(8172) Propagator in a box

In this exercise you are requested to find the propagator \( U(\theta|\theta_0) \) for a particle of mass \( M \) that is free to move in the interval \( 0 < x < a \), with hard wall boundary conditions, given the time \( t \). The exercise demonstrates the equivalence of two different strategies.

(1) Obtain a result for the propagator via spectral decomposition, where the summation is over the momentum \( k \).

(2) Obtain a result for the propagator via the Feynman path integral.

(3) Use the Poisson resummation formula to establish equivalence of the two results.