Tunneling through a square barrier

Quantum dynamics using the time-dependent Schrodinger equation 

Other side of the barrier. This probability is expressed in terms of a transmission coefficient, a property of the barrier/particle system that is defined below. 

Consider the solution of the time-independent Schrodinger equation flf = Ef for a given energy it. In regions I and III, where = 0, it is the free particle equations whose solutions are 

In Eqs (2.200), (2.201), and (2.205), the coefficients A, B, C, D, F, and G are constants that should be determined from the boundary conditions. Such conditions stem from the requirement that the wavefunction and its derivative should be continuous everywhere, in particular at the boundaries x = X] and x = xr, that is, iAiUl) = iAiiUl) [i/iAi(x)/t ]x=xL = (2.206) 

Note that we have only four conditions but six coefficients. The other two coefficients should be determined from the physical nature of the problem, for example, the boundary conditions at oo. In the present case we may choose, for example, D = 0 to describe a process in which an incident particle comes from the left. The wavefunction then has an incident (exp(zAx)) and reflected (exp(—zAx)) components in region I, and a transmitted component (exp(zAx)) in region III. Dividing the four equations (2.207) by. 4, we see that we have just enough equations to determine the four quantities B/A, C/A, F/A, and G/A. As discussed below, the first two are physically significant. We obtain for this case 

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As the semiclassical tunneling theory and Bardeen s original approach become inaccurate for potential barriers close to or lower than the energy level, the validity range of the MBA is much wider. In this subsection, the accuracy of the MBA is tested against an exactly soluble case, that is, the one-dimensional transmission through a. square barrier of thickness W=2 A (see Fig. 2.9). 

In [4] we have introduced a CA model for the NH3 formation which accounts only for a few aspects of the reaction system. In our simulations the surface was represented as a two-dimensional square lattice with periodic boundary conditions. A gas phase containing N2 and H2 with the mole fraction of t/N and j/h = 1 — j/n, respectively, is above this surface. Because the adsorption of H2 is dissociative an H2 molecule requires two adjacent vacant sites. The adsorption rule for the N2 molecule is more difficult to be described because experiments show that the sticking coefficient of N2 is unusually small (10-7). The adsorption probability can be increased by high energy impact of N2 on the surface. This process is interpreted as tunnelling through the barrier to dissociation [32]. 

Average square barrier models with [3=1.4 A (equation (C3.2.611 would predict the His 72 rate to be 1000 times faster. This equivalency of rates despite the great difference in distances is understood because the strongest pathways in the His 72 derivative contain a through-space tunnelling gap. 

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