An Exact Solution to the Time-dependent Schrodinger Equation for a Model One-dimensional Potential

Analytical solutions to the time-dependent Shr\"{o}dinger equation in one dimension are developed for time-independent potentials, one consisting of an infinite wall and a repulsive delta function. An exact solution is obtained by means of a convolution of time-independent solutions spanning the given Hilbert space with appropriately chosen spectral functions. Square-integrability and the boundary conditions are satisfied. The probability for the particle to be found inside the potential well is calculated and shown to exhibit non-exponential decay decreasing at large times as $t^{-3}$. The result is generalized for all square-integrable solutions to this problem.