Tunneling probability formulas

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. 

It should be noted that the parts of branches drawn by broken lines indicate noncontributing parts that make unphysical contributions, and they can be removed by the proper treatment of the Stokes phenomenon. After such a procedure, we can sum up all the contributions from the physically legal parts of the branches in the -set. We find the probabilistic weights of the branches together with the total probability obtained by the sum formula [Eq. (6)] in Fig. 5c. In Fig. 5d, the tunneling probability obtained by the sum formula is... 

In our opinion, the present evidence for tunneling in any dissociative chemisorption reaction is not overwhelming. Analyses have been made based upon one-dimensional tunneling formulas for the tunneling probability, such as the very simple inverse parabolic potential form ... 

In this range the formula (177.III) yields X— 1 and from (181.Ill) one gets P /P = X / << 1, which means that the tunneling probability is negligible compared to that of over-barrier transitions. 

The nuclear tunneling probability can be conveniently calculated by the formula (92 II)... 

This is the well-known Gamov formula of tunneling probability. It is now well understood that the connection formulas of Equations (2.49) and (2.50) are crucial. These formulas can be obtained from the Airy function, since the potential in the vicinity of the turning point can be approximated by a linear function of x for which the Airy function gives the exact analytical solution. 

This can be easily verified if we use formula (4.14) for an approximate calculation. The ratio of the tunneling probabilities for two different isotopes is given by the following equation ... 

The method is composed of the following algorithms (1) transition position is detected along each classical trajectory, (2) direction of transition is determined there and the ID cut of the potential energy surfaces is made along that direction, (3) judgment is made whether the transition is LZ type or nonadiabatic tunneling type, and (4) the transition probability is calculated by the appropriate ZN formula. The transition position can be simply found by... 

The factor k takes into acount the effects of nonadiabatic transition and tunneling properly. Also note that the electronic coupling //ad is assumed to be constant in the Marcus formula, but this is not necessary in the present formulation. The coupling Had cancels out in k of Eq. (126) and the ZN probability can be calculated from the information of adiabatic potentials. 

From the Gamov formula, calculate the probabilities for an electron and for a proton to tunnel through barriers of 1 and 10 A thickness with a height of 1 eV. 

To conclude, we shall estimate the characteristic distance r2, at which the electron tunnels during the ionization of the atoms in the external field. This estimation can be conducted with the help of the relation r2 cs I/F. Substituting into this formula, for instance, the energy of He ion ionization, 1 0.076 eV and F = 200kV cm-1, we find r2 = 40 A. According to eqn. (8), the probability of ionization by means of electron tunneling at a distance r2 at such values of I and F is equal to 5 x 109s. ... 

First of all, consider the case when all normal vibrations are classical. This takes place if the condition a)k -4 T works well for all frequencies. In the classical case the probability of tunneling can be calculated with the help of the general formula (18) using the Franck-Condon approximation and the well-known [10] properties of quasi-classical wave functions. We will not dwell upon the details of transition from the quantum description to the... 

A theoretical model of the low-temperature decay of etr in MTHF discovered in ref. 30 was suggested in ref. 31. According to this model, the disappearance of et in y-irradiated MTHF at 77 K is due to electron tunneling from a trap to a hole centre. The form of the potential barrier for electron tunneling used in ref. 31 to analyze the curves of the decay of etr is represented schematically in Fig. 9(a). To evaluate the probability of tunneling per unit of time, the Gamow formula... 

This formula demonstrates that the tunneling splitting is determined—like the imaginary part of metastable state energy (A.20)—as a normalized probability flux through the dividing line. In the present case this flux corresponds to coherent probability oscillations between the wells rather than exponential decrease of the survival probability in the well, so A is a real value. 

The first theory of hydrogen transfer wich takes into account the nuclear tunneling was developed by Levich et al., (1970). The authors calculated the transfer probability, Wif, using the general formula of the perturbation theory ... 

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