Quantum interference of position and momentum: A particle propagation paradox

Optimal simultaneous control of position and momentum can be achieved by maximizing the probabilities of finding their experimentally observed values within two well-defined intervals. The assumption that particles move along straight lines in free space can then be tested by deriving a lower limit for the probability of finding the particle in a corresponding spatial interval at any intermediate time $t$. Here, it is shown that this lower limit can be violated by quantum superpositions of states confined within the respective position and momentum intervals. These violations of the particle propagation inequality show that quantum mechanics changes the laws of motion at a fundamental level, providing a different perspective on causality relations and time evolution in quantum mechanics.

Figure 1

Probability density of position observed at $t_M=mL/B$ for an optimized overlap of $\langle L|B\rangle =2/\sqrt{3}-1 [BL\approx 0.024(2\pi \hslash \left)\right]$. The dotted line marks the envelope function of the interference pattern used in the optimization of the inequality violation. The rectangular line marks the interval $M=2L$ ($\left|x\right|\le L$) and the probability density corresponding to the minimal value of $P\left(M\right)$ needed to satisfy the propagation inequality Eq. (4). The observation of probability densities at less than half the value needed to satisfy the propagation inequality demonstrates its violation by the quantum state $|\psi \rangle$, with the defect probability $P_{\mathrm{defect}}$ given by the area between the rectangular line and the experimentally observed probability density.