The WKB Approximation

We seek the poles of the spectral function g(E) given by (3.7). In the WKB approximation the path integral in (3.7) is dominated by the classical trajectories which give an extremum to the action functional... 

As seen from this table, the WKB approximation is reasonably accurate even for very shallow potentials. At 7 = 0 the hindered rotation is a coherent tunneling process like that studied in section 2.3 for the double well. If, for instance, the system is initially prepared in one of the wells, say, with cp = 0, then the probability to find it in one of the other wells is P( jn, t) = 5sin (2Ar), while the survival probability equals 1 — sin ( Ar). The transition amplitude A t), defined as P( + t) = A t), is connected with the tunneling frequency by... 

FIG. 15. A comparison between the distance dependence of the tunneling barrier between a jellium tip and substrate immersed in solution versus vacuum under zero bias conditions. The apparent barrier height is derived from the WKB approximation. (From Ref. 110.)... 

The MFT equation of motion, Eq. (31), can be derived in many ways, including the WKB approximation [9], the eikonal method [13], a (semi)classical... 

A third matter to mention here is that the WKB approximation outlined above is limited in the realm in which it is valid. It is more applicable to protons than to electrons (Bockris and Sen, 1973). Other quantum mechanical methods of a quite different nature can be used13 (D. Miller, 1995) and have been applied to make numerical quantal calculations of the rate of redox reactions (Khan, Wright, and Bockris, 1977 Newton, 1986), but they depend on a knowledge of wave functions which, for electron levels in hydrated ions in solution, may still be too primitive for calculations of rate. 

We first follow the flow chart for the simple case of elastic scattering of structureless atoms. The number of internal states, Nc, is one, quantum scattering calculations are feasible and recommended, for even the smallest modem computer. The Numerov method has often been used for such calculations (41), but the recent method based on analytic approximations by Airy functions (2) obtains the same results with many fewer evaluations of the potential function. The WKB approximation also requires a relatively small number of function evaluations, but its accuracy is limited, whereas the piecewise analytic method (2) can obtain results to any preset, desired accuracy. 

The additional phases tt/2, where is the number of turning points encountered along the trajectory, emerge because of the breakdown of the WKB approximation near the turning points [Gutzwiller, 1967 Levit et al., 1980 McLaughlin, 1972]. Vibrations with energy E in the well have a period... 

Although the above explanation relied on a crude semiclassical estimate (with exponential accuracy), it can easily be refined either by exactly solving the Schrodinger equation for the one-dimensional potential (7.1) (see, for example, Press [1981]) or, for sufficiently high barriers (V0/h(o0> 2), by employing the WKB approximation. The eigenfunctions of stationary states A and E... 

Our discussion of solitons and breathers has been purely classical, since we implied the existence of excitations corresponding to the classical solutions (7.71) and (7.75). To see this correspondence it is natural to invoke the quasiclassical approximation. Dashen et al. [1975] quantized the sine-Gordon problem and showed that in the WKB approximation for a discrete chain with periodic boundary condition the soliton mass is renormalized to... 

So far, we did not apply the WKB approximation yet. The assumptions on which the above results are based, the strong vibronic coupling, k2 > A, and large energy gap in the electron spectrum, A 1, are typical for all tunneling problems in JT systems. If, as in the early papers of Bersuker [1,2], we substitute the oscillator ground-state wave functions in equation (17), then we come to the approximate results that can be obtained directly from the matrix element (12) with the oscillator functions. 

The WKB approximation was applied to the symmetric double-well potential so many times that it makes it difficult to provide a comprehensive list of references. In the context of the present paper, the most important references are the famous text of Landau and Lifshits [27], where tunneling in a symmetric double-well is given as a sample problem, and the paper of Auerbach and Kivelson [11] where the symmetric double-well potential is considered as one of the model cases. 

In this paper, we consider possible tunneling paths in the G ) (g h) and 7 0 (e t2) JT systems in which, in each case, a possible crossover has been suggested. Section 2 describes the WKB approximation in relation to JT systems. Sections 3 and 4 describe the analysis of the two systems and Section 5 summarizes the implications of these results. 

Many authors, for example [22-25], have described calculations of the tunneling splitting in the T (e f2) JT system. Interestingly, a recent numerical calculation [26] showed that, for the first time, there appeared to be a crossover from a T ground state to an A ground state when t2 vibrations dominate. Thus a new study of the tunneling paths under the WKB approximation is described here for the case in which the t2 vibrations in this problem dominate. 

The separation of the PES into a part determined by the reaction coordinate and a part described by a quadratic approximation in a subspace of the remaining coordinates has recently often been used, typically with the WKB approximation (236,237) Yamashita and Miller (238) utilized the reaction-path Hamiltonian method combined with the path-integral method to calculate the rate constant of the reaction of H + H2. 

The effect of solvent quality on the thickness of the layer L and free energy of the individual chains is demonstrated by the work of Milner et al. (1988) and Halperin (1988). The result for the chain configurations comprises an asymptotic solution of Eq. (31) for highly stretched chains, i.e., L n1/2l, via the WKB approximation. This yields the end segment probability as... 

Franck-Condon factor with an exponential factor with the form of the WKB approximation for timneling through intersecting parabolas. This is intrinsically a multidimensional problem, and a generalized bond-length displacement coordinate was introduced to make it tractable. In the process of reducing the coordinates, a measure of the relative munber of identical oscillators (same p and to) was included in the exponential factor, rj, and the timneling rate was expressed as (50)... 

Field emission results from tunneling of electrons from a metal into a vacuum under application of a strong electric field. The tunneling mechanism is described in the WKB approximation for emission from metal surfaces which leads to the well known Fowler-Nordheim equation ... 

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