http://mmg.tversu.ru ISSN 2311-1275

MMGMathematical Modelling

and Geometry

Volume 2, No 1, p. 33 – 53 (2014)

Algorithm for computing a wave packet evolution

of the time-dependent Schrodinger equation

A.A. Gusev1 O. Chuluunbaatar1,2,a S.I. Vinitsky1

and A.G. Abrashkevich3

1 Joint Institute for Nuclear Research, Dubna, Russia 2 National University of Mongolia,Ulaanbaatar, Mongolia 3 IBM Toronto Lab, Markham, Canada

e-mail: a chuka@jinr.ru

Received 18 April 2014, in final form 28 April 2014. Published 29 April 2014.

Abstract. The algorithm implemented as FORTRAN 77 program TIME6Twhich calculates, with controlled accuracy, the wave-packet evolution of the one-dimensional time-dependent Schrodinger equation on a finite time interval is pre-sented. Symmetric implicit operator-difference multi-layer schemes based on de-composition of the evolution operator up to the sixth-order of accuracy with re-spect to the time step are utilized. This decomposition is obtained via the explicittruncated Magnus expansion and Pade approximations. The additional gaugetransformations which provide the symmetry properties needed for discretization,within the framework of the high-order finite-element method, of the evolutionaryboundary problem on a finite spatial interval with the first and/or second typeboundary conditions are applied. Solution of time-dependent Schrodinger equa-tion for the Poschl-Teller two-center problem is used to illustrate an efficiencyof the proposed schemes by comparing the computational error and executiontime with those obtained by conventional symmetric splitting exponential opera-tor techniques using the Pade approximations and fast Fourier transform method.The program is applied to the benchmark calculations of the exactly solvablemodel of a one-dimensional time-dependent oscillator.

Keywords: time-dependent Schrodinger equation, finite-element method, partialdifferential equations, high-order accuracy approximations

PACS numbers: 02.30.Jr, 02.60.Lj, 31.15.Pf

The authors thank partial support RFBR Grants Nos. 14-01-00420 and 13-01-00668,and the theme 05-6-1119-2014/2016 “Methods, Algorithms and Software for ModelingPhysical Systems, Mathematical Processing and Analysis of Experimental Data”.

c©The author(s) 2014. Published by Tver State University, Tver, Russia

34 A.A. Gusev et al

.1 Introduction

Solutions of the time-dependent Schrodinger equation (TSDE) with a required accu-racy are needed for the control problems of the finite-dimensional quantum systems[1] and molecular processes [2], the decay problem in nuclear physics [3], the ioniza-tion problems of atomic and molecular physics in pulse fields [4, 5, 6, 7, 8], impactelectron-atom and electron-molecule ionization [9, 10] and others. For solving theTDSE in a finite-dimensional spatial domain [11, 12], it is common to seek a wave-packet solution expanded over an appropriate angular basis and then apply a certaindiscrete numerical scheme to the resulting system of the ordinary second-order dif-ferential hyperradial equations [13, 14] (e.g., finite-difference [15, 16, 17, 18], spline[19, 20] or finite-element [21, 22] method).

There are two main requirements for numerical methods that are used in practicefor numerical solution of the Cauchy problem for multidimentional time-independentSchrodinger equation. More specifically, such methods should be stable and guar-antee a high accuracy of discretization in the time-dependent and spacial variables.In that respect, splitting methods have important advantages, namely, unitarityof the evolution operator preserves the norm of a wave packet and guaranteespreservation of probability and unconditional stability of the method [23]. Thisapproach has been used in Refs. [24, 25, 26] to construct nonsymmetric evolution-ary schemes on the basis of the Magnus expansion of unitary evolution operatorusing the unitary [M/M ] Pade approximation up to the sixth-order of accuracyin the time-dependent variable. This method has been successfully applied toone-dimensional and three-dimensional time-independent Schrodinger equation, forexample [3, 6, 16]. Nevertheless, this scheme when used for numerical solution oftime-independent Schrodinger equation leads to systems of algebraic equations withnonsymmetric complex matrices of large dimensions which complicates achievementof a given accuracy of the required approximate solution.

Hence the construction of more economical and stable symmetric schemes is animportant and challenging problem [27, 28, 29, 30]. For its solution one could usean approximation of a wave packet in angular variables by the Kantorovich methodand finite element method (FEM) in the radial variable [31, 32]. Efficiency of suchcombination for the described class of problems has been proven in [21, 22, 33, 34,35].

In paper [13], a more efficient approach has been developed for constructionof operator-difference schemes with symmetric operators on a basis of the explicitMagnus expansion, i.e., expansion of logarithm of evolution operator into Taylorseries over a step of time-dependent variable on a uniform mesh [36, 37]. In this ap-proach, coefficients of evolution operator expansion are calculated in explicit formfrom a system of recurrence relations. Using partial splitting of evolution opera-tor with the help of gauge transformation dependent on the operators, multi-layeroperator-difference schemes with symmetric operators up to the fourth-order of ac-curacy in step of time-dependent variable have been constructed for Hamiltonianof a general type, as well as scheme up to the sixth-order of accuracy under the

Time-Dependent Schrodinger Equation 35

condition of equality to zero of a commutator of the first and second partial deriva-tives of the Hamiltonian in time-dependent variable. This circumstance allows usto use this approach for solving a wide range of evolutionary boundary problemswith a required accuracy.

