An algorithm for the numerical solution of the Schr\"odinger equation in the
case of a time dependent potential is proposed. Our simple modification
upgrades the well known method of Koonin while negligibly increasing the
computing time. In the presented test the accuracy is enhanced by up to an
order of magnitude.Comment: 4 pages, Revtex, Published in J. Comp. Physics 119, (1995) p. 206-20

Blendowske, R.

Schaefer, R.

English

arXiv

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An algorithm for the numerical solution of the Schr\"odinger equation in the case of a time dependent potential is proposed. Our simple modification upgrades the well known method of Koonin while negligibly increasing the computing time. In the presented test the accuracy is enhanced by up to an order of magnitude

Schäfer, R

Blendowske, R

CERN Document Server

On the Numerical Solution of the Time Dependent Schro¨dinger Equation
R. Scha¨fer
University of Siegen, Fachbereich Physik, D-57068 Siegen, Germany,
R. Blendowske
Leica Mikroskopie und Systeme GmbH, D-35530 Wetzlar, Germany
(September 1994)
An algorithm for the numerical solution of the Schro¨dinger equation in the case of a time de-
pendent potential is proposed. Our simple modication upgrades the well known method of Koonin
while negligibly increasing the computing time. In the presented test the accuracy is enhanced by
up to an order of magnitude.
02.60.+y, 25.85.-w
The microscopic description of many{body systems like atoms or nuclei is based on a many{body Hamiltonian.
The related wave functions are given by Slater determinants for fermions. In the case of time dependent processes like
atomic or nuclear collisions, nuclear ssion or fusion, however, the situation is too complex due to the great number of
degrees of freedom. Therefore, in most cases, a collective coordinate is introduced according to the essential physical
properties of the considered system. This procedure leads to a macroscopic model with one degree of freedom, which
is governed by an eective one{body Schro¨dinger equation including a time dependent potential in the considered
examples [1]. The well known coordinate representation reads
ih
@
@t
 (r; t) =

−
h2
2M
r2 + V (r; t)

 (r; t) : (1)
A momentum dependent potential V , leading to an integro-dierential equation, is excluded here. In almost all
practical cases eq. (1) has to be evaluated numerically. The standard technique essentially consists of the following
two steps: rst apply a suitable scheme for the space discretization and then perform the time integration.
The algorithm presented here is only related to the second step: The time integration in the case of an explicitly
time dependent potential V (r; t). Our proposed convenient modication upgrades the standard method to a much
more ecient version while negligibly increasing the computing time.
Our experience is due to the description of ternary ssion, i.e. a ssion process accompanied by the emission of an
{particle, with results to be published elsewhere. However, the presented algorithm might be of broader interest,
not necessarily restricted to this nuclear physics theme.
In this letter we will discuss the application for the case of a cylindrically symmetric, time dependent potential,
essentially following Koonin’s procedure [2], and present a test using an analytically solvable example.
As usual, in a cylindrically symmetric case, the angle dependence on ’ of the wave function is separated as
 =
1
p
2
 (; z; t) exp(i’) ; (2)
and a grid is dened by
j = (j −
1
2
)  ; j = 1; : : : ; N ;
zk = k z ; k = −Nz; : : : ; Nz : (3)
For simplication we introduce now
gj;k =
p
j  j;k (4)
leading to the space{discretized equation
ih
@
@t
gj;k = (Hg)j;k = (vg)j;k + (hg)j;k ; (5)
where the Hamiltonian H is split in a \vertical" (v) and a \horizontal" (h) part
1
(vg)j;k = −
h2
2M
cj gj+1;k − 2 gj;k + cj−1 gj−1;k
 2
+ Uj;k gj;k ;
(hg)j;k = −
h2
2M
gj;k+1 − 2 gj;k + gj;k−1
z 2
+ Uj;k gj;k ; (6)
with the abbreviations
cj =
jp
j2 − 1=4
;
Uj;k =
1
2
Vj;k +
2h2
4M2j
: (7)
Up to this point no changes are made in comparison to Koonin’s algorithm. We solve eq. (5) iteratively, with times
tn = nt, using a Taylor expansion of gjk(tn) = g
(n)
jk , up to and including (t)
2 and obtain
g(n+1) =

1−
i
h
H (t)1
−
1
2h2
H2 (t)2 −
i
2h
_V (t)2 +O
(
t 3

g(n) ; (8)
where all terms on the right hand side are evaluated at time tn. The time derivative _Vj;k = @Vj;k=@t is included to
ensure consistency up to second order contributions.
Inserting g(n) on the right hand side is not an advisable method to determine g(n+1), because of serious numerical
instabilities. Therefore, an \alternating direction implicit method" [3] is used, leading to a modied version of
Koonin’s algorithm
g(n+1) =

