A Generalized FDTD Method with Absorbing Boundary Condition for Solving a Time-Dependent Linear Schrödinger Equation

The Finite-Difference Time-Domain (FDTD) method is a well-known technique for the analysis of quantum devices. It solves a discretized Schrödinger equation in an iterative process. However, the method provides only a second-order accurate numerical solution and requires that the spatial grid size and time step should satisfy a very restricted condition in order to prevent the numerical solution from diverging. In this article, we present a generalized FDTD method with absorbing boundary condition for solving the one-dimensional (1D) time-dependent Schrödinger equation and obtain a more relaxed condition for stability. The generalized FDTD scheme is tested by simulating a particle moving in free space and then hitting an energy potential. Numerical results coincide with those obtained based on the theoretical analysis.


Introduction
The 1D time-dependent linear Schrödinger equation, which is the basis of quantum mechanics [1,2], can be expressed as follows [3,4]: where m is the mass of the particle (kg), 34 1.054 10     J•sec is Planck's constant, V is the potential (J),   , x t  is a complex number, and x t x t    indicates the probability of a particle being at spatial location x at time t.
It can be easily seen that the classic explicit two-level in time finite difference scheme, i.e., is unconditionally unstable, where is the approximation of .Here, x  and t  are the spatial grid size and time step, respectively, k Z  that denotes the set of all positive and negative integers, and 2 x  is a second-order central difference operator such that There are many numerical schemes developed for solving linear Schrödinger equations .In particular, Sullivan [3] and Visscher [4] applied the finite-difference time-domain (FDTD) method, which is often employed in simulations of electromagnetic fields, to solve the above Schrödinger equation.The application of FDTD technique for the analysis of quantum devices is often called the FDTD-Q scheme, which can be described as follows [3].
The variable


is first split into its real and imaginary components in order to avoid using complex numbers: Inserting Equation (4) into Equation ( 1) and then separating the real and imaginary parts result in the following coupled set of equations: and Thus, the second-order central finite difference approximations in space and time result in the FDTD-Q F. I. MOXLEY III ET AL. 164 schemes as follows: and Here, we assume that V is dependent only on x for simplicity.The computation of the above FDTD-Q scheme is very simple and straight-forward because one may obtain from Equation ( 7) and then from Equation (8).Previously, the second author analyzed the stability of the FDTD-Q scheme using the discrete energy method and obtained a condition for determining the time step, , so that the scheme is stable as follows [13]: where c is a constant.It should be pointed out that Soriano et al. [27] and Visscher [4] also used the eigenvalue method to analyze the stability of the FDTD-Q scheme and obtained a very similar condition of 2 max 1 2 However, as pointed out in [13], the numerical solution is still divergent.Equation (9) indicates that the condition must be less than 1 but not close to 1.The motivation of this study is to apply the idea of the FDTD method to develop a generalized FDTD method with absorbing boundary condition for solving the linear Schrödinger equation, so that a more relaxed condition for stability may be obtained.

Generalized FDTD Method
To develop a generalized FDTD scheme, we assume that x t  are sufficiently smooth functions which vanish for sufficiently large x and the potential V is dependent only on x.We first rewrite Equations ( 5) and (6) as , n  We then evaluate those derivatives in Equation ( 12) by using Equations ( 10) and (11) repeatedly: , , 2 , , 2 and so on.Substituting Equation (13) into Equation (12) gives Similarly, we employ the Taylor series method to expand Again, using Equations (10) and (11) repeatedly to evaluate those derivatives in Equation (15), we obtain , , 2 , , 2 and so on.Substituting Equation ( 16) into Equation (15) gives , .
  are approximated using some accurate finite differences, one may obtain a generalized FDTD scheme for solving the time-dependent linear Schrödinger equation as follows: It should be pointed out that in Equation (18a) , n x t    may be approximated by a higher-order accurate Lagrange polynomial or some other higher-order accurate approximations.Once is obtained from Equation (18a), one may construct a similar higherorder accurate Lagrange polynomial or some other higher-order accurate approximations for . Here, for simplicity, we limit ourselves to using finite difference approximations for the Laplace operator A. Furthermore, it can be seen from the above derivations that Equation ( 18) can be readily generalized to the multi-dimensional cases.For the case where the potential V is dependent on both temporal and spatial variables, the derivations are similar to those in Equation ( 16) except that the product rule of derivative with respect to t should be used.

Stability
In order to prevent the numerical solution from diverging, we need to analyze the stability of the generalized FDTD method in Equation (18).Here, we consider that the Laplace operator A is only approximated by either a second-order central difference operator , where and similar finite difference approximations for We assume that V is a constant and use the Von Neumann analysis [34] to analyze the stability of the generalized FDTD schemes.To this end, we first let Replacing A with where It can be seen that implying that, when Equation ( 24) is automatically satisfied, and, hence, the scheme with , is unconditionally stable.However, we cannot choose and, therefore, the generalized FDTD scheme should be imposed the condition in Equation (24).Noting that the condition in Equation (24) gives only where c is a constant.Using a similar argument, we may obtain the same inequality as that in Equation ( 26) for imag .
 Hence, we obtain the following theorem.Theorem 1.The generalized FDTD scheme is stable if Equation ( 26) is satisfied.
It can be seen that when N = 0 the condition in Equation (26) reduces to that in Equation (9).Also, the accuracy of the scheme is   x t    Similarly, for the fourth-order central difference and substitute them into Equation substituting Equat into Equation ( 18), and deleting the common factor quadratic equation for ion (28) , we obtain a real ik x e   as fols: low where Hence, we use a similar argument as before and o the following theorem.Theorem 2. The generalized FDTD scheme btain is stable if the following condition is satisfied where c is a constant.The accuracy of the scheme is It can be seen from the ger N, the evaluation for  