The main advantage of the Magnus proposal is that very often the truncatedseries still share important qualitative properties with the exact solution at variancewith other conventional methods of perturbation theory. For instance, in classicalmechanics the symplectic character of the time evolution is preserved at every orderof approximation. Similarly, the unitary character of the time evolution operatorin quantum mechanics is also preserved [38].

Another advantage of the symmetric operator-difference schemes considered isthat the number of exponential operators is considerably less than in similar conven-tional symmetric split-operator schemes (without commuting operators and withcommuting operators) with the same accuracy [39, 40, 41, 42, 43, 44, 45, 46]. Forexample, the sixth-order symmetric split-operator scheme [44, 45, 46] with a min-imal number of exponential operators has fifteen exponential operators includingeight exponential operators with Laplacian, while the scheme considered has onlythree exponential operators. In the framework of the unitary [M/M ] Pade ap-proximation, the sixth-order scheme considered contains only five layers, but thesymmetric split-operator scheme has 24 layers! Therefore for the high-order sym-metric split-operator schemes one usually uses the fast Fourier transform techniquesfor the exponential of the Laplacian on only uniform mesh in the spatial variable.

However, there are sets of evolutionary boundary problems in complex domainsand effective potentials with practical applications for solving of which one canuse the nonuniform mesh too, e.g., [47]. From this point of view the FEM andvariational methods will play a fundamental role in elucidating dynamical aspects ofboth basic quantum mechanics and quantum devices. The FEM is especially usefulin this respect due to its adaptability and flexibility in using the nonuniform meshes[48]. The application of the FEM in designing and modeling a new generation ofquantum devices will define the field of quantum wave function engineering [49].

The main purpose of this paper is to present the algorithm for solving one-dimensional TDSE using the second-, fourth- and sixth-order approximations inthe time-dependent variable and achieving the desired level of accuracy for a suf-ficiently smooth solution in spatial variable approximated on a nonuniform finiteinterval by means of the high-order FEM. This work presents the next step increating a set of programs for numerical solutions of the initial-boundary problemfor multi-dimensional equations of the Schrodinger type with a required accuracystarted in [24, 25]. It is based on our set of programs which implement both theKantorovich method and FEM [21, 22, 34, 35]. In this paper we present the al-gorithm implemented as the FORTRAN 77 program TIME6T for calculating anapproximate solution of the TDSE with controllable high accuracy and analysis ofits efficacy in comparison with conventional calculation schemes based on symmet-ric splitting exponential operator techniques using the Pade approximations and

36 A.A. Gusev et al

fast Fourier transform method.The paper is organized as follows. In Section 2 we give a brief overview of the

problem. In section 3 the general formulation and the high-order operator-differencemulti-layer calculation schemes in time variable are presented. An efficiency ofcomputational schemes and comparison with conventional ones are presented inSection 4. Test deck is discussed in Section 5.

.2 Statement of the problem

Consider the Cauchy problem for the time-dependent Schrodinger equation on timeinterval t ∈ [t0, T ]

ı∂ψ(x, t)

∂t= H(x, t)ψ(x, t), ψ(x, t0) = ψ0(x), (1)

H(x, t) = −12

∂2

∂x2+ f(x, t).

We require the solution ψ(x, t) to be continuous, have the general first derivativesthat are square integrable and belonging to the Sobolev space W1

2 ([xmin, xmax] ⊗[t0, T ]). Also we suppose that function f(x, t) is of sufficiently high smoothnessin spatial variable x, which has continuous partial derivatives up to the 2M-order(M = 1, 2, 3 is the order of approximation) in time variable t.

The first and second type boundary conditions and normalization condition inspatial variable x on finite interval [xmin, xmax] have the form

∂ψ(x, t)

∂x

∣

∣

∣

∣

x=xmin

= 0, or ψ(xmin, t) = 0, (2)

∂ψ(x, t)

∂x

∣

∣

∣

∣

x=xmax

= 0, or ψ(xmax, t) = 0, (3)

‖ψ(x, t)‖2 =∫ xmax

xmin

|ψ(x, t)|2dx = 1, t ∈ [t0, T ]. (4)

.3 High-order operator-difference multi-layer calculation

schemes in time variable

Eq. (1) can be rewritten in terms of unitary evolution operator U(t, t0) carryingthe initial state ψ0(x) to the solution ψ(x, t):

ı∂U(t, t0)

∂t= H(x, t)U(t, t0), U(t, t) = 1, (5)

which is considered on the uniform grid Ωτ [t0, T ] = t0, tk+1 = tk + τ, tK = T withtime step τ on the time interval [t0, T ]. Unitary operator U(tk+1, tk) carrying the

Time-Dependent Schrodinger Equation 37

solution ψ(x, tk) at t = tk (k = 0, . . . , K − 1) to the ψ(x, tk+1) at t = tk+1 can beexpressed in the form [13, 24, 25]

