Numerical solution of the time-dependent Schr�dinger equation for continuum states

Ritchie, Burke; Weatherford, Charles A.

doi:10.1002/1097-461x(2000)80:4/5<934::aid-qua42>3.0.co;2-o

Cited by 3 publications

(10 citation statements)

References 9 publications

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“…The TDSE results for the toal cross section are choppy (Figs. 4,5,7,9) because the cross section is calculated at temporal intervals greater than the temporal mesh size in order to reduce the expense of the calculation.…”

Section: Resultsmentioning

confidence: 99%

“…(2) for a scattering problem is described in Ref. [9]. Equation (2) reduces to the standard time-independent form of the integral Schrödinger equation in the steady-state limit for the case where the potential is independent of time, V( r, t ) = V( r).…”

Section: Theorymentioning

confidence: 99%

“…(5) because, once the operator integrating factor is evaluated as described in Ref. [9], one has simply to perform numerical quadrature over the time. On the other hand in Eq.…”

Section: Theorymentioning

confidence: 99%

“…In previous work a time-dependent method [9] was proposed to overcome this problem. This method was motivated by yet earlier work [10] which showed that the time-dependent Schrödinger equation, with time-independent potentials, could be used to evolve some initial trial wave function into the true, steady wave function.…”

Section: Introductionmentioning

confidence: 99%

“…The time evolution treatment in the former method suggested that one could simply solve the timeindependent integral Schrödinger or Lippmann-Schwinger equation by iteration on a trial wave function provided one, as in Ref. [9], evaluates the scattering kernel in momentum space where the free-electron Green's function is in diagonal form. These methods will be outlined mathematically and results presented in later sections of this article.…”

Section: Introductionmentioning

confidence: 99%

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Use of fast Fourier transform computational methods in electron scattering

Ritchie

Riley

2002

Int J of Quantum Chemistry

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Two methods are studied for elastic electron-atom and electron-molecule elastic scattering. The first is a numerical solution of the time-dependent Schrödinger equation suitable for continuum states; the second is a convergent modified Born series of the time-independent integral Schrödinger (Lippmann-Schwinger) equation. Both methods make use of fast Fourier transform (FFT) computational methods in order to bring the free-electron Green's function into diagonal form. A large advantage of this methodology is the ability to describe scattering from targets of any geometry and at any collision energy by using 3D FFTs to accomplish the labor of evaluating spatially complex collision kernels.

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Section: Resultsmentioning

confidence: 99%

Section: Theorymentioning

confidence: 99%

“…(5) because, once the operator integrating factor is evaluated as described in Ref. [9], one has simply to perform numerical quadrature over the time. On the other hand in Eq.…”

Section: Theorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Use of fast Fourier transform computational methods in electron scattering

Ritchie

Riley

2002

Int J of Quantum Chemistry

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Time‐dependent non‐wavepacket theory of electron scattering

Ritchie

Weatherford

2004

Int J of Quantum Chemistry

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ABSTRACT:A technique is given for solving the time-dependent Schrö dinger equation for electronϩpotential scattering at low energies (k 2 Յ 1.0 a.u.). This is accomplished by avoiding wavepackets, with the inevitable wavepacket spreading, and solving a time-integral form of the Schrö dinger equation directly. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem 100: 710 -712, 2004 Key words: electron scattering; time-dependent; non-wavepacket W e are motivated to overcome the limitation of time-dependent wavepacket theory due to wavepacket spreading, which effectively limits scattering-electron momenta (k) to values greater than 1 a.u. We present here results for k 2 ϭ 0.1 and k 2 ϭ 1.0 a.u. We solve the time-dependent Schröd-inger equationby direct solution of the integral equation version of the Schrö dinger equationwhich follows from Eq. (1). Equation (2) is solved by the following two steps: (a) evaluation of the operator-integrating factor in momentum space by Fourier-transforming the product V using the fast Fourier transform (FFT) algorithm; this step has the advantage that, for finite-range potentials, the product V is well removed from the boundaries of the numerical grid box while itself is not; (b) evaluation of the temporal integral by numerical quadrature using a plane wave as an initial wave function; a forwardbackward FFT is performed for each step in the quadrature. Greater detail is given in Refs. 1 and 2.

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