for k > 0 . Then, using (7) and (12) it is easy to see that and its corresponding inverse Laplace transform Similarly, it is easy to obtain Once the Green’s function is constructed by the above procedure, in order to obtain the wave function it is necessary to substitute equation (28) into the equation (11) resulting in the complete solution: Mathematica was employed for these computations and for plotting the results. As mentioned earlier, the QIVP can be treated as a propagation problem. These are initial-value problems governed by a parabolic Partial Differential Equation ( PDE ) of first order in the time. A familiar representation of a parabolic PDE in one dimension is the Diffusion Equation, C. NUMERICAL METHOD. f$_{t}$ = α f$_{xx}$. (29) 8 G ( x.x ′ , t ) = C √$_{t}$ [ − e ia t ( x $^{′}$− x ) 2 + e ia t ( x $^{′}$+ x ) 2 ] . (28) g$_{1}$$_{a}$ = $_{e}$− $^{(1}$− i ) √ ms ℏ ( x $^{′}$− x ) 2 (1 − i ) √ ms ℏ (24) G$_{1}$$_{b}$ = $_{e}$− 1 4 $_{t}$[ m h $^{(1}$− i ) $^{2}$( x $^{′}$+ x ) ] 2 (1 − i ) √ πt m ℏ . (25) G$_{2}$$_{a}$ = − $_{e}$− 1 4 $_{t}$[ m h $^{(1}$− i ) $^{2}$( x $^{′}$− x ) ] 2 (1 − i ) √ πt m ℏ (26) G$_{2}$$_{b}$ = $_{e}$− 1 4 $_{t}$[ m h $^{(1}$− i ) $^{2}$( x $^{′}$+ x ) ] 2 (1 − i ) √ πt m ℏ . (27) 1 m Defining the following constants C ≡ $^{2(1}$− i ) √ mπ ℏ , and a ≡ m 2 ℏ the total Green’s function is found to be: Ψ ( x, t ) = i αC √$_{t}$ ∫ ∞ 0 dx ′ ( − e ia t ( x $^{′}$− x ) 2 + e ia t ( x $^{′}$+ x ) 2 ) Ψ$_{I}$ ( x ′ , 0)