Wentzel - Kramers - Brillouin approximation

Wentzel Kramer-Brillouin approximation, tunneling, 1492 Whewefi, Reverend, 1050 Whiskers, 1327, 1336 White, and organic adsorption. 979 Wieckowski, A., 1146 Will, 1205... 

Fig. 10. Evolution of disturbances along the flame surface with time (i)—Wentzel-Kramers-Brillouin approximation.

Additionally to the thermally induced hopping the charge carries tuimel through the potential barrier. It can be determined within the Wentzel-Kramer-Brillouin approximation for one dimension as ... 

Wentzel-Kramers-Brillouin (approximation) Williams-Landel-Eerry (equation)... 

If on the other hand AFJ is much smaller than Ipv, the eigenstates 1), 12) essentially coincide with the handed states L), /2). In that case the absolute value of the difference E2 — 1] between the eigenstates is approximately 2 V v - We can estimate the size of the tunneling splitting for typical chiral molecules for instance within the Wentzel-Kramers-Brillouin approximation. If we assume a quartic double well potential with a barrier height of 200 kJ mol a barrier Avidth of 200 pm and a tunneling mass... 

At t=T the escape rate begins to depart from the above reported law, tending to a temperature-independent quantum transition rate (Fig. G.l), which has the same form, in the usual WKB (Wentzel-Kramers-Brillouin) approximation for quantum tunneling, as in the case of switching of magnetization via thermal activation ... 

With help of the Wentzel-Kramers-Brillouin approximation the field dependent transmission coefficient was calculated for the vacuum potential barrier. Integrating over all energies for a free electron gas at room temperature results in an exponentially increasing I-V curve that reaches 1 nA at 0.7 kV. With a band gap well above 2 eV proteins are good insulators, their electric breakdown is believed to be a conduction mechanism, which occurs at a voltage of about 200 V. The necessary potential for tip-emission can be further decreased to about 100 V using nano-fibers. To create transparent... 

The process of formation of a bubble having a critical radius, can be computed using a semiclassical approximation. The procedure is rather straightforward. First one computes, using the well known Wentzel-Kramers-Brillouin (WKB) approximation, the ground state energy Eq and the oscillation frequency //() of the virtual QM drop in the potential well U JV). Then it is possible to calculate in a relativistic framework the probability of tunneling as (Iida Sato 1997)... 

Field emission is a tunneling phenomenon in solids and is quantitatively explained by quantum mechanics. Also, field emission is often used as an auxiliary technique in STM experiments (see Part II). Furthermore, field-emission spectroscopy, as a vacuum-tunneling spectroscopy method (Plummer et al., 1975a), provides information about the electronic states of the tunneling tip. Details will be discussed in Chapter 4. For an understanding of the field-emission phenomenon, the article of Good and Muller (1956) in Handhuch der Physik is still useful. The following is a simplified analysis of the field-emission phenomenon based on a semiclassical method, or the Wentzel-Kramers-Brillouin (WKB) approximation (see Landau and Lifshitz, 1977). 

In the approximation of the Wentzel-Kramers-Brillouin method, we take fi = k(l)unf a) from the Landau theory on the hydrodynamic instability of the plane flame. The relative change in the disturbance because of the stretch-effect is equal to that in the tangential velocity component... 

Two coupled first order differential equations derived for the atomic central field problem within the relativistic framework are transformed to integral equations through the use of approximate Wentzel-Kramers-Brillouin solutions. It is shown that a finite charge density can be derived for a relativistic form of the Fermi-Thomas atomic model by appropriate attention to the boundary conditions. A numerical solution for the effective nuclear charge in the Xenon atom is calculated and fitted to a rational expression. 

Per-Olov Lowdin had a long and lasting interest in the analytical methods of quantum mechanics and my tribute to his legacy involves an application of the Wentzel-Kramers-Brillouin (WKB) asymptotic approximation method. It was the subject of a contribution(l) by Lowdin to the Solid State and Molecular Theory Group created by John C. Slater at the Massachusetts Institute of Technology. 

This quasiperiodicity can be explained as follows. By applying the fitting procedure, based on the WKB (Wentzel-Kramers-Brillouin) method, one can find that the smallest positive root x = x(jS+1) of the Hermite polynomial Hes+i(x) is approximately equal to... 

There are a number of other methods which may be used to obtain approximate wave functions and energy levels. Five of these, a generalized perturbation method, the Wentzel-Kramers-Brillouin method, the method of numerical integration, the method of difference equations, and an approximate second-order perturbation treatment, are discussed in the following sections. Another method which has been of some importance is based on the polynomial method used in Section 11a to solve the harmonic oscillator equation. Only under special circumstances does the substitution of a series for 4 lead to a two-term recursion formula for the coefficients, but a technique has been developed which permits the computation of approximate energy levels for low-lying states even when a three-term recursion formula is obtained. We shall discuss this method briefly in Section 42c. 

Refractive-Index Profile. The refractive-index profile was approximated by the conventional power law. The output pulse width from the GI POF was calculated by the Wentzel-Kramers-Brillouin (WKB) method (10) in which both modal and material dispersions were taken into account as shown in Equations (3), (4), and (5). Here, aintemodai cTintramodai, and CTtotai signify the root mean square pulse width due to the modal dispersion, intramodal (material) dispersion, and both dispersions, respectively. 

It is instructive to reproduce this result analytically. Within the Wentzel-Kramers-Brillouin (WKB) approximation, the inversion frequency is estimated as [71] ... 

For high radial number n or orbital momentum /, the Wentzel-Kramers-Brillouin (WKB) method is known to provide an excellent approximation. In fact, the WKB approximation works already surprisingly well for small values of n or /. We refer here to the review of Quigg and Rosner [17], who present a detailed study of the WKB approximation for confining potentials. Let us simply recall here some basic results. In the WKB approximation, the eigenenergies of S-waves are given by... 

It should be noted that, however, the treatment here implicitly assumes each elementary process as independent to each other, which is not always correct. For example, in case of a predissociation problem, there exist a curve crossing and classical turning points, both break the applicability of the Wentzel-Kramers-Brillouin (WKB) approximation. 

For a proper interpretation of STM, it is essential to understand how the tunneling current works. Figure 7.1b shows a simple model of a tunneling junction consisting of a metal (tip)-vacuum-metal (surface) in which the tunnehng current can be derived bythe Fermi golden rule and Wentzel-Kramers-Brillouin (WKB) approximation. The tunneling current is then expressed by ... 

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