Using the Feynman-Kac Path Integral Method in Computing Eigenvalues and Matrix Elements for the Infinite Square Well with a Negative Delta Potential

The exact analytical solution of the Feynman-Kac path integral for the Infinite Square Well with a negative value delta function potential at the origin is presented and compared with numerical calculations approximated by random walk simulations on a discrete grid. In addition, a method using parity symmetry on the matrix element is presented that allows higher order eigenstates to be computed. The method provides exact values in the limit of infinitesimal step size and infinite time for the lowest eigenstates.