Solution of the Time-Dependent Schrodinger Equation Using a Continuous Fourier Transform

WILFRED0 RODRIGUEZ Departamento de Tecnologia Electrhica, Universidad de Puerto Rico en Bayamh,

Bayambn, Puerto Rico 00620

Abstract

We discuss the solution of the time-dependent Schriidinger equation. The split operator method is used to propagate an initial wave function and the continuous Fourier transform is applied to obtain the energy eigenvalues. High resolution in the energy is obtained without line shaping techniques.

Introduction

The solution of the time-dependent Schrodinger equation has been performed by various numerical techniques: second-order differencing by Kosloff and Kosloff [ 11, the split operator method by Feit et al. [ 2 1, the short iterative Lanczos propagator by Lanczos [3] , and the Chebychev scheme by Tal-Ezer and Kosloff [ 4 ] . These schemes differ in the way the initial wave function propagator is represented.

If one is interested in the eigenvalues, the initial wave function W r , 0) has to be propagated in order to compute the correlation function,

( 1 )

Once a record of P( t ) is obtained, its Fourier transform has sharp resonances at the eigenvalues of the system under study.

In the split operator method ( SPO) the correlation function P( t ) is Fourier trans- formed using the fast Fourier algorithm (m) and line shaping of the discrete spectrum is used [ 21 to localize the resonances. We present an alternate method of performing the Fourier transform of P( t ) , the continuous Fourier ( CFT ) [ 51. The correlation function is represented by B splines for which there is an analytical Fourier transform. Now the function P ( E ) is continuous, thus enabling an easy localization of the local maxima and improving the energy resolution beyond the discrete Fourier transform scheme.

P ( t ) = ( W r , 0 ) I w r , 0 ) ) *

Review of the Split Operator Method

The time-dependent Schrodinger equation,

has the formal solution

International Journal of Quantum Chemistry: Quantum Chemistry Symposium 25, 107-1 1 1 (1991) 0 1991 John Wiley & Sons, Inc. CCC 0020-760819 1/01 0107-5$04.00

108 RODRIGUEZ

\k(r, t ) = Qt)\k(r, o ) ,

O( t ) = exp(-ifiSt/h), (4)

( 3 )

where the evolution operator for a time-independent Hamiltonian is given by

where \k( r , 0) is the wave function at t = 0, and 6t is the time step. Several schemes have been proposed [ 61 to approximate the exponentiation in Eq. (4) . In the split operator method [ 21 the propagator is approximated in the following way:

O(t) = exp(-iFat/2h) exp(-iF%t/h) exp(-iFbt/2h), ( 5 )

where the truncation error is determined by the next higher commutator between the potential and kinetic energy. The immediate advantage of this technique has been that the kinetic energy operator T = P 2 / 2 m is diagonal in the momentum representation while the potential energy operator is diagonal in the position representation.

Propagation of an initial wave function \k( r , 0) is achievid in the following steps: Fourier transform 9( r , 0) into momentum representation:

U(P, 0 ) = 3 [ W r , 0 ) l . ( 6 )

exp(-iF6t/2h)~(p, 0 ) . (7)

Apply the kinetic energy operator:

The kinetic energy operator provides a free particle propagation. Fourier transform U( p, 6t/2) into position representation:

\k(r, 6t/2) = 3[U(p , 6t/2)]. ( 8 )

exp(-iht/h)\k(r, 6 t / 2 ) . (9)

Apply the potential energy operator:

The potential operator provides a phase shift to the free propagated wave function.

(10)

Fourier transform into momentum space:

~ ( p , 6 t / 2 ) = s[exp(-iPGt/h)\k(r, 6t/2)1.

Apply the kinetic energy operator:

exp(-iF6t/2h)~(p, 6 t / 2 ) . (11)

(12)

The propagated wave function is obtained by a Fourier transform:

\k(r, st) = 3[exp(- iF6t /2h)~(p, 6t/2)1.

The computational procedure is facilitated if the exponential operators are initially stored. A convenient way to perform the transforms is by using the FFT algorithm.

The numerical solution *( r , t ) of the time-dependent Schrodinger equation can be used to obtain the energy eigenvalues. The correlation function P ( t ) can be computed for a sequence of time steps,

TIME-DEPENDENT SCHRODINGER EQUATION 109

P ( t i ) ; ti = isat, i = 0, 1, 2 , . . . , M . (13) The numerical Fourier transform of P( t, ), P( E ) will exhibit sharp local maxima f o r E = E n ; n = 0 , 1 , 2 , . . . , N .

