Solution of the time-dependent Schrödinger equation employing a basis of explicit discrete-coordinate eigenfunctions

spherical and azimuthal symmetry, adiabaticity, and multiphoton excitation of a rotating Morse oscillator

F. J. Lin, James Muckerman

Research output: Contribution to journalArticle

20 Citations (Scopus)

Abstract

We have recently presented a method for solving the time-dependent Schrödinger equation by collocation on various members of a family of spatial grids and have shown that this method is interpretable in terms of the expansion of the wave function in a basis of discrete coordinate eigenfunctions. The method is applicable to molecular systems with one or multiple spatial dimensions, and also to bound or unbound molecular systems. We now place the method in the more general setting of spatial grids belonging to the family of scaled Gauss-Markov quadrature points. Further, we now treat molecular systems described in terms of the more general case of a curvilinear coordinate system. This allows us to treat, for example, a system with spherical or azimuthal symmetry and to employ the information regarding the symmetry, i.e. transform from Cartesian coordinates to spherical polar coordinates and obtain the Schrödinger equation for a wave function for a configuration space with a lower number of dimensions. In addition, we describe a modified adiabatic approach for decreasing the number of spatial dimensions in the Hamiltonian appearing in the time-dependent Schrödinger equation for the wave function. We then obtain a coupled set of differential equations describing the time evolution of the molecular system, including the time-dependent Schrödinger equation in terms of the Hamiltonian with the lower number of spatial dimensions, and we apply our method for solving the time-evolution problem for the wave function by employing the Hamiltonian with that smaller number of spatial dimensions. Finally, we demonstrate the application of this extension of our collocation method to the nonlinear problem of multiple-photon excitation of a rotating anharmonic diatomic molecule by the explicitly time-dependent term describing irradiation of the molecule by an intense classical electromagnetic field. We compare our results for the quantal approach with our results for two approximate approaches: a modification of the adiabatic quantal approach and a variation of the Monte Carlo classical trajectory approach.

Original languageEnglish
Pages (from-to)538-568
Number of pages31
JournalComputer Physics Communications
Volume63
Issue number1-3
DOIs
Publication statusPublished - 1991

Fingerprint

Wave functions
Eigenvalues and eigenfunctions
Hamiltonians
eigenvectors
oscillators
symmetry
wave functions
excitation
spherical coordinates
collocation
Molecules
grids
Electromagnetic fields
polar coordinates
Differential equations
Photons
Cartesian coordinates
Trajectories
Irradiation
diatomic molecules

ASJC Scopus subject areas

  • Computer Science Applications
  • Physics and Astronomy(all)

Cite this

@article{8d633a3a956e45af9ae5264bd7ba6657,
title = "Solution of the time-dependent Schr{\"o}dinger equation employing a basis of explicit discrete-coordinate eigenfunctions: spherical and azimuthal symmetry, adiabaticity, and multiphoton excitation of a rotating Morse oscillator",
abstract = "We have recently presented a method for solving the time-dependent Schr{\"o}dinger equation by collocation on various members of a family of spatial grids and have shown that this method is interpretable in terms of the expansion of the wave function in a basis of discrete coordinate eigenfunctions. The method is applicable to molecular systems with one or multiple spatial dimensions, and also to bound or unbound molecular systems. We now place the method in the more general setting of spatial grids belonging to the family of scaled Gauss-Markov quadrature points. Further, we now treat molecular systems described in terms of the more general case of a curvilinear coordinate system. This allows us to treat, for example, a system with spherical or azimuthal symmetry and to employ the information regarding the symmetry, i.e. transform from Cartesian coordinates to spherical polar coordinates and obtain the Schr{\"o}dinger equation for a wave function for a configuration space with a lower number of dimensions. In addition, we describe a modified adiabatic approach for decreasing the number of spatial dimensions in the Hamiltonian appearing in the time-dependent Schr{\"o}dinger equation for the wave function. We then obtain a coupled set of differential equations describing the time evolution of the molecular system, including the time-dependent Schr{\"o}dinger equation in terms of the Hamiltonian with the lower number of spatial dimensions, and we apply our method for solving the time-evolution problem for the wave function by employing the Hamiltonian with that smaller number of spatial dimensions. Finally, we demonstrate the application of this extension of our collocation method to the nonlinear problem of multiple-photon excitation of a rotating anharmonic diatomic molecule by the explicitly time-dependent term describing irradiation of the molecule by an intense classical electromagnetic field. We compare our results for the quantal approach with our results for two approximate approaches: a modification of the adiabatic quantal approach and a variation of the Monte Carlo classical trajectory approach.",
author = "Lin, {F. J.} and James Muckerman",
year = "1991",
doi = "10.1016/0010-4655(91)90275-P",
language = "English",
volume = "63",
pages = "538--568",
journal = "Computer Physics Communications",
issn = "0010-4655",
publisher = "Elsevier",
number = "1-3",

