## 3 Results
### Quantum Mechanical Motion of a K\({}^{+}\) Ion
#### 3.2.2 The Effect of QM Coherence on K\({}^{+}\) Conductance  
As has been found in classical MD simulations, the K\({}^{+}\) current through the selectivity filter consisted of a series of bursts with long, intermittent halting periods reaching up to several hundred nanoseconds (Figure 2); again, these halting periods are characterized by some typical configurations of K\({}^{+}\) and H\({}_{2}\)O particles. Figure 5 demonstrates two such examples. As before, the most stable configurations were S1-0-S3 or with an intermittent water S1-W-S3. A break up of this situation can occur if the K\({}^{+}\) ion in site S1 moves up to the next potential valley. In this situation, particles from the lower positions can follow, and this leads to vacancies for additional ions to enter the selectivity filter from the inner cavity. This observation opens the question whether a quantum mechanical treatment of the K\({}^{+}\) ion at site S1 in such a configuration can predict shorter halting periods and, thus, an enhanced K\({}^{+}\) current. Signatures for this to occur are indicated by the evolution of the wave packet as shown in Figure 3. For the time t = 0.5 ps, both examples of wave evolutions show an appreciable probability amplitude of the K\({}^{+}\) ion in a region well beyond the carbonyl oxygens of Tyr78 located at around 0.65 nm. This indicates some possibility that the K\({}^{+}\) ion can also be found at site S0 at this very instance of time. In the classical situation the ion has to remain at site until a peak in oxygen vibrational motion provides the force for transition. Opposed to the classical situation in the QM version there are at least two reasons why the wave packet can propagate more easily into site S0. One reason is tunneling [24]. The other reason, which we have suggested in a previous report [7], is that the front part of the wave packet drags the oxygen atoms upward, reducing the confining potential for the lower part of the wave function, which can then propagate more easily. At this stage it cannot be said with certainty which of the two strategies will dominate, but evidence from previous calculations, as presented in [7], is strongly suggestive for the second option. In order to estimate the transition time of the ion to site S0, we inspected the probability distribution that emerged in course of decoherence. Let \(W_{z}(t)\) be the probability to find the particle above a certain position \(z\) at time \(t\) as calculated from a non-decohering wave function. Our interest focuses on positions \(z\) in the range of site S0. If coherence time is \(\tau\), then the probability for the wave function to lose coherence within an interval \(dt\) is given by  
\[p_{inc}(t)dt=\frac{e^{-t/\tau}}{\tau}dt \tag{9}\]  
The average probability \(P_{z}\), to find the K\({}^{+}\) ion after decoherence at locations above \(z\) is then given by  
\[P_{z}=\int_{0}^{\infty}p_{inc}(t)W_{z}(t)dt. \tag{10}\]  
At this stage we recall that the environment of ions acts as a detector for the positional degree of freedom of ions within a temporal scale provided by coherence time. Over a longer period of time, the evolution of an ion is composed from a series of short quantum evolutions with random moments of decoherence. Each formation as a new wave packet at a new position will of course satisfy energy and momentum conservation. Consequently, it is the location of the ion at the moment of decoherence which becomes decisive for the further development of its wave function. Should this location be above the threshold \(z\) we expect its translocation within one coherence interval \(\tau\). If this location happens to be below the value of \(z\), the particle's wave function has to evolve again setting out from the initial location. If after two attempts the particle translocates, we assign a transition time of \(2\tau\).