%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 }