  The weak equivalence principle (WEP) in the quantum regime has been the
subject of many studies with a broad range of approach to the problem. Here we
tackle the problem anew through the time of arrival (TOA) operator approach by
constructing the time of arrival operator for a non-relativistic and
structureless particle that is projected upward in a uniform gravitational
field with an intended arrival point below the classical turning point. The
TOA-operator is constructed under the constraint that the inertial and
gravitational masses are equivalent, and that Galilean invariance is preserved.
These constraints are implemented by Weyl-quantization of the corresponding
classical time of arrival function for the projectile. The expectation value of
the TOA-operator is explicitly shown to be equal to the classical time of
arrival plus mass-dependent quantum correction terms, implying incompatibility
of the weak equivalence principle with quantum mechanics. The full extent of
the violation of the WEP is shown through the mass dependence of time of
arrival distribution for the projectile.
