The plane waves can be purely either positive or ``negative energy'', however,
localized states have uncertainty in the
momentum and tend to have both positive and ``negative energy'' components.
As the momentum components become relativistic, the ``negative energy'' amplitude becomes appreciable.
The cross terms between positive and ``negative energy'' will give rise to very rapid oscillation of the expected values of both velocity and position. The amplitude of the oscillations is small for non-relativistic electrons but grows with momentum (or with localization).
Jim Branson 2013-04-22