In the double-slit experiment, when single electrons are used, it is important to understand that the behavior of individual electrons is inherently probabilistic in nature. The experiment does not allow us to predict the exact path of a single electron; instead, it provides a statistical distribution of where electrons are likely to be detected.
If the double-slit experiment with single electrons is performed in North America, the probability of detecting an electron in Europe would be extremely low. Electrons are particles with mass and are subject to various interactions and disturbances in the environment. These interactions would cause the electron to scatter, lose coherence, and be influenced by external factors, making it highly unlikely for it to travel across the Atlantic and reach Europe without being significantly altered or detected along the way.
Additionally, it is important to note that the concept of "the same" electron becomes somewhat blurred in quantum mechanics. In quantum theory, particles such as electrons do not possess well-defined trajectories like classical objects. Instead, they exist in a superposition of states, represented by a wave function, until they are measured or interact with their surroundings. When an electron is detected, it collapses into a specific state, losing the coherence it had before.
Therefore, even if an electron somehow managed to traverse the distance from North America to Europe without being detected or significantly altered, it would not be possible to definitively claim that it is "the same" electron that was launched in the experiment. The electron's wave function would have collapsed during its journey or at the moment of detection, making it impossible to establish a direct connection between the initial electron and the final detection in Europe.
In summary, while the probability of an electron being detected in Europe in the described scenario is incredibly low, it is not possible to establish a direct correlation between the launched electron and any specific detection event due to the probabilistic and indeterminate nature of quantum mechanics.