  Polymer Quantum Mechanics is based on some of the techniques used in the loop
quantization of gravity that are adapted to describe systems possessing a
finite number of degrees of freedom. It has been used in two ways: on one hand
it has been used to represent some aspects of the loop quantization in a
simpler context, and, on the other, it has been applied to each of the infinite
mechanical modes of other systems. Indeed, this polymer approach was recently
implemented for the free scalar field propagator. In this work we compute the
polymer propagators of the free particle and a particle in a box; amusingly,
just as in the non polymeric case, the one of the particle in a box may be
computed also from that of the free particle using the method of images. We
verify the propagators hereby obtained satisfy standard properties such as:
consistency with initial conditions, composition and Green's function
character. Furthermore they are also shown to reduce to the usual Schr\"odinger
propagators in the limit of small parameter $\mu_0$, the length scale
introduced in the polymer dynamics and which plays a role analog of that of
Planck length in Quantum Gravity.
