III. RESULTS. Since we use the implicit method called Crank-Nicholson with a degree of approximation through second order that is unconditionally stable, there is no formal restriction on the step sizes, and we expected little difference between the results obtained by both methods. The real and imaginary parts of the wave function are plotted for a set of different values of the time. For simplicity, dimensionless units are used where m = ℏ = 1 . In these units we choose the length of the region to be L = 2 . The set begins with two small values of time and ends with a relatively longer value compared with the value of the natural period. In dimensionless units, T = 2 π ω = 5 . 092 , for the first ground state with a value of ω = 1 . 233 . The exact, time dependent wave function on the semi-infinite domain is obtained by replacing the value of the initial condition in the equation (29) as follows: After performing the integration using Mathematica Software, the explicit form of the wave function is where Erfi is the error function with imaginary argument $^{12}$. 13 A. ANALYTICAL METHOD. Ψ ( x, t ) = i αC √$_{t}$ ∫ L 0 $_{dx}$′ ( − e ia t ( x $^{′}$− x ) 2 + e ia t ( x $^{′}$+ x ) 2 ) sin ( $_{kx}$′$_{)}$ (47) Ψ ( x, t ) = 1 √$_{2}$ [ (( 1 4 − i 4 ) ( − 1) 1 4 $_{e}$− iπ ( πt +2 aLx ) 2 aL 2 ) ( Erfi [( 1 2 + i 2 ( − πt + aL ( L − x )) √ $_{aL}$√$_{t}$ ] − e 2 iπx L Erfi [( 1 2 + i 2 ) ( πt + aL ( L − x )) √ $_{aL}$√$_{t}$ ] − e 2 iπx L Erfi [( 1 2 + i 2 ) ( − πt + aL ( L + x )) √ $_{aL}$√$_{t}$ ] + Erfi [( 1 2 + i 2 ) ( πt + aL ( L + x )) √ $_{aL}$√$_{t}$ ] )] (48)