On the mutual exclusiveness of time and position in quantum physics and the corresponding uncertainty relation for free falling particles

Abstract

The uncertainty principle is one of the characteristic properties of quantum theory, where it signals the incompatibility of two types of measurements. In this paper, we argue that measurements of time-of-arrival $T_x$ at position $x$ and position $X_t$ at time $t$ are mutually exclusive for a quantum system, each providing complementary information about the state of that system. For a quantum particle of mass $m$ falling in a uniform gravitational field $g$, we show that the corresponding uncertainty relation can be expressed as $ΔT_x ΔX_t \geq \frac{\hbar}{2mg}$. This uncertainty relationship can be taken as evidence of the presence of a form of epistemic incompatibility in the sense that preparing the initial state of the system so as to decrease the measured position uncertainty will lead to an increase in the measured time-of-arrival uncertainty. These findings can be empirically tested in the context of ongoing or forthcoming experiments on measurements of time-of-arrival for free-falling quantum particles.

Figures (2)

• Figure 1: Schematic representation of the two protocols A and B to measure the position-distribution and time of arrival-distribution. On the left panel, we represent the position distribution that is constructed from the measurement of the position of a particle using detectors located at positions $x_k,\ k=1,\cdots,n$ switched on at time $t$. We repeat the same experiment $N$ times and record each time the location where the particle is detected. On the contrary, the protocol B shown on the right panel, which allows for the empirical construction of the time-of-arrival distribution, consists of measuring the presence of a particle with one single detector located at a fixed position $x$ that is switched on at a given time $t_1$. If the particle is detected, we record it and repeat the experiment $N$ times to increase the power of the associated statistical measure. We then repeat the same experiment but choose a different time of detection $t_2$, then $t_3$, then $t_4$, ..., $t_n$. Notice that repeated measurements should not be allowed with the detector being switched on at $t=t_1,\ t_2,\ ..,\ t_n$ for the same experiment, which explains why we need a larger total number of trials compared to protocol A. Repeated measurement would indeed destroy the quantum coherence of the particle, and would not allow to construct an empirical distribution for the time of a first measurement at position $x$.
• Figure 2: TOA-position uncertainty relation and the mean value of TOA for a free-falling particle. On the left panel, we show the value of the product $\Delta T_x \Delta X_0$ as a function of the initial standard deviation $\Delta X_0 = \sigma$ obtained from the numerical integration of the standard deviation of the stochastic variable given by equation \ref{['Eq:TxSol']} (continuous-red-line), as well as the values obtained in the far-field regime as per equation \ref{['Eq:Uncertainty:Farfield']} for $q\ll 1$ (dashed-blue-line) and equation \ref{['Eq:Uncertainty:Farfield:qlarge']} for $q\gg 1$ (dashed-dotted-blue-line), and in the near-field regime \ref{['Eq:Uncertainty:Nearfield']} (dotted-blue-line). As we see from this graph, the bound $\frac{\hbar}{2mg}$ (bottom dashed-blue line) is universal for all regimes. On the right panel, we display the mean values of the TOA (continuous-red-line) as well as the classical TOA $t_c= \sqrt{\frac{2x}{g}}$ (dashed-blue-line) and the two asymptotes corresponding to the near-field (dotted-blue-line) and the far-field with $q\gg 1$ (dashed-dotted-blue-line). Notice that the mean value of the TOA is always greater than the classical TOA, showing the discrepancy between the two values of TOA obtained from experience A (classical TOA) and B (red-continuous curve). In these two graphs, we took the values for $x=10^{-5} \text{m},\ g=9.8 \text{m}\cdot\text{s}^{-2}$ and $m=1.67 \cdot 10^{-27}\text{kg}$ (hydrogen atom). We added a secondary $x-$axis at the top to visualize the evolution of $q$ as a function of $\sigma$ as well.