## 3 Results
### Quantum Mechanical Motion of a K\({}^{+}\) Ion  
#### 3.2.1 Coherence of the Wave Function  
At absolute zero temperature (0 K), the wave packet of the K\({}^{+}\) ion maintains its coherence permanently. In this idealized situation, all interactions between the ion's wave packet and the atomic structure of its surrounding would evolve in a completely deterministic way. However, since the ion is exposed to physiological temperatures, the surrounding thermal bath is subject to random interactions between ions and the backbone atoms of the filter. The evolution of the ion wave function obtains a random element. Eventually, there is a loss of coherence between different spatial regions of the wave packet and a quantum-classical transition is initiated. It seems to be important to obtain estimates about coherence time \(\tau\), which is the typical time it takes until coherence is lost to a significant degree. The loss of coherence generally follows an exponential decay law [21], and if we set the maximum of coherence to 1, this can be seen as  
\[C(t)=e^{\frac{-t}{\tau}} \tag{8}\]  
Empirically, coherence time \(\tau\) can be estimated in the following way. First a classical MD simulation is initiated where several K\({}^{+}\) ions enter and leave the selectivity filter until a specific configuration of K\({}^{+}\) and H\({}_{2}\)O particles is found after some time interval t\({}_{\text{start}}\). Following this initiation, a K\({}^{+}\) ion is selected for QM simulation, and this simulation is continued over some time interval \(\Delta t\) during which the chosen K\({}^{+}\) wave function evolves quantum mechanically according the Schrodinger equation. All other particles behave classically, as before. The evolution of the QM wave function is recorded, and the conditions at time t\({}_{\text{start}}\) are stored. Following this step, the evolution during time interval \(\Delta t\) is repeated. This time, due to random perturbations from the thermal bath, the particle wave function follows a slightly different path. The two emerging versions of the wave partly overlap giving rise to interference terms, similar to the situation expected from a double-slit experiment [22]. Finally, the normalized magnitudes of this interference provide an estimate for coherence. Figures 3 and 4 provide a typical example for this situation. Figure 3 shows the real part of the wave function at the beginning and after 0.5 ps for two separate evolutions of the wave function. Figure 4 provides the coherence terms of two such separate evolutions during an interval of \(\Delta t=1\) ps. Figure 3: **Top**: Real part of the wave function at the beginning of the quantum mechanical calculation. The length of one oscillation corresponds to the mean de Broglie wavelength given by the mean momentum of the K\({}^{+}\) ion. (Since the particle starts as a localized Gaussian wave packet, it actually contains a wide spectrum of de Broglie wavelengths.) The particle is on the lower side of site S1 (see, e.g., Figure 5a). The blue scale near the horizontal axis shows the range of sites S1 and S0, as defined by the charge centers of the respective lone pairs of the oxygen atoms of the carbonyl groups of Gly77, Tyr78, and Gly79, respectively. **Bottom**: Real part of two separate evolutions (one red, one green) of the wave function after 0.5 ps, which happened under identical conditions except for random thermal noise introduced by the oxygen atoms of the carbonyl groups. The mean positions are already slightly shifted. In both cases, a significant part of the wave function extends over almost one nanometer and reaches over more than two sites. As can be seen form the inserted examples, coherence terms are subject to perturbations from oxygen atoms adding random phases to the evolving wave function. A least-squares fit to the present data yielded a coherence time of 0.295 ps in the cases shown and can be expected to lay between 0.1 and 0.3 ps.