Causality and relativistic localization in 1-d Hamiltonians

Superluminal motion and its connection to causality is explored in the context of quantum mechanical systems. It is shown that, while the mean velocity of a quantum wave packet may be less than c, certain portions of the wavepacket may still spread superluminally. Criteria are developed that can be used to determine if a wavepacket is spreading superluminally, and these criteria are used to show that the relativistic Schrodinger equation is non-causal and that up to 8\% of the wavepacket may violate causality. It is also verified that the Dirac and Klein-Gordon equations are causal, and extensions of this work to quantum field theories are briefly discussed.