first derivative is given by: The second space derivative is The algebraic relationship develops a tri-diagonal system represented by the following equation The code presented in this work was based on the design of DTBCs derived by the work of Arnold 8,9 with the inclusion of modifications that adapt DTBCs to the particular example of the escape problem. The original problem as conceived by Arnold and derived in the dissertation of his student, Earhardt $^{10}$, evaluates the transport of a quantum particle that enters one side of a finite domain and exits from the opposite side. The Right TBC is: 10 of a set of imax simultaneous linear equations, where imax is the maximum number of grid points , imax − 1 equations come from the interior scheme and one equation comes from the Right Transparent Boundary Condition (at i = imax ). The left boundary at i = 0 is still a Dirichlet Boundary Condition and it is set to zero in order to force the function to stay in the domain. ̂ A Ψ n = ⃗ b, (37) The future time level occurs at n, the past time level at n − 1 and the algebraic relationship for the one-dimensional approximation of the Schrödinger equation is: f$_{xx}$ | n + 1 2 i = 1 2 ( f$_{xx}$ | n +1 i + f$_{xx}$ | n i ) . (35) Then the Crank-Nicholson difference equation is: f$_{t}$ | n + 1 2 i = f n +1 i − f n i △ t . (34) − Ψ n i +1 + C Ψ n i − Ψ n i − 1 = Ψ n − 1 i +1 − C ′$_{Ψ}$ n − 1 i + Ψ n − 1 i 1 (36) with C = (2 − iρ ) , C ′ = (2 + iρ ) , and ρ = $^{4}$△ x 2 △ t . i ( Ψ n i − Ψ n − 1 i △ t ) = − 1 4 ( Ψ n i +1 − 2Ψ n i + Ψ n i − 1 △ x 2 + Ψ n − 1 i +1 − 2Ψ n − 1 i + Ψ n − 1 i 1 △ x 2 ) .