method for solving the Schrödinger equation is required. Since it is extremely difficult to develop a numerical method for a two dimensional integro-differential equation with nonlocal transparent boundary conditions and skew boundaries, this work will develop an essential numerical method for obtaining the solution of the non-relativistic one-dimensional Schrödinger equation. The most demanding part of this work is the implementation of Transparent Boundary Conditions ( TBCs ) $^{8,9}$. The TBCs , recently developed by mathematicians, arise in the ne- cessity to deal numerically with the natural infinite domain of the wave function. That is, due to the limitation of the computer core size, the infinite wave solution of the Schrödinger equation has to be solved in a finite domain by imposing artificial boundary conditions. The criterion of transparency is that the incident wave at the boundary has the smallest reflection coefficient possible. If the solution with these artificial conditions agrees with the infinite solution, the artificial boundary conditions are said to be transparent. From the several available approaches to derive TBCs , this work is based on the excellent treatment carried out by Anton Arnold $^{8}$and his student Mathias Earhardt9,10 that concerns the transport of a quantum particle that enters one side of a finite domain and exits from the opposite side. In the following, in the next section, we first set up the “ Escape Problem , i.e. the escape of a particle from a finite region, as a Quantum Initial Value Problem ( QIVP ). We then develop two different approaches for solving the EP . The first is the analytic method that provides the certainty of the result. The second is the numerical method, where an algorithm will be meticulously presented by introducing Discrete Transparent Boundary Conditions (DTBC) $^{10}$, based on the implicit Crank-Nicholson method. Subsequently, the consistency of both methods is confirmed in section III by comparing the solutions for the real and imaginary parts of the wave function at different times, showing an excellent agreement. In the last section, conclusions are drawn from the work that has been accomplished, stressing the valuable physical information provided by the wave function as well as the numerical importance of the method developed in this work. 3