ψ(x, tk+1) = U(tk+1, tk)ψ(x, tk), U(tk+1, tk) = exp (−ıτAk) , (6)

Here the effective time-independent Hamiltonian is related to the original oneH(x, t) by the Magnus expansion [36, 37],

Ak =1

τ

∫ tk+1

tk

dt1H(x, t1) +ı

2τ

∫ tk+1

tk

dt1

∫ t1

tk

dt2[H(x, t2), H(x, t1)] + . . . ,(7)

where [ , ] is the operator commutator.To solve the Cauchy problem (1)–(4) numerically at each step by transforming

ψ(x, tk) into ψ(x, tk+1), we use the the explicit truncated Magnus expansion (7) ofthe evolution operator U(tk+1, tk) up to order O(τ 2M+1)

U(tk+1, tk) = exp(

−ıτA(M)k

)

+O(τ 2M+1). (8)

Now, we would like to express the truncation A(M)k in terms of H(x, t) and its time

partial derivatives. Substituting the Taylor expansion of H(x, t) in the vicinity oftc = tk + τ/2

H(x, t) =2M−1∑

j=0

(t− tc)jj!

∂jtH(x, tc) +O(

τ 2M)

(9)

into the integrals, one can derive an analytical expression of the operators A(1)k ,

A(2)k , . . . , A

(M)k by means of the symbolic algorithm GATEO (Generation of Ap-

proximations of the Time-Evolution Operator) [28]. To show the complexity ofcalculations, we present the first three approximations of the exponential (8) for

the final effective Hamiltonians A(M)k in the form A

(M)k = A

(M)k + ıA

(M)k

A(1)k = H,

A(1)k = 0,

A(2)k = A

(1)k +

τ 2

24

..H, (10)

A(2)k = A

(1)k +

τ 2

12[H,.H ],

A(3)k = A

(2)k +

τ 4

1920

....H − τ 4

720[H, [H,

..H ]]− τ 4

240[.H, [

.H,H ]],

A(3)k = A

(2)k −

τ 4

480[...H ,H ] +

τ 4

480[..H,.H ] +

τ 4

720[H, [H, [H,

.H]]],

where H ≡ H(x, tc),.H≡ ∂tH(x, t)|t=tc

, . . .

38 A.A. Gusev et al

Table 1: The real and imaginary parts of the coefficients α(M)ζ , ζ = 1, . . . ,M ,

M = 1, 2, 3.

M ζ Reα(M)ζ Imα

(M)ζ

1 1 +0 −12 1 −1/

√3 ≈ −0.577350269189625 −1

2 2 +1/√3 ≈ +0.577350269189625 −1

3 1 −0.814799554248922 −0.8540567306516633 2 +0.0 −1.2918865386966733 3 +0.814799554248922 −0.854056730651663

After the application of the known generalized [M/M ] Pade approximation [50]to the exponential operator (8), the scheme (6) yields to the following implicitoperator-difference [13, 24, 25]

ψ0k = ψ(x, tk),(

I +τα

(M)ζ A

(M)k

2M

)

ψζ/Mk =

(

I +τα

(M)ζ A

(M)k

2M

)

ψ(ζ−1)/Mk , ζ = 1, . . . ,M, (11)

ψ(x, tk+1) = ψ1k,

where I is the unit operator. The coefficients, α(M)ζ (ζ = 1, . . . ,M), stand for the

roots of the polynomial equation 1F1(−M,−2M, 2Mı/α) = 0, where 1F1(a, b, x) isthe confluent hypergeometric function and the overline indicates the complex con-jugate. Table 1 lists the values of the coefficients α

(M)ζ for M = 1, 2, 3.

Note, that this approach preserves the unitarity of the approximate evolutionoperator, since the truncated A

(M)k is always self-adjoint. Imα

(M)ζ 6= 0 yields that

all the functions ψζ/Mk have an equal norm, ‖ψ0

k‖ = ‖ψ1/Mk ‖ = · · · = ‖ψ1

k‖.The scheme (11) has two disadvantages. Firstly, this scheme contains the non-

symmetric operator A(M)k atM ≥ 2. Secondly, this scheme contains the third-order

differential operator by the spatial variable for implementation at M = 3, andrequires more difficult and long calculations.

To generate the schemes with a symmetric operator A(M)k (also free of the dif-

ferential operator by the spatial variable with third-order at M = 3), we apply a

gauge transformation ψ = exp(

ıS(M)k

)

ψ, that yields a new operator

A(M)k = exp

(

ıS(M)k

)

A(M)k exp

(

−ıS(M)k

)

. (12)

The unknown symmetric operator S(M)k is found from the additional condition [13]:

exp(

ıS(M)k

)

A(M)k exp

(

−ıS(M)k

)

= O(τ 2M), (13)

Time-Dependent Schrodinger Equation 39

in accordance with the well-known formula

exp(A)B exp(−A) =∑

j=0

1

j!

