## 3 Results
### Temporal Characteristics  
At first sight, (11) would predict an increase of \(\overline{T}\) with increasing coherence times. However, there emerges a characteristic quantum effect strongly influencing the pattern behind time \(\overline{T}\). If \(\tau\) is very small, the quantum wave packet exists only for a very short time and small spatial spread, so that the probability \(P_{z}\) for extending into the next site is also small (e.g., initial expansion terms are quadratic in time). As a result, time \(\overline{T}\) is extended and coherence time \(\tau\) is small, the situation approaching a classical situation. However, halting time \(\overline{T}\) can also be large with long coherence times as \(P_{z}\) converges to a value determined by the height of the separating potential barrier between two neighboring sites. As a result, we observed halting time \(\overline{T}\) passing through a minimum when \(P_{z}\) was still rising for intermediate values of coherence time. In most instances this minimum was reached when \(\tau\) acquired typical values for K\({}^{+}\) ions in the SF, which was around a few tenths of a picosecond (Figure 4). In order to see how this quantum phenomenon could change the conductance of K\({}^{+}\) ions through the selectivity filter, we compared the situation with classical MD simulation over a period of 660 ns and recorded the time of each emission of a particle simultaneously with the alignment of filter occupations of ions and waters. Within one series of runs a total of 218 K\({}^{+}\) ions were emitted, and roughly the same number of H\({}_{2}\)O molecules. The two configurations which produced the longest halts of current are shown in Figure 5. For these two configurations we again simulated runs in which the top K\({}^{+}\) ion was represented as a quantum mechanical wave packet. Short period (2 ps) time intervals were repeated many times to yield estimates of coherence and halting times prior to a transition from site S1 to S0. We found that during the 660 ns of MD simulation, the configuration shown in Figure 5 occurred three times, and the corresponding halting times were 6.5, 21.9, and 87.4 ns, respectively. The average classical halting time was, thus, 38.6 ns. For the quantum mechanical simulations we selected four different starting times. For each situation we calculated a number of evolutions which differed in the width of the initial wave packet. These widths varied between 0.005 and 0.05 nm (defined as the width characterized by a probability density above 1/e of peak values). The evolution, which gave the longest halting time, was taken as the most credible one because it implied that with this width the wave packet had the smallest exponential penetration into the classically forbidden zones of the potential. We also calculated the halting times for different coherence times. We obtained an average halting time of 1.16 ns for a coherence time of 0.4 ps and of 10.16 ns for a coherence time of 0.1 ps. Therefore, even if we assume a relatively short coherence time of 0.1 ps, the quantum mechanical halting time was only around 25% of the classical one. The configuration in Figure 5 also occurred three times with halting times of 21.6, 47.7, and 236.1 ns, respectively, thus providing an average of 101.8 ns. Under the same conditions as above, we obtained an average halting time of 0.60 ns for a coherence time of 0.4 ps, and of 33.90 ns for a coherence time of 0.1 ps. For the shorter coherence, the halting time in the quantum case was now around 33% of the classical one. Figure 6 depicts the classical and the quantum mechanical evolution of a wave packet with an original width of 0.05 nm and the corresponding classical configuration at the beginning and after an elapsed time of 0.227 ps. One notes that the QM wave packet has already an appreciable presence within site S0.