A Quantum Distance Operator and its Use in Measuring the Weak Equivalence Principle

We introduce a new non-relativistic quantum operator for the distance traveled by a particle in a given interval of time. Our operator measures the integrated expected trajectory of the particle. If its expectation value depends on the particle's mass we can infer that the particle's motion is also dependent on its mass and thus violates the Weak Equivalence Principle (WEP). As a proof of concept we use the operator to analyze the expected distance traveled by a free Gaussian wavepacket in free space with some initial momentum. The distance such a particle travels becomes close to light-like as its mass vanishes and agrees with the classical result for macroscopic masses. This result shows that different versions of the WEP, while equivalent to each other in the classical theory, are, in fact, incompatible in quantum mechanics. Also considered is the similar problem of a particle in a spherical well potential.