In a recent paper Makarov, D. E.; Metiu, H. J. Chem.
Phys. 1998, 108, 590, makarov:1999:fpes:sfsdGP
we developed a directed genetic programming approach
for finding the best functional form that fits the
energies provided by ab initio calculations. In this
paper, we use this approach to find the analytic
solutions of the time-independent Schrodinger equation.
This is achieved by inverting the Schrodinger equation
such that the potential is a functional depending on
the wave function and the energy. A genetic search is
then performed for the values of the energy and the
analytic form of the wave function that provide the
best fit of the given potential on a chosen grid. A
procedure for finding excited states is discussed. We
test our method for a one-dimensional anharmonic well,
a double well, and a two-dimensional anharmonic
oscillator.
http://pubs.acs.org/journals/jpcafh/index.html
directed genetic programming (DGP), monte Carlo,
"straightforward GP...leads to poor results" p8451
DGP adds form of solution? Fset=+,-<*,/ pop=100
G<=250. Ekart well best(?) -1.5576eV, most within 0.5
percent. Even better with Bessel function. Also tried
Gaussian. NB "proper choice of the grid is
important" p852. Asymptotic region dominates
tunnelling. "we believe that...more readily find the
solution that has the simplest functional form" p8542
(ie the lowest energy eigenstate).
Excited states. Harmonic oscillator creation operator.
problems with second excited state? Hartree
approximation (p8544) , separate x and y dimensions,
use same bell curve for both x and y. x,y back
together? Seeded run? Fset now also includes exp First
excited state E=2.534.
%0 Journal Article
%1 makarov:2000:JPCA
%A Makarov, Dmitrii E.
%A Metiu, Horia
%D 2000
%J Journal of Physical Chemistry A
%K DGP, algorithms, genetic mathematica programming,
%P 8540--8545
%R doi:10.1021/jp000695q
%T Using Genetic Programming To Solve the Schrodinger
Equation
%V 104
%X In a recent paper Makarov, D. E.; Metiu, H. J. Chem.
Phys. 1998, 108, 590, makarov:1999:fpes:sfsdGP
we developed a directed genetic programming approach
for finding the best functional form that fits the
energies provided by ab initio calculations. In this
paper, we use this approach to find the analytic
solutions of the time-independent Schrodinger equation.
This is achieved by inverting the Schrodinger equation
such that the potential is a functional depending on
the wave function and the energy. A genetic search is
then performed for the values of the energy and the
analytic form of the wave function that provide the
best fit of the given potential on a chosen grid. A
procedure for finding excited states is discussed. We
test our method for a one-dimensional anharmonic well,
a double well, and a two-dimensional anharmonic
oscillator.
@article{makarov:2000:JPCA,
abstract = {In a recent paper [Makarov, D. E.; Metiu, H. J. Chem.
Phys. 1998, 108, 590], \cite{makarov:1999:fpes:sfsdGP}
we developed a directed genetic programming approach
for finding the best functional form that fits the
energies provided by ab initio calculations. In this
paper, we use this approach to find the analytic
solutions of the time-independent Schrodinger equation.
This is achieved by inverting the Schrodinger equation
such that the potential is a functional depending on
the wave function and the energy. A genetic search is
then performed for the values of the energy and the
analytic form of the wave function that provide the
best fit of the given potential on a chosen grid. A
procedure for finding excited states is discussed. We
test our method for a one-dimensional anharmonic well,
a double well, and a two-dimensional anharmonic
oscillator.},
added-at = {2008-06-19T17:35:00.000+0200},
author = {Makarov, Dmitrii E. and Metiu, Horia},
biburl = {https://www.bibsonomy.org/bibtex/271f296656c05c99a0de131e77d5b7e02/brazovayeye},
doi = {doi:10.1021/jp000695q},
interhash = {b42e00673ce4de7d508745252dcbeaf4},
intrahash = {71f296656c05c99a0de131e77d5b7e02},
issn = {1089-5639},
journal = {Journal of Physical Chemistry A},
keywords = {DGP, algorithms, genetic mathematica programming,},
notes = {http://pubs.acs.org/journals/jpcafh/index.html
directed genetic programming (DGP), monte Carlo,
{"}straightforward GP...leads to poor results{"} p8451
DGP adds form of solution? Fset={+,-<*,/} pop=100
G<=250. Ekart well best(?) -1.5576eV, most within 0.5
percent. Even better with Bessel function. Also tried
Gaussian. NB {"}proper choice of the grid is
important{"} p852. Asymptotic region dominates
tunnelling. {"}we believe that...more readily find the
solution that has the simplest functional form{"} p8542
(ie the lowest energy eigenstate).
Excited states. Harmonic oscillator creation operator.
problems with second excited state? Hartree
approximation (p8544) , separate x and y dimensions,
use same bell curve for both x and y. x,y back
together? Seeded run? Fset now also includes exp First
excited state E=2.534.},
pages = {8540--8545},
timestamp = {2008-06-19T17:46:13.000+0200},
title = {Using Genetic Programming To Solve the Schrodinger
Equation},
volume = 104,
year = 2000
}