Time of arrival of electrons in the double-slit experiment

Using Nelson’s stochastic mechanics, quantum motion of electrons in the double-slit experiment is studied numerically. It is found that not only the distribution of arrival positions but also that of arrival times at the screen forms an interference pattern. It is demonstrated that the presence time and arrival time are well interpreted in terms of stochastic mechanics.

Figure 1

Sample paths in the double-slit experiment. $N$ denotes the number of sample paths. Two slits are placed on the $X$ axis. Atomic units (a.u.) have been used.

Figure 2

First counting scheme and the multiple counting scheme. Detector’s width and area for the point $(X,Y)$ are $dX$ and $dXdY$, respectively. The traversing time in the first counting scheme is $(T_3+T_4)∕2$ for sample path 1 while the existing times in the multiple counting scheme are $(T_1+T_2+T_3+T_4+T_5)∕5$ for the sample path 1 and $(T_1^{\prime }+T_2^{\prime })∕2$ for the sample path 2.

Figure 3

(Color online) Number of electrons arrived at the screen (upper figure) and the averaged time of arrival (lower figure). The multiple counting scheme (magenta, $\circ$), first counting scheme (green, $+$), presence time (red solid line), and arrival time (blue broken line) are plotted. The window in the lower figure indicates the region shown in Fig. 4.

Figure 4

(Color online) Time of arrival distributions at the screen positions $X=28$, 29, 30, 31 $(Y=600)$. Distributions are calculated by the multiple counting scheme (red, $○$) and the first counting scheme (blue, $+$). The solid vertical lines and broken lines represent $⟨T{⟩}_{X,Y}^m$ and $⟨T{⟩}_{X,Y}^f$, respectively.