Aj , B

, (14)

with A0, B = B and Aj+1, B = [A, Aj, B]. We seek for S(M)k in the form of

a power series with respect to τ

S(M)k =

2M−1∑

j=0

τ jS(j). (15)

Substituting the expansion of S(M)k into condition (13) and equating the terms with

the same powers of τ in (14), we obtain a set of algebraic (or operator) recurrencerelations for evaluating the unknown operator coefficients S(j) with the initial con-dition S(0) = 0. The first three approximations of the operators (12) and (15) havethe form

A(M)k = − ∂

∂xf(M)1 (x, tc)

∂

∂x+ f

(M)2 (x, tc),

S(M)k = − ∂

∂xg(M)1 (x, tc)

∂

∂x+ g

(M)2 (x, tc). (16)

The functions f(M)1 (x, tc), f

(M)2 (x, tc) and g

(M)1 (x, tc), g

(M)2 (x, tc) are defined as fol-

lows

f(1)1 (x, tc) = f

(2)1 (x, tc) =

1

2, f

(3)1 (x, tc) =

1

2+

τ 4

720

∂2..f

∂x2,

f(1)2 (x, tc) = f(x, tc), f

(2)2 (x, tc) = f(x, tc) +

τ 2

24

..f ,

f(3)2 (x, tc) = f(x, tc) +

τ 2

24

..f +

τ 4

1920

....f +

τ 4

1440

(

∂.f

∂x

)2

− τ 4

720

∂..f

∂x

∂f

∂x− τ 4

2880

∂4..f

∂x4, (17)

g(1)1 (x, tc) = g

(2)1 (x, tc) = 0, g

(3)1 (x, tc) = −

τ 4

720

∂2

∂x2

.f,

g(1)2 (x, tc) = 0, g

(2)2 (x, tc) =

τ 2

12

.f,

g(3)2 (x, tc) =

τ 2

12

.f +

τ 4

480

...f +

τ 4

720

∂.f

∂x

∂f

∂x+

τ 4

2880

∂4.f

∂x4,

where f ≡ f(x, tc),.f≡ ∂tf(x, t)|t=tc

, . . . and tc = tk + τ/2.

40 A.A. Gusev et al

As the result, we obtain the following symmetric implicit operator-differencescheme:

ψ0k = exp

(

ıS(M)k

)

ψ(x, tk),(

I +τα

(M)ζ A

(M)k

2M

)

ψζ/Mk =

(

I +τα

(M)ζ A

(M)k

2M

)

ψ(ζ−1)/Mk , ζ = 1, . . . ,M, (18)

ψ(x, tk+1) = exp(

−ıS(M)k

)

ψ1k.

Note that, in the case of M = 1, this scheme corresponds to well-known Crank-Nicolson scheme [51]. One can see, that for M = 3, operator S

(M)k in a general

case contains the second-order differential operator ∂2/∂x2. Hence in this case

we use again the generalized [L/L] Pade approximations for exp(

±ıS(M)k

)

. This

approximation has the order of O(τ 4L+2), while 4L+2 ≥ 2M , so that we can chooseL = 1 at M = 3. In this case we lead to the following implicit operator-differencescheme [13]:

ψ0k = ψ(x, tk),(

I −α(1)η S

(3)k

2

)

ψ1k =

(

I − α(1)1 S

(3)k

2

)

ψ0k,

ψ0k = ψ1

k,(

I +τα

(M)ζ A

(M)k

2M

)

ψζ/Mk =

(

I +τα

(M)ζ A

(M)k

2M

)

ψ(ζ−1)/Mk , ζ = 1, . . . ,M, (19)

ψ0k = ψ1

k,(

I +α(1)1 S

(3)k

2

)

ψ1k =

(

I +α(1)1 S

(3)k

2

)

ψ0k,

ψ(x, tk+1) = ψ1k.

Let us present operator H(t) ≡ H(x, t) as

H(t) = H0 + g(t), (20)

where H0 is positively defined linear self-adjoint operator that is time-indepen-dent and satisfying condition ‖ψ‖ ≤ ‖H0ψ‖, and g(t) is a function with boundedderivatives in time-dependent variable up to the order 2M :

∥

∥

∥

∥

∂m

∂tmg(t)

∥

∥

∥

∥

≤ c1, 0 ≤ m ≤ 2M. (21)

Suppose further that on each time-dependent interval ti ∈ [tk, tk+1], i = 1, . . . , s +1 ≤ 2M the following estimates are valid at 0 ≤ m ≤ 2M :

∥

∥

∥

∥

(adH(ts+1)) · · · (adH(t2))∂mg(t1)

∂tmψ

∥

∥

∥

∥

≤ c2‖Hs/20 ψ‖, (22)

τ‖H1/20 ‖ < c3, τ < 2Mµ||A(M)

k (t)||−1. (23)

Time-Dependent Schrodinger Equation 41

Here ψ belongs to Sobolev space Ws2([xmin, xmax] ⊗ [t0, T ]); constants c1, c2, c3 do

not depend on time-dependent step τ , and µ ≈ 0.28 is a root of transcendentalequation µ exp(µ+ 1) = 1.

