A. DESCRIPTION OF THE SYSTEM. II. THEORY The system representing the one-dimensional Escape Problem ( EP ) is a group of particles restricted inside a region of size L delimited by boundaries and free of external influences where, abruptly, one of the boundaries becomes transparent, allowing particles to escape. The group of particles is a dilute gas rarefied enough to be considered as a “ Knudsen Gas ” 11 whose density is so small that only the interactions with the boundaries are relevant. From the perspective of quantum mechanics, the EP is a QIVP . Thus, the intention is to solve the 1-D Schrödinger equation from a given initial condition. The EP is similar to a diffusion type problem; it has the same structure as propagation problems in which a partial differential equation is solved with the aid of a known initial value. The evolution of the gas inside the region 0 < x < L will be determined by the wavefunction at the initial time. To demonstrate the approach, here the initial wavefunction is chosen to be the ground state of an Infinite Well Potential of length L . Therefore, the initial value or the initial condition inside of the box is with k = π L . Of course, outside of the box the value of the initial wave function vanishes. The evolution of the EP is represented in the figure 1. The non-relativistic, time dependent, one-dimensional Schrödinger equation is with the initial condition: The Laplace Transform Method is one of the many appropriate techniques to solve differ- ential equations. The first task is to determine the Laplace transform of the wave function and its time derivative. For this particular example the definition of the Laplace transform 4 B. ANALYTICAL METHOD. Ψ$_{I}$ ( x, 0) = sin ( kx ) (1) ∂ Ψ ( x, t ) ∂t = i ℏ 2 m ∂ $^{2}$Ψ ( x, t ) ∂x 2 (2) Ψ ( x, t ) | t =0 = Ψ$_{I}$ ( x, 0) . (3)