  In quantum mechanics, the operator representing the displacement of a system
in position or momentum is always accompanied by a path-dependent phase factor.
In particular, two non-parallel displacements in phase space do not compose
together in a simple way, and the order of these displacements leads to
different displacement composition phase factors. These phase factors are often
attributed to the nonzero commutator between quantum position and momentum
operators, but such a mathematical explanation might be unsatisfactory to
students who are after more physical insight. We present a couple of simple
demonstrations, using classical wave mechanics and classical particle
mechanics, that provide some physical intuition for the phase associated with
displacement operators.