Theorem. [13] Suppose that τA(M)k , the explicit Magnus expansion of linear

self-adjoint operator H(t) = H0 + g(t) of order O(τ 2M+1), M = 1, 2, 3, at themoment of time tc = tk+τ/2 on uniform grid Ωτ [t0, T ] = t0, tk+1 = tk+τ, tK = T,satisfies conditions (21), (22), (23) at fixed time-dependent step τ . Then errorsof numerical solutions ψ(tk+1) ≡ ψ(x, tk+1) at the moment of time t = tk+1 forsymmetric operator-difference multi-layer schemes (18) and (19) are bounded withestimate

ǫM = ‖ψext(tk+1)− ψ(tk+1)‖ ≤ Cτ 2M (tk+1 − t0)‖H2M+10 ψ0‖, (24)

where ψext(tk+1) is the exact solution of evolutionary equation and C is some con-stant independent on τ and M .

Proof is straightforward following the schemes of proof in accordance with[26, 52].

From estimate (24) it follows that the symmetric operator-difference multi-layerschemes with step τ = τM < 1 at M = 1, 2, 3 have almost the same errors ǫM , ifthe following conditions are fulfilled

τ1 = τ 33 , τ2 =√

τ 33 . (25)

In each transition from ψ(tk) to ψ(tk+1) we get the 1, 2, 5-layered schemes at M =1, 2, 3, i.e., atM = 2, 3 calculation times on the same grids are about 2 and 5 timeslonger than at M = 1, respectively. Nevertheless, according to (25) in order toreach the same accuracy when selecting time-dependent steps for a scheme withM = 1, 2 the total computation time is 1/(5τ 23 ) and 2/(5

√τ3) times longer than at

M = 3, respectively. For example, if τ3 = 0.01, then the total computation timeusing schemes M = 1 and M = 2 is 2000 and 4 times longer than using schemewith M = 3, respectively. This result following from the theorem is confirmed bynumerical experiments performed in [3, 13] and in the test calculations of Section5 of the present paper.

.4 Efficiency of the computational schemes and comparison

with conventional ones

Let us consider the TDSE (1)–(4) on the finite time interval t ∈ [t0, T ] for the two-center problem with Poschl-Teller potentials [8] similar to an ionization problem[7]. We choose a special case of a resting well, A0 < 0, and a barrier, A1 > 0 (or awell A1 < 0), that moved with velocity v with respect to the resting well:

ı∂ψ(x, t)

∂t= H(x, t)ψ(x, t),

H(x, t) = −12

∂2

∂x2+

A0

cosh2(x)+

A1

cosh2(x− x0(t)), (26)

42 A.A. Gusev et al

-30 -20 -10 0 10 20 30-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

E2

En(t

)

t

E1

a)-30 -20 -10 0 10 20 30

0.0

0.2

0.4

0.6

0.8

1.0

pc(t)

pd(t)=p

1(t)+p

2(t)

p1(t)

pc(t)

p2(t)

pn(t

)

t

p1(t)

b)

Figure 1: a) Eigenenergies En(t) of the instant Hamiltonian in dependence of the time as

a parameter t. b) The bound state probabilities p1(t), p2(t) and the ionization probability

pc(t) vs the time t. Here v = 1/2, x0(t0) = vt0 = −15, t0 = −30.

where H(x, t) is an instant Hamiltonian and x0(t) = vt is an position of the centerof moving barrier. For numerical calculation with required accuracy the initialboundary problem in a spatial axis is reduced to a sufficiently large finite intervalx ∈ (xmin, xmax), with boundary and normalization conditions

ψ(xmin, t) = 0, ψ(xmax, t) = 0, ‖ψ(x, t)‖2 =∫ xmax

xmin

|ψ(x, t)|2dx = 1. (27)

We consider an example of the evolution of the wave packet in the time-intervalt ∈ [t0, T ], induced by the moving barrier (A1 = 15/8) with velocity v, with respectto the resting well (A0 = −15/8) supporting two bound states n0 = 2 with energiesE1(t = t0) ∼= EW

1 = −9/8 = −1.125 and E2(t = t0) ∼= EW2 = −1/8 = −0.125. At

v > 0 we choose the initial time t0 and the final time T from the following valuesof initial x0(t0) = vt0 = −15 and final x0(T ) = vT = 15 positions of the centerof moving barrier to preserve with required accuracy the discrete spectrum statessupported by the resting well in both the initial time t0 and the final time T . So, westart from the initial state that corresponds with required accuracy to the groundstate supported by the resting well

ψ(x, t0) ∼= ψW1 (x) = N1(cosh x)

−(√1−8A0−1)/2. (28)

Note that in the case A1 + A0 = 0 at t = 0 the potential of the problem (26) isequal to zero on the whole axis and the instant Hamiltonian H(x, t) at t = 0 haspure continuous spectrum that provide full ionization of the considered quantumsystem and capture to the discrete spectrum states during further evolution.

