Transition from discrete to continuous time-of-arrival distribution for a quantum particle

We show that the Kijowski distribution for time of arrivals in the entire real line is the limiting distribution of the time-of-arrival distribution in a confining box as its length increases to infinity. The dynamics of the confined time-of-arrival eigenfunctions is also numerically investigated and demonstrated that the eigenfunctions evolve to have point supports at the arrival point at their respective eigenvalues in the limit of arbitrarily large confining lengths, giving insight into the ideal physical content of the Kijowsky distribution.

Figure 1

The top figure shows the accumulated probability of arrival vs time at the origin for a Gaussian wave packet with mean momentum $⟨P⟩=100$, full width at half maximum ${\sigma }_x=0.05$, and initial expected value $⟨X⟩=-1$ (all the quantities in atomic units). The dots correspond to confined motion with $l=10$, and the solid line to the full line. In the lower figure we depict the corresponding probability densities (with the same notation), defined as explained in the text.

Figure 2

Accumulated probability of arrival (upper figure) vs time at the origin for two Gaussian wave packets with mean momentum $⟨P_1⟩=200$ and $⟨P_2⟩=100$ with maximum interference at the origin, e.g., $⟨X_1⟩=-1$ and $⟨X_2⟩=-0.5$. The full width at half maximum is ${\sigma }_x=0.05$ for both Gaussians. The dots correspond to confined motion with $l=10$, while the solid line corresponds to the full line. For completeness the dotted line depicts the quantum probability flux at the origin, integrated up to the relevant instant. The inset shows the zone of maximal discrepancy between integrated flux and accumulated Kijowski’s probability, where the accumulated probability for confined motion matches Kijowski’s. In the lower figure we show the corresponding probability densities (with the same notation) defined as explained in the text, as well as the quantum probability flux at the origin: in this situation there is a backflow effect [19].

Figure 3

The upper figure shows the accumulated probability of arrival at the origin vs time for two Gaussian wave packets with mean momentum $⟨P_1⟩=5$ and $⟨P_2⟩=1.5$ with $⟨X_1⟩=-1$ and $⟨X_2⟩=-0.5$. The full width at half maximum is ${\sigma }_x=0.05$ for both Gaussians. The distinct lines correspond to different values of $l$: $l=3$, long dashed line; $l=5$, dotted-dashed line, and $l=15$, circles. The solid line corresponds to Kijowski’s distribution. In the lower figure we show the corresponding probability densities (with the same notation), as defined in the text.

Figure 4

Width at half maximum (WHM) of the evolved odd and even eigenfunctions (upper and lower lines) at the corresponding eigenvalues closest to $t=0.01$ vs length $l$.

Figure 5

Probability density ${\mid {\phi }_n\left(\tau \right)\mid }^2$ vs position at the corresponding closest eigenvalue ${\tau }_n$ to $t=0.01$, for the even (upper figure) and odd (lower figure) eigenfunctions. The different lines are associated with $l=1$ (thick solid line), $l=2$ (dashed line), $l=3$ (long-dashed line), $l=4$ (dotted-dashedline), and $l=5$ (thin solid line).