The Continuous Fourier Transform

One is interested in the following transform:

P( E) = J+m dt P( t ) exp( iEt) , --m

where P( t ) is the correlation function; in practice, one only has a discrete record of P ( t j ) .

In the CFT method [ 51 the correlation function is represented by B splines, N-k

P ( t ) = 2 ajBj,k(t) 3 ( 1 5 ) j= 1

where the kth-order B spline B,,k( t ) is a polynomial of degree (k - 1 ) in the non- empty interval ( t j , tj+k) and can be obtained from the recurrence relation [ 8,9],

where ti I tj+k and Bj,l ( t ) = 1 if tj I t I tj+l and zero otherwise. The set of coefficients aj are computed by solving the set of linear equations given by

The Fourier transform of P( t ) is obtained by substituting Eq. ( 15) in ( 14),

P( E) = aj s”” dt Bj,k( t ) exp( iEt) . (18) j =1 XJ

The integral in ( 18) for a uniform mesh of spacing h and N points is given by N-k

P( E) = 2 ajaj( E ) exp( iEtj) , j = I

where * aj(E) = h((exp(-iEh) - l)/iEh)k, j = k to ( N + 1 - 2k) . (20)

Case Study. The 1-D Asymmetric Double Well Potential

We calculated the energy eigenvalues for the one-dimensional double well po- tential, which is very well documented [ 2,7-9 ] in order to compare the resolution obtained through the CFT.

* See [ 5 ] for the endpoints, j = 1 to k - 1 a n d j = N - 2(k - 1 ) to ( N - k ) .

110 RODRIGUEZ

The SPO was used to generate \k(x, t ) for a potential of the form

V ( X ) = ko - kzX2 + k3X3 + k4X4 (21)

q ( x , 0 ) = exp[-{(x - ~ ) / 2 a } ~ ] + exp[-{(x + a ) / 2 ~ } ~ ] , (22) where ko = -132.7074997, k2 = 7, k3 = 0.5, k4 = 1, a = 1.9, and (r = 0.87. The parameters for propagation are 6t = 5.73 for 10,000 time steps and 6x = 0.825 with 5 12 grid points. The numerical integration was performed with the trapezoidal method, which uses a second-order B-spline fit to the data.

The results in Table I show an agreement to three decimals with F&F [ 21 ; our results have eight significant figures throughout due to the trapezoidal integration for the correlation function. The reduced energy is taken from the potential at x = 0.0. The CFT allows a continuous identification of the energy eigenvalues without the need of line shaping.

and initial wave function

TABLE I. Energy eigenvalues En for asymmetric double well potential.

l3-T PI CFT

n Unreduced Reduced Unreduced Reduced

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

-144.96594 -138.75331 - 137.66686 -133.35413 -1 32.01629 - 128.65342 -125.33856 -121.47198 -1 17.27558 -112.76882 -107.98548 -102,94872

-97.67768 -92.18760 -86.49 180 -80.60168 -74.52721 -68.27733 -61.8601 1 -55.28325 -48.55450 -41.68244 -34.67820 -27.5581 1 -20.35076 -13.11431 -6.001 53

-12.258438 -6.04541 8 -5.286089 -0.646627

0.691204 4.053229 7.368937

1 I .23552 1 15.43 1918 19.938752 24.7220 16 29.758660 35.029816 40.519900 46.21 5701 52.105824 58.180283 64.430 I75 70.847383 77.424233 84.152963 9 1.024978 98.029195

105.14908 I 112.356276 119.592224 126.704585

-144.96584 -138.75333 -137.99454 -133.35411 - 132.01630 - 128.65341 -125.33845 -121.47172 - 1 17.27544 - 1 12.76879 - 107.98549 - 102.94867 -97.67775 -92.18770 -86.49160 -80.60156 -74.52718 -68.27714 -61.8601 8 -55.28327 -48.55458 -41.68222 -34.67814 -27.55802 -20.35045 -13.11409 -6.00169

-12.258340 -6.045830 -5.287040 -0.646610

0.69 I I99 4.054089 7.369049

11.235779 15.432059 19.938709 24.722009 29.758829 35.029749 40.519799 46.215899 52.105939 58.180319 64.430359 70.847319 77.424229 84.152919 9 1.025279 98.029359

105.149479 112.357049 119.593409 126.705809

TIME-DEPENDENT SCHRODINGER EQUATION 111

Acknowledgments

We give special thanks to Ronald Martinez and Mayra V6lez for their support. These calculations were performed in a Sun Sparc 390 and IBM RISC-6000. This work was supported by the Resource Center and for Science and Engineering and the University of Puerto Rico at Bayam6n.

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Received April 1 7, 199 1