WKB formula

Fig. 5. Tunneling describes the quantum mechanical delocalization of a particle into the region where the total energy (dotted line) is less than the potential energy V(x). The tunneling barrier V(x) — E enters directly into the WKB formula for the wavefunction tail, whereas the zeroth order wavefunctions are typically expressed in a basis set expansion which may or may not be optimized for its tunneling characteristics...

We recovered here the usual WKB formula for tunneling probability, which exhibits an exponentially decaying behavior. On the other hand, from Eq. (2.10), we observed immediately that resonances occur when the thickness of the barrier equals integer multiples of one half of the de Broglie wavelength in the barrier region. 

Very few potential barrier models, including the rectangular barrier model discussed above, yield exact results for the tunneling problem. In general one needs to resort to numerical calculations or approximations. A very useful approximation is the WKB formula, which generalizes the solution exp( zhr) of the free particle Schrodinger equation to the form... 

The identity of eqs. (2.6) (at T = 0) and (3.47) for the cubic parabola is also demonstrated in appendix A. Although at first glance the infinite determinants in (3.46) might look less attractive than the simple formulas (2.6) and (2.7), or the direct WKB solution by Schmid, it is the instanton approach that permits direct generalization to dissipative tunneling and to the multidimensional problem. 

To obtain a more penetrating view on A as an interaction between the original single wells, we compare two analytical routes to the evaluation of the basic two-center charge resonance integrals in this limit WKB and the lowdimensional perturbation formula. 

The connection between the classical and quantum formulations of the transport coefficients has been studied by applying the WKB method to the quantum formulation of the kinetic theory (B16, B17). In this way it was shown that at high temperatures the quantum formulas for the transport coefficients may be written as a power series in Planck s constant h. When the classical limit is taken (h approaches zero), then the classical formulas of Chapman and Enskog are obtained. 

Formula (3.53) demonstrates that the decay rate for a metastable state is equal to the inverse period of classical vibrations in the well (attempt frequency) times the barrier transparency. The more traditional treatment of metastable-state decay using the one-dimensional WKB approximation is given in Appendix A. 

Each I-U curve is obtained within fractions of 37) The formula is valid for low bias voltages in a second, therefore thermal drift can be WKB (Wentzel-Kramers-Brillouin)... 

The ordinary Airy function A,(z) corresponds to this solution with A = 0. Equation (85) represents the famous connection formula for the WKB solutions crossing the turning point. As can now be easily understood, once we know all the Stokes constants the connections among asymptotic solutions are known and the physical quantities, such as the scattering matrix, can be derived. However, the Airy function is exceptionally simple and the Stokes constants are generally not known except for some special cases (40). 

This semiclassical wave function does not require any special matching condition at the boundary between classically allowed and classically forbidden regions, which represents the main obstacle in the ordinary WKB theory. The tunneling splitting Aq can be calculated from the Herring formula [54],... 

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