1 +
i
2h
v t+
i
8h
_V (t)2
−1

1−
i
2h
h t−
i
8h
_V (t)2


1 +
i
2h
h t+
i
8h
_V (t)2
−1

1−
i
2h
v t−
i
8h
_V (t)2

g(n) ; (9)
where again all terms on the right hand side are evaluated at time tn. Our result may be veried by an expansion in
terms of t up to second order. There are no restrictions on the commutation of [v; h]. By neglecting terms dependent
on _Vj;k, our expression reduces to the one used by Koonin et al. in [2].
For numerical convenience we now dene w(n) by
1 +
i
2h
h t+
i
8h
_V (t)2

w(n) =

1−
i
2h
v t−
i
8h
_V (t)2

g(n) (10)
leading to 
1 +
i
2h
v t+
i
8h
_V (t)2

g(n+1) =

1−
i
2h
h t−
i
8h
_V (t)2

w(n) : (11)
In order to determine w(n), and thereafter g(n+1), one has to invert tridiagonal matrices only, which can be performed
by a suitable algorithm [4].
A central point for the eciency of our modication is that the determination of
_V
(n)
j;k =
V
(n)
j;k − V
(n−1)
j;k
t
(12)
up to order (t)0 poses no appreciable eort. Anyway, the values of Vj;k have to be calculated for all time steps in
order to determine v and h. It is deemed unsuitable to compensate this by determining v and h at an intermediate
2
time step t+ t=2 because one has to calculate the potential at this additional time step, as well. In many cases this
produces appreciable eort. For example, in our calculations concerning the ternary ssion it took approximately the
same time to calculate V (n) for one value of n on the whole grid as to perform one iteration step from g(n) to g(n+1).
To test our algorithm, with regard to the modication due to keeping terms dependent on _V , we consider the
following, explicitly time dependent potential
V (r; t) =
1
2
M!2 r 2 − 2h!2 t : (13)
At time t = 0 this is the potential of the harmonic oszillator with frequency !. For the initial wave function we choose
an eigenfunction of this harmonic oszillator
 (r; t = 0) = c‘ H‘(
p
 z)
p
= exp

−

2

2 + z2

; (14)
where  = M!=h, ‘ 2 f0; 1; 2; : : :g, H‘ = Hermite polynomial, and c‘ = constant of normalization.
The analytical solution is given by
 (r; t) =  (r; t = 0) exp
(
− i! [‘+ 3=2] t + i!2 t2

: (15)
The results are displayed in Table I for dierent radial quantum numbers ‘ (dierent numbers of nodes of the wave
function). Both parts of the wave function|real and imaginary|are considered separately. We use ! = 2, t = 1=60
with 60 time steps and grid parameters N = Nz = 64 and  = z = 0:25. The enhancement in accuracy becomes
as large as an order of magnitude. The norm of the wave function is (nearly) conserved, since the algorithm|modied
or not|is nearly unitary [2].
We note that the considerations for a one dimensional time dependent potential are quite the same. One obtains
 (x; t+ t) =

1 +
i
2h
H t+
i
4h
_V (t)2
−1

1−
i
2h
H t−
i
4h
_V (t)2

 (x; t) +O
(
t 3

:
(16)
Again the calculation of _V (x; t) poses nearly no eort. In contrast to the ordinary method, where the _V terms are
neglected [4], the above expression is accurate up to order (t)2. The approximation (16) is unitary; it automatically
maintains the normalization of . We note that no alternating direction method is needed in the one dimensional
case.
We summarize: In the case of a time dependent potential the standard algorithm of Koonin can easily be modied
to obtain a greater accuracy. Therefore, terms depending on _V have to be included to get a consistent time expansion
up to order (t)2. The related algorithm has been shown to be superior in the presented numerical test.
[1] K. Langanke, J. A. Maruhn, S. E. Koonin, Computational Nuclear Physics 2, Nuclear Reactions (Springer-Verlag, New
York 1993)
[2] S. E. Koonin, K. T. R. Davies, V. Maruhn-Rezwani, H. Feldmeier, S. J. Krieger, J. W. Negele, Phys. Rev. C15 (1977) 1359
[3] R. S. Varga, Matrix Iterative Analysis (Prentice-Hall, Englewood Clis 1962)
[4] W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge University Press 1989)
3
TABLE I. Results of the numerical test. The relative error of the real and imaginary part of the wave function is displayed
for dierent numbers of nodes. The modication of the Koonin’s standard algorithm lead to an enhancement in the accuracy
of up to an order of magnitude.
nodes Relative error of Relative error of
the real part: the imaginary part:
standard modied standard modied
‘ = 0 10% 0:4% 4% 0:2%
‘ = 1 10% 0:6% 4% 0:4%
‘ = 2 1% 0:3% 40% 5%
‘ = 3 20% 4% 1:5% 0:6%
4


On the Numerical Solution of the Time-Dependent Schroedinger Equation

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