Calculations were performed on the spatial variable interval x ∈ (−512, 512)that is sufficient to avoid reflection from its boundaries on considered time intervalt ∈ [t0, T ]. The wave functions ψn(x; t) of discrete spectrum En < 0, and ψν

E(x; t) ≡ψ←→E (x; t) of continuous spectrum E ≥ 0 of the instant Hamiltonian H(x, t) from (26)

Time-Dependent Schrodinger Equation 43

0.00.5

-40 -20 0 20 40

-0.4

0.4

-0.4

0.4

-0.4

0.4

-0.20.2

-0.20.2

-0.20.2

-0.2

0.2

-0.2

0.2

-40 -20 0 20 40

-0.2

0.2

x

t=30

t=18

t=12

t= 6

t= 4

t= 1

t= 0

t=-1

t=-2

t=-6

a)

0.000

0.0011E-3 0.01 0.1 1

0.00

0.04

0.0

0.2

0

20

0.0

0.5

0.0

0.5

0.0

0.4

0.0

0.4

0.0

0.4

1E-3 0.01 0.1 10.0

0.4 t=30

t=18

t=12

t= 6

t= 4

t= 1

t= 0

t=-1

t=-2

t=-6

E b)

Figure 2: a) Real (solid line), image (dashed line) parts of the wave function and b)

the distribution of ionization probability pE(t) vs the time t for fixed value of velocity

v = 1/2 and initial position of the moving barrier x0(t0) = vt0 = −15, t0 = −30.

dependent on t as a parameter are calculated in spatial interval x ∈ (xmin, xmax)with the corresponding homogeneous third-type boundary conditions by modified[33] version of KANTBP program [34] using appropriate asymptotes of solutions.The subscript ν equals to → or ← that corresponds to the positive or negativedirections of the final momentum q = ±

√2E, respectively. After attaching the

asymptotes on whole axis x ∈ (−∞,+∞) these functions satisfy the conventionalorthogonality and completeness relations

∫ +∞

−∞dx(ψν′

E (x; t))∗ψν

E′(x; t) = (2π)δ(E −E ′)δνν′ , (29)

∫ +∞

−∞dx(ψν

E(x; t))∗ψn(x; t) = 0, (30)

n0∑

n=1

ψn(x; t)ψn(x′; t) +

∑

ν=→←

∫ +∞

0

dE(ψνE(x; t))

∗ψνE(x

′; t) = δ(x− x′). (31)

Dependence of eigenenergies En < 0 of the instant Hamiltonian on time pa-rameter t is shown in Fig. 1a. One can see that in a vicinity of time value t = 0the Hamiltonian has only one eigenvalue E1 < 0 and at time t = 0 it has onlycontinuous spectrum.

44 A.A. Gusev et al

0 1 2 3 4

0,0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1,0

pn(T

), p

c(T

) p1(T)

p2(T)

pc(T)

v

p1(T)

Figure 3: The probabilities p1(T ), p2(T ) of ground and first excited states, and ionization

probability pc(T ) vs velocity v for time T = 15/v and initial time t0 = −15/v when the

position of the center of barrier is equal x0(T ) = 15 and x0(t0) = −15, correspondingly.

The probabilities pn(t) and pc(t) of transition to the bound and continuumstates and its distribution pE(t) vs energy E ≥ 0 of continuum spectrum in theabove capture and ionization processes are calculated by formulas

pn(t) = |tn0(t)|2, tn(t) =

∫ xmax

xmin

dx(ψn(x, t))∗ψ(x, t) (32)

pE(t) =|t→E (t)|2 + |t←E (t)|2

2π, t

→←E (t) =

∫ xmax

xmin

dx(ψ→←E (x, t))∗ψ(x, t), (33)

and as follows from (31), at Emax ≫ 1 they satisfy, with required accuracy, thecondition

n0∑

n=1

pn(t) + pc(t) = 1, pc(t) =

∫ Emax

0

pE(t)dE. (34)

As it is mentioned above, at time moment t = 0 the effective potential is equalzero, and eigenfunctions of the instant Hamiltonian correspond to continuous spec-trum wave functions. After that the effective potential becomes again nonzero andprocesses of captures to the exited and ground states may occur which is shownby evolution of probabilities pE(t) and pc(t) in Figs. 1b and 2. In Fig. 2 it isshown that at t ≥ 1 the maxima of energy distribution pE∼1 ∼ 0.5 correspond tothe forward and backward ionization waves with similar frequencies moved to leftand right. The maxima of pc(t) at t ∼ 4 on Fig. 1b correspond to the maxima ofpE∼0.01 ∼ 1 at frame t = 4 on Fig. 2b and coincide with ionization and captureprocesses. With increasing velocity the probability densities of the excited statesat large velocity tend to zero (see Fig. 3). The wave function and the distributionof ionization probability pE(t) for some values of velocity are shown in Fig. 4. Asit is seen from Fig. 4 with increasing v the forward ionization waves dominate andtheir energy increase.

Time-Dependent Schrodinger Equation 45

0,0

0,2

0,4-80 -60 -40 -20 0 20 40 60 80

-0,4

-0,2

0,0

0,2

-0,2

0,0

0,2

-80 -60 -40 -20 0 20 40 60 80

-0,4

0,0

0,4

v=0.1

v=1

v=2

v=0.5

X a)

0,0

0,5

1,0

1,5

2,0

0,01 0,1 1 10

0,0

0,2

0,4

0,0

0,3

0,6

0,01 0,1 1 100,0

0,4

0,8

B

v=0.1

v=1

v=2

v=0.5

C

D

E

E b)

Figure 4: a) Real (solid line), image (dashed line) parts of the wave function and b)

the distribution of ionization probability pE(T ) by energy E for fixed values of velocity

v = 0.1, 0.5, 1, 2 for values of parameters given in Fig. 3.

