  We introduce a new semi-relativistic quantum operator for the length of the
worldline a particle traces out as it moves. In this article the operator is
constructed in a heuristic way and some of its elementary properties are
explored. The operator ends up depending in a very complicated way on the
potential of the system it is to act on so as a proof of concept we use it to
analyze the expected distance traveled by a free Gaussian wavepacket with some
initial momentum. It is shown in this case that the distance such a particle
travels becomes light-like as its mass vanishes and agrees with the classical
result for macroscopic masses. This preliminary result has minor implications
for the Weak Equivalence Principle (WEP) in quantum mechanics. In particular it
shows that the logical relationship between two formulations of the WEP in
classical mechanics extends to quantum mechanics. That our result is
qualitatively consistent with the work of others emboldens us to start the task
of evaluating the new operator in non-zero potentials. However, we readily
acknowledge that the looseness in the definition of our operator means that all
of our so-called results are highly speculative. Plans for future work with the
new operator are discussed in the last section.
