# Quantum interference of position and momentum: a particle propagation paradox

## TL;DR

This paper demonstrates that quantum superpositions can violate classical bounds on particle propagation, revealing fundamental differences in causality and motion laws in quantum mechanics.

## Contribution

It introduces a new particle propagation inequality and shows its violation by quantum superpositions, challenging classical notions of particle motion.

## Key findings

- Quantum superpositions violate the derived propagation inequality.
- Classical straight-line motion assumptions are challenged.
- Quantum mechanics alters fundamental causality relations.

## 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.