  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.
