in an unpredictable way what is supposed to be measured. This results from the fact that, prior to the measurement process, the system does not really possess this property. In this model for describing nature, it is the measurement process itself that, out of a large number of possibilities, creates the physical observable properties of a quantum system. 

The last line, Table 5.1, reports the purely classical moments. The zeroth classical moment is a little smaller than the zeroth quantum moment, because of the wave mechanical tunneling of the collisional pair into the classically forbidden region which enhances the intensities. All odd moments of classical profiles are, of course, zero. The second and fourth moments are significantly smaller than the quantum moments, because... 

Fig. 4A The vibrational levels of the H2 ground electronic state. The levels are drawn between the classical limits of vibration, but there is a small probability for vibration to extend into the classically forbidden region.

Thus, the energy levels are equally spaced starting with the zero-point energy hv0 (Fig. E.4). Note the solutions have the property that there is some probability of finding the particle in classically forbidden regions, that is, the particle penetrates into the walls. 

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