Classically 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. 

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...

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. 

This expression is nothing but the Bohr-Sommerfeld quantization rule (see, e.g., Landau and Lifshitz [1981]). In the metastable potential of Figure 3.3 there are also imaginary-time periodic orbits satisfying (3.41) that develop between the turning points inside the classically forbidden region. It is these trajectories that are responsible for tunneling [Levit et... 

For finite systems, such as atoms and molecules, the asymptotic structure of the KS exchange potential vx(r) in the classically forbidden region is due entirely to Pauli correlations5 as described by Wx (r). Thus,... 

In the classically forbidden region we can write the normalized density as... 

This means that quantum-mechanical waves can "tunnel" under a potential barrier, but decay exponentially within it. For quantum particles the classically "forbidden" region (what a Teutonic expression ) is somewhat penetrable, but is impenetrable if the barrier is infinitely high. 

The connection formula for tracing a phase-integral solution of the differential equation (4.1) from the classically allowed to the classically forbidden region is... 

When l —1/2 there is on the positive part of the real axis a classically forbidden region in the generalized sense delimited by the origin. It is assumed to be delimited also by a generalized classical turning point t, i.e., a first-order zero of Q2(z). In this region the wave function is... 

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