Wentzel - Kramers - Brillouin

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)... 

DIV.2a. I. Prigogine et G. Garikian, Sur le calcul des niveaux energetiques par la methode de Wentzel-Kramers-Brillouin et son application a I hydrogene liquide (On the calculation of the energy levels by the Wentzel-Kramers-Brillouin method and its application to liquid hydrogen), J. Phys. 330-332 (1947). 

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). 

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

Still another method, namely use of the WKB (Wentzel-Kramers-Brillouin) theory, has been suggested by Englman and Ranfagni (1980) for use with multidimensional or complex energy surfaces. 

That this condition is a generalization of the standard WKB (Wentzel-Kramers-Brillouin) condition can be seen by considering a one-degree-of-freedom system, where we have... 

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

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. 

The effects of X7 may be treated using ordinary non-degenerate perturbation theory or, as in Dunham s original work, by means of the Wentzel-Kramers-Brillouin method... 

In another series of papers [26] Shin has used the WKB (Wentzel-Kramers-Brillouin) method for evaluating the vibrational transition matrix element, employing various forms of interaction potential. Comparisons are made with quantum-mechanical solutions. 

This is supported by computed rate constants with tunneling contributions using the semiclassical Wentzel-Kramers-Brillouin (WKB) theory. The WKB procedure requires a carefully described intrinsic reaction path, and... 

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)... 

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... 

The calculation of the electronic-transmission factor currently involves three different methods, viz. the Landau-Zener formula, Fermi s golden mle [35], and electron tunneling formalism such as the Wentzel-Kramer-Brillouin method [36]. We used the Landau-Zener formula [37,38] to calculate it ... 

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. 

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... 

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