Quantum interference of position and momentum: A particle propagation paradox

Abstract

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 new perspective on causality relations and time evolution in quantum mechanics.Despite its great success in explaining a wide range of phenomena, quantum mechanics fails to provide a convincing explanation of how particles move in free space. In the standard formalism, the quantum state of a particle is represented by a wavefunction, and this wavefunction spreads out in space as a particle propagates. The spatial distribution of the wavefunction represents the probability of finding the particle at a given position, assuming that it is detected at the appropriate time t. However, the wavefunction at time t does not convey any information about the position of the particle at any other time. There have been a number of approaches that attempt to identify elementary trajectories which might explain the statistical patterns of quantum dynamics [1][2][3][4], but all of them suffer from untraceable additional assumptions, and there remains a great deal of ambiguity in the interpretation of quantum statistics in space and time [5][6][7][8]. Nevertheless, there is some evidence that the evolution of wavefunctions in time and space is not consistent with the assumption of a constant velocity determined by the momentum. In particular, it has been shown that the current of probability described by the Schrödinger equation can point in the opposite direction of all available momentum components [9,10]. It is therefore reasonable to ask whether quantum theory might require modifications to Newton's first law [11][12][13]. Unfortunately, it is difficult to identify such modifications using only a statistical analysis of the available data. As pointed out by Schleich et al. [6], the Wigner function can actually be interpreted as a reconstructed probability distribution over straight line trajectories defined by initial position and momentum. The deviations from classical physics caused by quantum interference then result in negative values of the quasi-probability expressed by the Wigner function. Similarly, the measurement of the wavefunction presented by Lundeen et al. [5] characterize quantum interferences in terms of the complex-valued quasi-probability of position and momentum expressed by the Dirac distribution [14]. In the light of these unresolved ambiguities, it is unclear whether the attribution of trajectories has any...

Control of particle propagation beyond the uncertainty limit by interference between position and momentum

“…A similar result can be obtained for states localized in the momentum interval |p| ≤ B/2. As shown in [15], constructive quantum interferences between two states will enhance both P (L) and P (B), resulting in a value of the probability sum that exceeds the uncertainty limit suggested by Eq. (18).…”

“…Since the condition M is a necessary consequence of motion in a straight line for particles that satisfy both the spatial condition L and the momentum condition B, it is possible to derive an experimentally testable criterion for Newton's first law based on the probabilities of the conditions L, B, M. This condition is the particle propagation inequality introduced in [15],…”

“…As has been shown recently, it is possible to derive a statistical limit for free space propagation in a straight line that refers only to initial position, initial momentum, and a second position obtained at a later time t [15]. The advantage of this approach is that it relates to the practical problem of controlling the position at time t by choosing the appropriate combination of position and momentum at time zero.…”

“…Moreover, the preparation of a rectangular wavefunction is not uncommon in optical systems, as demonstrated by experiments on spatial qubits [16][17][18][19]. However, it may be important to test whether the observed effect is robust against changes of the wavefunction, and how wavefunctions that are not perfectly confined in the respective intervals change the statistics used to demonstrate the failure of Newton's first law in [15]. In the following, I will therefore consider the violation of the statistical limits for free space propagation along straight lines for a wider class of superposition states, evaluating the localization in the position and momentum intervals for arbitrary wavefunctions.…”

Determining complementary properties using weak-measurement: uncertainty, predictability, and disturbance

“…Such states have not received much attention in the literature, which is not surprising given that it was hitherto unclear how to prepare them or project onto them. One exception is [42] which shows that a particle in a state like equation (5) can violate Newton's first law. The coherence between the particle's position and momentum allows for interference between the two properties.…”