Absorbing Boundary Condition
When t expansive.Therefore, he particle travels and hits the boundary, it will on.This will to create an it is our suggestion to choose reflect back to the domain under considerati distort the wave packet solution.It is ideal absorbing boundary condition so that the particle will not reflect back.Here, we develop a second-order absorbing boundary condition (ABC) which is obtained from analyzing the group velocity of the wavepacket at the boundaries [15].To this end, we assume group velocities of the traveling particle to be obtained as Thus, the differential form of the wavenumber can be   , 0.
Since a wave maintains various components with different group velocities, we impose a highe ary condition as follows: It should be pointed out that for a wave traveling towards the left, and are substituted b v is equal to 2 v the compont of the wave with group velocity 1 v (or 2 v ) will be absorbed to the second order With Equations ( 5), ( 6) d ( 34), the w efunctions at the left and right boundaries can be determined s .

 
, , , 0, , the upper signs in Equation ( 35) apply to the left boundary, whereas the lower signs apply then use the second-order finite difference schemes to ap , , Upon substituting Equations ( 37) and (38) into Equation (35), we obtain discrete absorbing bou tions as follows:

Numerical Examples
To test the stability of the generalized FDTD schem Equation (27) and Equation ( 30) with discrete absorbing boundary conditions, Equation (39), we employed the present schemes and the original FDTD scheme to simuspace and then hitting an es in late a particle moving in free energy potential as tested in [3].To this end, we initiated a particle at a wavelength of  in a Gaussian envelop of width  with the following two equations: and e is the center of the pulse.We chose a mesh of id points and the following values for parame : Jsec, , wher 1600 0 k spatial gr ters [3] where and Based on the above formula, the electron mov space and then hits an energy potential with a total gy of about 150 eV.The energy is purely kinetic due to the fact that there is no potential energy available before the energy barrier is reached.With an increase time, the electron will propagate in the positive spatial direction.The waveform begins to spread, but the total kinetic energy remains constant.After the electron st es in free ener in rikes the potential barrier, part of the energy will be converted to potential energy.The waveform indicates that there is some probability that the electron is reflected and some probability that it penetrates the potential barrier.However, the total energy should remain constant.
In our computations, we chose N = 2 in Equation ( 27) and Equation (30), and let where  is a parameter used in [3].Using Equation (43), we rewrite the conditions in Equation (26) and Equation (31) for N = 2 as  Figures 1 and 2 show the simulation of an electron moving in free space and then hitting a potential of 100 eV, which was obtained by using the original scheme (N = 0) with μ = 0.46875.It can be seen that when μ = 0.46875 (in which  is stable and indeed the numerical , the FDTD-Q scheme ondition milar results solution does not diverge.Figure 1 shows that when the absorbing boundary condition is not imposed, the wavepacket is distorted at are obtained when we used the generalized FDTD scheme (N = 2) with μ = 0.46875.It is noted that when μ = 0.5 the original FDTD-Q scheme produces a divergent solution, because which violates the stability condition.Thus, we employed the generalized FDTD scheme, Equation (27) with N = 2 and Equation (30) with N = 2 for this case.It is noted that when μ = 0.5, implying the stability condition Equa and tion ( 26) is satisfied, implying the stability condition Equation ( 31) is satisfied.
Figures 3 and 4 show the simulation of an electron moving in free space and then hitting a potential of 100 eV, which was obtained using the generalized FDTD scheme,  Equation (27) with N = 2 and μ = 0.5.It can be seen from Figure 3 that when the absorbing boundary condition is not imposed, the wavepacket is distorted at On the other hand, when an absorbing boun is imposed, the wavepacket disappears at as shown in  as shown The above numerical example indicates that both generalized FDTD schemes break through the limita 0.5) of the original FD inted out that one may obtain a larger value of μ if N is chosen to be larger in the generalized FDTD scheme.

Conclusion
We have developed a generalized FDTD method with absorbing bo dition for solving the 1D timedependent Schrödinger equation and obtain a more re- approximations are employed for spatial derivatives Numerical results coincide with those obtained based on the theoretical analysis.
and substitute them into Equation (19a).This gives 18), substituting Equation (20) into the resulting equations, and then deleting the common factor ik x e  , we obtain

2 x
indicate whether or not there is a double root with real 1   in Equation (23) (for this case, the numerical solution may still blow up), we choose the maximum value of 2 sin  and require

k
the next 800 grid points.Two quantities of importance in quantum mechanics are the expected values of the kinetic energy and the potential energy.They are cal in the simulation as follows,

4 5 4 5Figure 1 .Figure 2 .
Figure 1.Simulation of an electron moving in free spac and then hitting a potential.The original FDTD-Q schem was employed with µ = 0.46875 and no absorbing boundary condition.Here, the horizontal coordinate is k and the vertical coordinate is ψ real .e e

Figure 4 .
Figure 4. Simulation of an electron moving in free space and then hitting a potential.The second-order FDTD scheme was employed with µ = 0.5 and absorbing boundary condition.

Figure 4 . 4 5
Figures5 and 6show the simulation of an electro from Figure5that when the absorbing oundary condition is not imposed, the wavepacket is in Figure6.tion (μ < TD-Q scheme.It should be po undary con laxed condition for stability when central difference space and then hitting a potential of 100 eV, which was obtained using the generalized FDTD scheme, Equation(30) with N = 2 and μ = 0.5.Again, it can be

Figure 5 .Figure 6 .
Figure 5. Simulation of an electron moving in free space and then hitting a potential.The fourth-order FDTD scheme was employed with µ = 0.5 and no absorbing boundary condition.