The reason why we have chosen this problem is related to the fact that it hassuitable properties to demonstrate the efficiency of different computational unitaryschemes mentioned in introduction during an evolution of initial state in stronglyreformatted effective barrier potential that leads to excitation, ionization of thestates and capture to the states considered above.

In framework for the symmetric split-step exponential methods we have usedthe following approximations of the unitary evolution operator [46]:

U(tk+1, tk) ≡ U(tk + τ, tk) = S2M+1(tk+1, tk) +O(τ 2M+1), (35)

where M = 1, 2, 3 and the symmetric unitary operators S3, S5 and S7 have theforms

S3(tk+1, tk) = exp

(

ıtk+1 − tk

2

1

2

∂2

∂x2

)

exp

(

−ı∫ tk+1

tk

f(x, s)ds

)

× exp

(

ıtk+1 − tk

2

1

2

∂2

∂x2

)

, (36)

S5(tk+1, tk) =3∏

j=1

S3(aj , aj+1), S7(tk+1, tk) =7∏

j=1

S3(bj , bj+1),

46 A.A. Gusev et al

Table 2: Comparison of the discrepancy functions Er(t = 20, 1) for the approxima-

tions of the order O(τ 3), O(τ 5) and O(τ 7) with the time step τ = τ 33 , τ =√

τ 33 and

τ = τ3 = 0.0625, respectively.

O(τ 3) O(τ 5) O(τ 7)

Er(t, 1) CPU,s Er(t, 1) CPU,s Er(t, 1) CPU,s

Magnus 0.2521(-6) 3841 0.2296(-7) 105 0.1262(-8) 69

SSOS+Pade 0.6765(-7) 4355 0.3210(-6) 327 0.2246(-6) 208

SSOS+FFT 0.2776(-3) 1475 0.3104(-6) 62 0.2225(-6) 33

depending on the set of parameters

a1 = tk, a2 = a1 + ωτ, a3 = a2 + (1− 2ω)τ, a4 = tk+1,

b1 = tk, b2 = b1 + ω3τ, b3 = b2 + ω2τ, b4 = b3 + ω1τ, (37)

b5 = b4 + ω0τ, b6 = b5 + ω1τ, b7 = b6 + ω2τ, b8 = tk+1,

ω0 = 1− 2(ω1 + ω2 + ω3),

with the following numerical values given with the double precision accuracy:

ω = 1

2− 3√2, ω1 = −1.177679984178871, (38)

ω2 = 0.235573213359358, ω3 = 0.784513610477557.

The unitary operator S3 has three exponential operators including two expo-nential operators with Laplacian, S5 has seven exponential operators including fourexponential operators with Laplacian, S7 has fifteen exponential operators includ-ing eight exponential operators with Laplacian. According to [44, 45], the unitaryoperators S3, S5 and S7 have a minimal number of exponential operators for theindicated leading errors. We have considered two different expansions of the expo-nential of the Laplacian.

For the first one, we have used generalized [M/M ] Pade approximation [50](denoted here by SSOT+Pade). In this case the second-order scheme has twolayers, the fourth-order scheme has eight layers, and the sixth-order scheme has24 layers. For the second one, we have used the straightforward and inverse fastFourier transform (FFT) techniques for the exponential of the Laplacian, using auniform mesh (the number of grids is a power of 2) for the spatial variable (denotedhere by SSOT+FFT).

Calculation by all three schemes were performed on the spatial variable inter-val x ∈ (−512, 512) that was sufficient to avoid reflection from its boundaries onconsidered time interval t ∈ [−20, 20]. For the schemes with the Pade approxima-tion, we have used the nonuniform finite-element grid Ωx[xmin, xmax] = xmin =−512, (256),−128, (512),−32, (512), 32, (512), 128, (256), xmax = 512, where the

Time-Dependent Schrodinger Equation 47

Table 3: The same results Table 2, but only for Magnus schemes with the time step

τ = τ3 = 0.125.

O(τ 3) O(τ 5) O(τ 7)

Er(t, 1) CPU,s Er(t, 1) CPU,s Er(t, 1) CPU,s

Magnus 0.1613(-4) 456 0.1450(-5) 37 0.3635(-6) 34

number in the brackets denotes the number of finite-element in the intervals. Be-tween each two nodes we have applied the Lagrange interpolation polynomials upto the order p = 8. For FFT, the uniform spatial grid Ωx[xmin, xmax] = xmin =−512, (2048), xmax = 512 is used.

In Table 2 we present the discrepancy functions of the numerical wave packet asa function of the computation time in CPU seconds for various orders of accuracy inthe time increments. The Table shows that the fourth- and sixth-order SSOS+Padeand SSOS+FFT schemes give similar errors, nevertheless the SSOS+Pade schemesrequires much more CPU time. The fourth- and sixth-order Magnus schemes givethe best results, but require about twice CPU time than SSOT+FFT. We observedthat error of the second-order SSOS+Pade scheme is smaller than of the second-order Magnus scheme. Also the second-order SSOS+FFT scheme gives the lowaccuracy, because a norm of the wave packet at the moment t = 20 equals 0.999(i.e., correct only up to 3 digits after the decimal point). Note that both calcula-tion schemes with Pade approximations saved the norm of the wave packet duringcalculation process with the accuracy that is not less than 10−13.