}

TY - JOUR

T1 - Solution of the time-dependent Schrödinger equation employing a basis of explicit discrete-coordinate eigenfunctions

T2 - spherical and azimuthal symmetry, adiabaticity, and multiphoton excitation of a rotating Morse oscillator

AU - Lin, F. J.

AU - Muckerman, James

PY - 1991

Y1 - 1991

N2 - We have recently presented a method for solving the time-dependent Schrödinger equation by collocation on various members of a family of spatial grids and have shown that this method is interpretable in terms of the expansion of the wave function in a basis of discrete coordinate eigenfunctions. The method is applicable to molecular systems with one or multiple spatial dimensions, and also to bound or unbound molecular systems. We now place the method in the more general setting of spatial grids belonging to the family of scaled Gauss-Markov quadrature points. Further, we now treat molecular systems described in terms of the more general case of a curvilinear coordinate system. This allows us to treat, for example, a system with spherical or azimuthal symmetry and to employ the information regarding the symmetry, i.e. transform from Cartesian coordinates to spherical polar coordinates and obtain the Schrödinger equation for a wave function for a configuration space with a lower number of dimensions. In addition, we describe a modified adiabatic approach for decreasing the number of spatial dimensions in the Hamiltonian appearing in the time-dependent Schrödinger equation for the wave function. We then obtain a coupled set of differential equations describing the time evolution of the molecular system, including the time-dependent Schrödinger equation in terms of the Hamiltonian with the lower number of spatial dimensions, and we apply our method for solving the time-evolution problem for the wave function by employing the Hamiltonian with that smaller number of spatial dimensions. Finally, we demonstrate the application of this extension of our collocation method to the nonlinear problem of multiple-photon excitation of a rotating anharmonic diatomic molecule by the explicitly time-dependent term describing irradiation of the molecule by an intense classical electromagnetic field. We compare our results for the quantal approach with our results for two approximate approaches: a modification of the adiabatic quantal approach and a variation of the Monte Carlo classical trajectory approach.

AB - We have recently presented a method for solving the time-dependent Schrödinger equation by collocation on various members of a family of spatial grids and have shown that this method is interpretable in terms of the expansion of the wave function in a basis of discrete coordinate eigenfunctions. The method is applicable to molecular systems with one or multiple spatial dimensions, and also to bound or unbound molecular systems. We now place the method in the more general setting of spatial grids belonging to the family of scaled Gauss-Markov quadrature points. Further, we now treat molecular systems described in terms of the more general case of a curvilinear coordinate system. This allows us to treat, for example, a system with spherical or azimuthal symmetry and to employ the information regarding the symmetry, i.e. transform from Cartesian coordinates to spherical polar coordinates and obtain the Schrödinger equation for a wave function for a configuration space with a lower number of dimensions. In addition, we describe a modified adiabatic approach for decreasing the number of spatial dimensions in the Hamiltonian appearing in the time-dependent Schrödinger equation for the wave function. We then obtain a coupled set of differential equations describing the time evolution of the molecular system, including the time-dependent Schrödinger equation in terms of the Hamiltonian with the lower number of spatial dimensions, and we apply our method for solving the time-evolution problem for the wave function by employing the Hamiltonian with that smaller number of spatial dimensions. Finally, we demonstrate the application of this extension of our collocation method to the nonlinear problem of multiple-photon excitation of a rotating anharmonic diatomic molecule by the explicitly time-dependent term describing irradiation of the molecule by an intense classical electromagnetic field. We compare our results for the quantal approach with our results for two approximate approaches: a modification of the adiabatic quantal approach and a variation of the Monte Carlo classical trajectory approach.

UR - http://www.scopus.com/inward/record.url?scp=0026105986&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0026105986&partnerID=8YFLogxK

U2 - 10.1016/0010-4655(91)90275-P

DO - 10.1016/0010-4655(91)90275-P

M3 - Article

VL - 63

SP - 538

EP - 568

JO - Computer Physics Communications

JF - Computer Physics Communications

SN - 0010-4655

IS - 1-3

ER -