WKB approximation

Finally, one problem still remains. There are complex temis which need to be associated with the detemimant. The complex temis (Maslov indices) have to do with the square root of tlie detenninant, which may be negative, and also appear in the related WKB approximation. They can be calculated, albeit with difficulty... 

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... 

Often overlooked, the phase of continuum wavefunctions contains valuable information. It is conveniently illustrated by consideration of the form of the wavefunction within the quasiclassical (WKB) approximation [55],... 

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)... 

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... 

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). 

The potential curve for the electrons near the tip surface is shown in Fig. 1.38. The relevant dimensions are much smaller than the radius of the tip end. Therefore, a one-dimensional model is adequate. In the metal, the energy level of the electrons is lower than the vacuum level by the value of the work function c ). From the point of view of classical mechanics, the electrons cannot escape from the metal even with a very high external field, that is, the potential barrier is impenetrable. From the point of view of quantum mechanics, there is always a finite probability that the electrons can penetrate the potential barrier. In the semiclassical (WKB) approximation, the transmission coefficient for a general potential barrier is (Landau and Lifshitz, 1977) ... 

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. 

Early applications of WKB approximations to the Coulomb problem in Schrodinger theory demonstrated the necessity and expediency of the Kramers modification ) ... 

Appendix A Decay of a Metastable State and Tunneling Splitting in Terms of the One-Dimensional WKB Approximation Appendix B Equivalence of the Instanton Approach to Semiclassical TST... 

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... 

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. 

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... 

Exact Calculation WKB Approximation (7.9) Gaussian Packet Approximation (7.10)... 

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... 

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