In Table 3 we show the same results for Magnus schemes with doubled timestep. In this case the discrepancy function of the sixth-order Magnus scheme hasthe same order as SSOS+Pade and SSOS+FFT schemes, and calculation CPUtimes are similar to the SSOS+FFT’s ones. Analysis of these results show thatMagnus’s scheme with optimal choice of nonuniform mesh has the efficiency com-parable with SSOT+FFT, and much more efficient than SSOT+Pade. Presentedresults clearly show that calculation CPU times needed to obtain a solution withthe same accuracy are essentially lower for all sixth-order schemes than for thecorresponding lower-order schemes.

.5 Test calculations

The TDSE from (1)–(4) for a one-dimensional harmonic oscillator with an explicittime-dependent frequency ω(t) on the finite time interval t ∈ [0, T ] has the form [1]

ı∂ψ(x, t)

∂t=

(

−12

∂2

∂x2+ω2(t)x2

2

)

ψ(x, t). (39)

48 A.A. Gusev et al

0 2 4 6 8 1010

-1810

-1710

-1610

-1510

-1410

-1310

-1210

-1110

-1010

-910

-810

-710

-610

-510

-410

-310

-210

-1

2M=6

3=5.99

2M=4

2=3.99

Er(

t,j)

t

2M=2

1=1.99

Figure 5: The test results of the discrepancy functions Er(t, j), j = 1, 2, 3 (dash-dotted,

dashed and solid curves) for the approximations of order 2M = 2, 4, 6 with the time step

τ = 0.009765625.

Time-dependent frequency ω(t) and initial wave packet ψ0(x) are chosen following[13, 24]

ω2(t) = 4− 3 exp(−t), ψ0(x) =4

√

1

πexp

(

−12(x−

√2)2)

. (40)

For M = 3, functions f(M)1 (x, t), f

(M)2 (x, t) and g

(M)1 (x, t), g

(M)2 (x, t) are defined

as follows

f(3)1 (x, t) =

1

2− τ 4

240exp(−t),

f(3)2 (x, t) =

4− 3 exp(−t)2

x2 − τ 2

16exp(−t)x2

− τ 4

3840exp(−t) (24 exp(−t)− 61)x2, (41)

g(3)1 (x, t) = − τ 4

240exp(−t),

g(3)2 (x, t) =

τ 2

8exp(−t)x2 − τ 4

960exp(−t) (12 exp(−t)− 19)x2.

For M = 1 terms with orders τ 2 and τ 4 should be neglectied while at M = 2 onlyterms of order τ 4.

To analyze the convergence on a sequence of three doubly-condenced time grids,we define the auxiliary time dependent discrepancy functions

Er2(t, j) =

∫ xmax

xmin

|ψ(x, tk)− ψτj (x, tk)|2dx, j = 1, 2, 3, (42)

and the Runge coefficient

β(t) = log2

∣

∣

∣

∣

Er(t, 1)−Er(t, 2)Er(t, 2)−Er(t, 3)

∣

∣

∣

∣

, (43)

Time-Dependent Schrodinger Equation 49

where ψτj (x, t) are the numerical solutions with the time step τj = τ/2j−1. For thefunction ψ(x, t) one can use the numerical solution with the time step τ4 = τ/8.Hence, we obtain the numerical estimates for the convergence order of the numericalscheme (19), that strongly correspond to theoretical ones β(t) ≡ βM(t) ≈ 2M .

For performing an accuracy control over the numerical solution in step τ , theRunge coefficient (43) is calculated on four doubly condensed grids using the addi-tional subroutine RUNGE.

To approximate the solution ψ(x, t) in the variable x, we make use of the finite-element grid Ωx[xmin, xmax] = xmin = −11, (220), xmax = 11 and the time stepτ = 0.009765625, where the number in the brackets denotes the number of fi-nite elements for the intervals. Between each two nodes we apply the Lagrangeinterpolation polynomials up to the order of p = 8. Fig. 5 displays the behav-ior of discrepancy functions Er(t; j), j = 1, 2, 3 (dash-dotted, dashed and solidcurves, respectively) and convergence rate βM(t) for the approximations of theorder 2M = 2, 4, 6 with the time step τ = 0.009765625. To demonstrate highefficiency of sixth-order scheme, we have used the quadruple precision version ofTIME6T fortran code.

The double precision version of TIME6T fortran code with the above test cal-culations as well as with benchmark calculations [14] of the Cauchy problem forthe multidimensional time-dependent Schrodinger equation is available for user inJINRLIB [53].

The considered examples clearly demonstrate the advantages of the new methodand its efficiency. The proposed technique and software will be asked-for and willfind applications in solving wave-packed problems in physics of atomic, molecularand quantum-dimensional systems as well as in quantum calculations.

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